{
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  "generator": "readable-paper 0.1.4",
  "paperId": "6353145ba5aa",
  "version": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
  "sha256": "745d038ee6c94dcfa4338029a8e93f162bd419c5d7e302fe4c30d7529bdb8774",
  "title": "The p-adic section conjecture",
  "status": "Unverified reading map",
  "nodes": [
    {
      "id": "theorem-1",
      "kind": "theorem",
      "label": "Theorem 1.1",
      "source": {
        "blockId": "p2-14f3098072",
        "page": 2,
        "quote": "Theorem 1.1. For every prime p, every finite extension k/Qp, and every smooth proper\ngeometrically connected curve X/k of genus at least two, the map"
      },
      "statement": [
        {
          "blockId": "p2-14f3098072",
          "page": 2,
          "quote": "Theorem 1.1. For every prime p, every finite extension k/Qp, and every smooth proper\ngeometrically connected curve X/k of genus at least two, the map"
        },
        {
          "blockId": "p2-3127881a48",
          "page": 2,
          "quote": "(\ns : Gk →π1(X) a continuous group homomorphism"
        },
        {
          "blockId": "p2-68bccc8cc9",
          "page": 2,
          "quote": "X(k) −→"
        },
        {
          "blockId": "p2-3c30ed1a23",
          "page": 2,
          "quote": ")\nπ1(X¯k)"
        },
        {
          "blockId": "p2-b83772449f",
          "page": 2,
          "quote": "such that (π1(X) →Gk) ◦s = idGk"
        },
        {
          "blockId": "p2-e4b01d80dd",
          "page": 2,
          "quote": "is bijective. The quotient is by conjugation by the full geometric étale fundamental group."
        }
      ],
      "proof": [
        {
          "blockId": "p17-6aa1f30eeb",
          "page": 17,
          "quote": "Proof of the surjectivity in Theorem 1.1. Let s : Gk →ΠX be a continuous group-homomorphic\nsection. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26],\nthere is a valuation w of K extending the p-adic valuation, together with a prolongation ew to\nf\nK, such that\ns(Gk) ⊆De\nw(f\nK/K)."
        },
        {
          "blockId": "p17-26f38acfac",
          "page": 17,
          "quote": "We classify the possibilities for w.\nThe valuation transcendence inequality gives"
        },
        {
          "blockId": "p17-9b0b095290",
          "page": 17,
          "quote": "ratrank(Γw/Γk) ≤1."
        },
        {
          "blockId": "p17-7b7f5a8a8a",
          "page": 17,
          "quote": "Since Γk has rational rank one and ordered rank is at most rational rank, w has rank at most\ntwo. Rank one is excluded by Lemma 7.2. If w has rank two, let u be its rank-one coarsening.\nThe nonzero proper convex subgroup of the rank-two value group has a unique extension to\nthe value group of ew: it consists of those values some positive multiple of which lies in the\noriginal subgroup. This follows because an algebraic valued extension has torsion quotient of\nvalue groups. Every automorphism stabilizing ew preserves this subgroup, so s(Gk) stabilizes the\nresulting prolongation of u. If u is nontrivial on k, it can be normalized to extend the p-adic\nvaluation and is excluded by the same lemma."
        },
        {
          "blockId": "p18-9a49bae2ba",
          "page": 18,
          "quote": "It remains that u is trivial on k. Properness gives it a center on X. This center cannot be\ngeneric because u is nontrivial, so it is a closed point a. The valuation ring of u dominates the\ndiscrete valuation ring OX,a. Every element of K× is a power of a uniformizer times a unit of\nthis local ring; hence u is equivalent to orda. Its residue field is the finite extension k(a)/k. The\nsecond valuation in the composition w is the unique prolongation of the p-adic valuation to k(a).\nThus w is of type 2h.\nThe chosen horizontal prolongation to K¯k selects a geometric point above a. Its stabilizer\nacts on constants through Gk(a) ⊆Gk, with the embedding determined by that point. Since\ns(Gk) projects onto Gk, necessarily k(a) = k. For the full étale tower the horizontal inertia\nis trivial, and the horizontal decomposition group is the point section at a. Equivalently, the\ntype-2h conclusion of [22, preprint version, Section 6.1] gives s = sa after the corresponding\ngeometric choice of base point. This proves surjectivity.\nNotice that a type-2h valuation can lie over a non-rational closed point. Rationality here\nfollows from the surjection to Gk; no unrestricted equivalence between “not type 2h” and “not\noriginating from a rational point” is used."
        },
        {
          "blockId": "p18-796f538cd7",
          "page": 18,
          "quote": "Proof of the injectivity in Theorem 1.1. Suppose a, b ∈X(k) have the same section class modulo\nthe full geometric fundamental group. Let J be the Jacobian of X, and use a as base point for\nthe Abel–Jacobi map\nia : X −→J,\nx 7−→[x −a]."
        },
        {
          "blockId": "p18-d2c2d7a7c2",
          "page": 18,
          "quote": "Their induced sections on J are again geometrically conjugate. For every positive integer n,\nmultiplication [n] : J →J is a finite étale torsor under J[n], also when p | n since k has\ncharacteristic zero. Conjugate sections give isomorphic Gk-actions on its geometric fiber, so the\nexistence of a fixed point is unchanged by conjugation. The section at ia(a) = 0 has the fixed\nlift 0. Therefore the fiber over ia(b) has a Gk-fixed point, which is a k-rational point of that\nfiber. Thus\nia(b) ∈\n\\"
        },
        {
          "blockId": "p18-8e35977d71",
          "page": 18,
          "quote": "n≥1\nnJ(k)."
        },
        {
          "blockId": "p18-bb146811cd",
          "page": 18,
          "quote": "This is also the usual Kummer argument: the class of the multiplication torsor pulled back along\na section is unchanged by geometric conjugation.\nThe compact p-adic analytic group J(k) has an open subgroup U ∼= Zg(X)[k:Qp]\np\nby the local\nlogarithm. If q = [J(k) : U], then qJ(k) ⊆U. For each r ≥0, take n = qpr in the displayed\nintersection. Any element of that intersection can then be written as pr(qy) with y ∈J(k), and\nhence belongs to prU. As U is a finite free Zp-module,"
        },
        {
          "blockId": "p18-f81daf828d",
          "page": 18,
          "quote": "\\"
        },
        {
          "blockId": "p18-2755e2003c",
          "page": 18,
          "quote": "r≥0\nprU = 0,"
        },
        {
          "blockId": "p18-f9740ddbdc",
          "page": 18,
          "quote": "and hence ia(b) = 0. If a ̸= b, the linear equivalence of the degree-one divisors a and b would\ngive a degree-one morphism X →P1, forcing genus zero. Hence a = b, as required."
        }
      ],
      "guide": [
        {
          "text": "The assertion is a bijection between rational points and continuous splittings, modulo conjugation by the full geometric étale fundamental group. Retain this quotient and all the hypotheses when restating the claim.",
          "sources": [
            {
              "blockId": "p2-14f3098072",
              "page": 2,
              "quote": "Theorem 1.1. For every prime p, every finite extension k/Qp, and every smooth proper\ngeometrically connected curve X/k of genus at least two, the map"
            },
            {
              "blockId": "p2-3127881a48",
              "page": 2,
              "quote": "(\ns : Gk →π1(X) a continuous group homomorphism"
            },
            {
              "blockId": "p2-68bccc8cc9",
              "page": 2,
              "quote": "X(k) −→"
            },
            {
              "blockId": "p2-3c30ed1a23",
              "page": 2,
              "quote": ")\nπ1(X¯k)"
            },
            {
              "blockId": "p2-b83772449f",
              "page": 2,
              "quote": "such that (π1(X) →Gk) ◦s = idGk"
            },
            {
              "blockId": "p2-e4b01d80dd",
              "page": 2,
              "quote": "is bijective. The quotient is by conjugation by the full geometric étale fundamental group."
            }
          ]
        },
        {
          "text": "The introduction distinguishes an étale section from a birational lift. The latter is additional information; do not treat it as automatic when reading the proposed argument. This is a reading guide, not verification of the lifting step.",
          "sources": [
            {
              "blockId": "p2-2b25b1a064",
              "page": 2,
              "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
            }
          ]
        }
      ],
      "guideStatus": "Agent-authored reading guide; not independently verified."
    },
    {
      "id": "theorem-2",
      "kind": "theorem",
      "label": "Theorem 1.2",
      "source": {
        "blockId": "p3-de0986e624",
        "page": 3,
        "quote": "Theorem 1.2. Let p be a prime, let k/Qp be finite, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Put K = k(X), choose a separable closure Ksep, and let\nf\nK ⊂Ksep be the compositum of the function fields of all connected finite étale covers of X, with\nall their K-embeddings. For every real-valued rank-one norm on K extending the normalized\np-adic norm on k, and every prolongation to Ksep, set"
      },
      "statement": [
        {
          "blockId": "p3-de0986e624",
          "page": 3,
          "quote": "Theorem 1.2. Let p be a prime, let k/Qp be finite, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Put K = k(X), choose a separable closure Ksep, and let\nf\nK ⊂Ksep be the compositum of the function fields of all connected finite étale covers of X, with\nall their K-embeddings. For every real-valued rank-one norm on K extending the normalized\np-adic norm on k, and every prolongation to Ksep, set"
        },
        {
          "blockId": "p3-5c3066fd86",
          "page": 3,
          "quote": "E = [\nKsep,\nH = c\nf\nK ⊂E,"
        },
        {
          "blockId": "p3-102d183ed0",
          "page": 3,
          "quote": "where the completions use that fixed prolongation. Then the specified subfield Ksep of E is\ncontained in H."
        }
      ],
      "proof": [
        {
          "blockId": "p15-0ad83e953c",
          "page": 15,
          "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
        },
        {
          "blockId": "p15-7af76a853a",
          "page": 15,
          "quote": "Db < −logp |T| < ab −Db"
        },
        {
          "blockId": "p15-acbc14cc1c",
          "page": 15,
          "quote": "the coordinate T has an analytic mb-th root. Pull this cover back by the nonconstant morphism\nX¯k →P1\n¯k given by T = tb/βb, and take a normalized connected component. Denote the resulting\ncover by Wb →X¯k.\nLemma 3.4 shows that its horizontal indices still divide ℓ. Roots pull back to every point of\nthe normalized component, so tb has an analytic mb-th root on the entire inverse image in W an\nb,Ω\nof the collar\n|βb|p−(ab−Db) < |tb| < |βb|p−Db"
        },
        {
          "blockId": "p15-8ac499fdb2",
          "page": 15,
          "quote": "inside Bb. Here we multiplied the pulled-back root of T by an mb-th root of βb in Ω.\nEmbed the finitely many function fields ¯k(Wb) into the fixed Ksep over K0 and form their\ncompositum. Let W →X¯k be the associated smooth proper connected curve. It dominates\nevery Wb; its horizontal indices divide ℓby Lemma 3.4. All collar roots therefore pull back to\nthe full inverse images in W an\nΩ.\nProposition 3.2, with Y = WΩ, produces a connected finite étale cover VΩ→WΩand an\nanalytic map from its exterior region to Can\nΩover Xan\nΩ. Since W is proper over the algebraically\nclosed field ¯k, invariance of finite étale covers under the algebraically closed extension ¯k ⊂Ω[10]\ngives a connected finite étale cover V →W and an isomorphism"
        },
        {
          "blockId": "p15-3a65c0c9c6",
          "page": 15,
          "quote": "V ׯk Ω≃VΩ\nover WΩ."
        },
        {
          "blockId": "p15-e981bdf0e0",
          "page": 15,
          "quote": "No descent assertion is needed for the exterior analytic map.\nAlthough no such bound was imposed on C →X¯k, the composite V →X¯k still has horizontal\nindices dividing ℓ. By Lemma 3.4, its function field embeds over K0 in Lℓ. This embedding\ncan, if desired, be chosen to extend the already chosen embedding of ¯k(W): extend that\nembedding to ¯k(V ) inside Ksep and then take its Galois closure, which still has indices dividing\nℓ. Proposition 4.4 places its image in H."
        },
        {
          "blockId": "p16-7e7543d8a3",
          "page": 16,
          "quote": "Lemma 3.3 now gives a K0-embedding F = ¯k(C) ,→H ⊂E. Because F/K0 is normal, this\nembedding has image the chosen subfield F of E. Indeed a primitive element of F is sent to a\nroot of its minimal polynomial, all of whose roots already belong to F; the same holds for every\nelement. The image, having degree [F : K0], is therefore F itself. Consequently F ⊂H.\nEvery element of Ksep lies in a finite Galois extension of K0 inside Ksep: this is an algebraic\nclosure of the characteristic-zero field K0.\nTaking the union over those extensions proves\nKsep ⊂H inside E."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-3",
      "kind": "lemma",
      "label": "Lemma 3.1",
      "source": {
        "blockId": "p6-4a7d2c52b5",
        "page": 6,
        "quote": "Lemma 3.1. Let f : C →X¯k be a finite surjective morphism of smooth proper connected curves,\nand let b ∈X(¯k). Fix a rational function tb which is a local parameter at b. There is an open\ndisk Bb about b in Xan\nΩ, with coordinate tb, such that its full inverse image in Can\nΩis a disjoint\nunion of open disks. On the disk about c ∈f−1(b) there is a coordinate zc for which"
      },
      "statement": [
        {
          "blockId": "p6-4a7d2c52b5",
          "page": 6,
          "quote": "Lemma 3.1. Let f : C →X¯k be a finite surjective morphism of smooth proper connected curves,\nand let b ∈X(¯k). Fix a rational function tb which is a local parameter at b. There is an open\ndisk Bb about b in Xan\nΩ, with coordinate tb, such that its full inverse image in Can\nΩis a disjoint\nunion of open disks. On the disk about c ∈f−1(b) there is a coordinate zc for which"
        },
        {
          "blockId": "p6-4a4d6d3a74",
          "page": 6,
          "quote": "tb = zec\nc ,"
        },
        {
          "blockId": "p6-c0041351b3",
          "page": 6,
          "quote": "where ec is the ramification index at c. No condition that ec be prime to p is required. For finitely\nmany distinct points b, the disks can be chosen disjoint and disjoint from the fixed point ξ."
        }
      ],
      "proof": [
        {
          "blockId": "p6-af5d5cc85a",
          "page": 6,
          "quote": "Proof. Choose a local analytic parameter u at c. The pullback of tb is ueca(u) with a(0) ̸= 0.\nAfter choosing an ec-th root of a(0) in Ωand shrinking the disk, a(u)/a(0) is sufficiently close to\n1 for its ec-th root to be given by"
        },
        {
          "blockId": "p6-a82a52e120",
          "page": 6,
          "quote": "exp\n\ne−1\nc\nlog(a(u)/a(0))\n\n."
        },
        {
          "blockId": "p6-0f4a412be2",
          "page": 6,
          "quote": "Here it is enough to make the valuation of a(u)/a(0) −1 greater than vp(ec) + 1/(p −1). Thus\nzc = u a(u)1/ec is analytic with a nonzero derivative at the center. Analytic inversion gives a\nsmaller disk on which it is a coordinate and tb = zec\nc .\nDo this in disjoint neighborhoods of all the finitely many points above b. The complement of\nthese neighborhoods in the proper analytic curve Can\nΩis compact. Its image under f is compact,\nclosed, and does not contain b. Consequently a sufficiently small disk about b has its entire\ninverse image in the chosen neighborhoods. Shrink it once more so that, in every zc-chart, that\ninverse image is the disk defined by the corresponding bound on |zc|ec.\nThe point ξ does not equal any point of X(¯k), since its projection to X¯k is generic. The\nshrinking disks at b form a neighborhood basis there. Hausdorffness therefore permits all disks\nto avoid ξ and one another. We may choose their radii in |Ω×|. The argument also applies when\nξ is an Ω-rational point whose projection to X¯k is generic."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-4",
      "kind": "proposition",
      "label": "Proposition 3.2",
      "source": {
        "blockId": "p6-bba07b773a",
        "page": 6,
        "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
      },
      "statement": [
        {
          "blockId": "p6-bba07b773a",
          "page": 6,
          "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
        },
        {
          "blockId": "p6-e5c0993e40",
          "page": 6,
          "quote": "Ab = {rb < |tb| < Rb}."
        },
        {
          "blockId": "p6-b993750202",
          "page": 6,
          "quote": "Set\nU = Xan\nΩ\\\n["
        },
        {
          "blockId": "p6-cc1c5411e6",
          "page": 6,
          "quote": "b∈B\n{|tb| ≤rb},"
        },
        {
          "blockId": "p6-709fe15125",
          "page": 6,
          "quote": "where each indicated closed disk is taken inside Bb. Then there is a smooth proper connected\ncurve VΩand a finite étale surjective map VΩ→Y , together with an analytic map"
        },
        {
          "blockId": "p6-6863f2684b",
          "page": 6,
          "quote": "(V an\nΩ)|U −→Can\nΩ\nover Xan\nΩ."
        },
        {
          "blockId": "p6-966efaa943",
          "page": 6,
          "quote": "The notation on the left denotes the inverse image of U under the composite map to Xan\nΩ."
        }
      ],
      "proof": [
        {
          "blockId": "p7-51fd4b0c57",
          "page": 7,
          "quote": "Proof. The inverse image YU of U and the inverse images Yb of the disks {|tb| < Rb} form an\nopen cover of Y an. Distinct Yb are disjoint, and YU ∩Yb = g−1(Ab).\nOn YU, take the pullback of Can\nΩ→Xan\nΩ. It is finite étale of degree d, since U contains no\nbranch point. On each Yb, take the trivial degree-d cover. These two covers are isomorphic\nover their overlap. Indeed, write qmb\nb\n= g∗tb for the assumed analytic root. In the local factor\ncorresponding to c ∈f−1(b), put ac = qmb/ec\nb\n. The polynomial defining that factor splits as"
        },
        {
          "blockId": "p7-bdb0fbfe65",
          "page": 7,
          "quote": "Zec −g∗tb =\nY"
        },
        {
          "blockId": "p7-15d8e7fa2f",
          "page": 7,
          "quote": "ζ∈µec\n(Z −ζac)."
        },
        {
          "blockId": "p7-15cdd177d5",
          "page": 7,
          "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
        },
        {
          "blockId": "p7-b00214ee1a",
          "page": 7,
          "quote": "We record explicitly how this analytic map produces an embedding into the fixed completion\nH. This step concerns the entire H-valued point, not just its underlying analytic point."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-5",
      "kind": "lemma",
      "label": "Lemma 3.3",
      "source": {
        "blockId": "p7-7113083a50",
        "page": 7,
        "quote": "Lemma 3.3. In Proposition 3.2, suppose that the composite VΩ→XΩis isomorphic, over XΩ,\nto the base change of a finite morphism V →X¯k from a smooth proper connected curve, and\nassume that ξ ∈U. If ¯k(V ) admits a K0-embedding into H, then ¯k(C) admits a K0-embedding\ninto H."
      },
      "statement": [
        {
          "blockId": "p7-7113083a50",
          "page": 7,
          "quote": "Lemma 3.3. In Proposition 3.2, suppose that the composite VΩ→XΩis isomorphic, over XΩ,\nto the base change of a finite morphism V →X¯k from a smooth proper connected curve, and\nassume that ξ ∈U. If ¯k(V ) admits a K0-embedding into H, then ¯k(C) admits a K0-embedding\ninto H."
        }
      ],
      "proof": [
        {
          "blockId": "p7-a80d8ea63e",
          "page": 7,
          "quote": "Proof. The given embedding determines an H-valued point of V . Together with the fixed\ninclusion Ω⊂H, it determines an H-valued point of V ׯk Ω, and the chosen isomorphism\ntransports this point to VΩ. Its image in XΩ(H) is exactly the fixed point induced by K0 ,→H\nand Ω,→H. In particular its underlying analytic image is ξ ∈U.\nAn H-valued point of an algebraic variety over Ωis equivalently an H-valued point of its\nanalytification [29, Fact 5.1.3]. More explicitly, it gives an analytic point v together with an\nisometric embedding of the completed residue field H(v) into H. The analytic map over U in\nProposition 3.2 induces a compatible map on completed residue fields, and hence an H-valued\npoint of CΩover the same point of XΩ(H). Project this point to C. Its image on X¯k is the\ngeneric point with the specified map K0 →H. Since C →X¯k is finite, its support on C is also\ngeneric. The point therefore induces the required field embedding ¯k(C) ,→H over K0.\nNeither the analytic map over U nor a chosen local trivialization has been asserted to descend\nto ¯k."
        },
        {
          "blockId": "p7-c5be7fa760",
          "page": 7,
          "quote": "We shall form auxiliary covers by pullback and compositum, then pass to a finite étale\nupper cover. The following lemma shows that these operations preserve the bound on horizontal\nramification needed for membership in Lℓ."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-6",
      "kind": "lemma",
      "label": "Lemma 3.4",
      "source": {
        "blockId": "p8-c7ae99a541",
        "page": 8,
        "quote": "Lemma 3.4. Fix a positive integer n. Among finite extensions of K0, the condition that every\nramification index at a closed point of X¯k divide n is preserved by taking subextensions, finite\ncomposita, and Galois closures. It is also preserved by normalized pullback along a finite morphism\nof smooth proper curves and by composition on top with a finite étale cover. Consequently every\nfinite cover of X¯k with indices dividing ℓhas a K0-embedding of its function field into Lℓ."
      },
      "statement": [
        {
          "blockId": "p8-c7ae99a541",
          "page": 8,
          "quote": "Lemma 3.4. Fix a positive integer n. Among finite extensions of K0, the condition that every\nramification index at a closed point of X¯k divide n is preserved by taking subextensions, finite\ncomposita, and Galois closures. It is also preserved by normalized pullback along a finite morphism\nof smooth proper curves and by composition on top with a finite étale cover. Consequently every\nfinite cover of X¯k with indices dividing ℓhas a K0-embedding of its function field into Lℓ."
        }
      ],
      "proof": [
        {
          "blockId": "p8-c97559e01e",
          "page": 8,
          "quote": "Proof. All valuations in this statement are the horizontal, discrete closed-point valuations,\nwhich are trivial on ¯k. Their residue characteristic is zero. The completed base field at such\na point is A = ¯k((t)). Every finite extension of A has residue field ¯k and is of the form ¯k((u))\nfor a uniformizer u: successive subtraction of residue representatives gives the power-series\nexpansion in its complete valuation ring. If its ramification index is e, then t = uea(u) for a unit\na(u) ∈¯k[[u]]×. The unit has an e-th root in ¯k[[u]], by Hensel’s lemma in residue characteristic\nzero. Changing the parameter gives t = ze. Thus, inside a fixed algebraic closure, the local field\nis A(t1/e), which is cyclic over A since ¯k contains all roots of unity.\nFor e | n, this is a subfield of A(t1/n). All local factors of a finite compositum, or of a normal\nclosure formed from conjugates, therefore remain subfields of this same cyclic extension. This\nproves the assertions about subextensions, composita, and Galois closures.\nFor pullback, write the local base map as t = sra(s), where r ≥1 and a(s) is a unit.\nExtracting its e-th root in ¯k[[s]] shows that every normalized local factor after pullback has\nramification index e/ gcd(e, r), which divides e. Finally, ramification indices multiply in a tower,\nand an étale upper cover contributes index 1. Apply the Galois-closure assertion to a cover\nwith indices dividing ℓand embed that closure into the fixed Ksep. Its field lies in the defining\ncompositum Lℓ."
        }
      ],
      "guide": []
    },
    {
      "id": "remark-7",
      "kind": "remark",
      "label": "Remark 3.5",
      "source": {
        "blockId": "p8-1aa850b304",
        "page": 8,
        "quote": "Remark 3.5. Lemma 3.4 makes no assertion about tame reduction in residue characteristic p, nor\nabout composing two arbitrarily ramified covers. Its last assertion about composition requires\nthe upper cover to be étale."
      },
      "statement": [
        {
          "blockId": "p8-1aa850b304",
          "page": 8,
          "quote": "Remark 3.5. Lemma 3.4 makes no assertion about tame reduction in residue characteristic p, nor\nabout composing two arbitrarily ramified covers. Its last assertion about composition requires\nthe upper cover to be étale."
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "lemma-8",
      "kind": "lemma",
      "label": "Lemma 4.1",
      "source": {
        "blockId": "p8-36813d5eeb",
        "page": 8,
        "quote": "Lemma 4.1 (Units on disks and annuli). An analytic unit on an open disk over Ωhas constant\nabsolute value in |Ω×|. If A = {r < |S| < R} is an open annulus and u ∈O(A)×, then"
      },
      "statement": [
        {
          "blockId": "p8-36813d5eeb",
          "page": 8,
          "quote": "Lemma 4.1 (Units on disks and annuli). An analytic unit on an open disk over Ωhas constant\nabsolute value in |Ω×|. If A = {r < |S| < R} is an open annulus and u ∈O(A)×, then"
        },
        {
          "blockId": "p8-6b360aa50e",
          "page": 8,
          "quote": "u = cSd(1 + h),\nc ∈Ω×,\nd ∈Z,\nh ∈O(A),\n|h(x)| < 1\n(x ∈A),"
        },
        {
          "blockId": "p8-fcb18b4841",
          "page": 8,
          "quote": "where cSd is a term of the Laurent expansion of u. The function 1 + h has an analytic ℓ-th root\non A. Consequently, on any analytic cover of A on which S has an ℓ-th root, so does u."
        }
      ],
      "proof": [
        {
          "blockId": "p8-62d6c4abb0",
          "page": 8,
          "quote": "Proof. On a circle |S| = ρ of radius ρ ∈|Ω×|, the Laurent Gauss norm is multiplicative.\nNormalize u and its inverse to have norm one. Their reductions are inverse units in eΩ[s, s−1], so"
        },
        {
          "blockId": "p9-7df61f8099",
          "page": 9,
          "quote": "the reduction of u is a monomial. Thus exactly one Laurent term dominates at that radius. Its\nexponent cannot change inside the radius interval. Indeed, convergence on compact subannuli\nreduces any proposed change to finitely many competing terms; a tie between two terms occurs\nat a radius in |Ω×|, since the additive value group is Q. Such a tie contradicts the preceding\nreduction argument. A single term cSd therefore dominates throughout A, giving the asserted\nfactorization. The same argument on closed subdisks uses that the only units in eΩ[s] are\nconstants and proves the disk assertion. Its constant norm belongs to |Ω×| by evaluation at a\nrational point of the disk.\nSince ℓis a unit in the valuation ring, the binomial series for (1 + h)1/ℓconverges on every\ncompact subannulus; the resulting functions agree on overlaps. A root of c lies in Ω, and a root\nof Sd follows from a root of S, also when d < 0."
        },
        {
          "blockId": "p9-ee00c7509d",
          "page": 9,
          "quote": "Recall the data attached to a split stable model of a proper curve. Its skeleton is a finite\nmetric graph. Each vertex carries the smooth normalization of the corresponding special-fibre\ncomponent, with one marked point for each incident edge end. A loop has two distinct marks\non the same normalization. The tube of a node is an open annulus, and the tube of a smooth\nspecial-fibre point is an open disk. The graph with these marked residue curves is its metrized\ncomplex. Stability means\n2g(v) −2 + val(v) > 0"
        },
        {
          "blockId": "p9-f62b2664e5",
          "page": 9,
          "quote": "at each vertex v, where g(v) is the genus of its residue curve."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-9",
      "kind": "lemma",
      "label": "Lemma 4.2",
      "source": {
        "blockId": "p9-a7590a88c3",
        "page": 9,
        "quote": "Lemma 4.2 (Roots on stable node annuli). Let C/Ωbe a smooth proper connected curve of\ngenus at least two, equipped with its stable skeleton. Let A be a node annulus and S an annular\ncoordinate on A. There is a connected finite étale cover C′ →C such that S has an analytic ℓ-th\nroot on the entire inverse image of A. A single such cover can be chosen for any finite collection\nof node annuli."
      },
      "statement": [
        {
          "blockId": "p9-a7590a88c3",
          "page": 9,
          "quote": "Lemma 4.2 (Roots on stable node annuli). Let C/Ωbe a smooth proper connected curve of\ngenus at least two, equipped with its stable skeleton. Let A be a node annulus and S an annular\ncoordinate on A. There is a connected finite étale cover C′ →C such that S has an analytic ℓ-th\nroot on the entire inverse image of A. A single such cover can be chosen for any finite collection\nof node annuli."
        }
      ],
      "proof": [
        {
          "blockId": "p9-ef3971a04b",
          "page": 9,
          "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
        },
        {
          "blockId": "p10-187bd21fdc",
          "page": 10,
          "quote": "vertices. All lengths belong to Q. At a lifted circuit vertex each base direction has total degree\nℓ; at an identity-copy vertex it has total degree one. The local Riemann–Hurwitz formula is"
        },
        {
          "blockId": "p10-8cb1fe65f9",
          "page": 10,
          "quote": "2g(D′) −2 = ℓ\n2g(D) −2\n + 2(ℓ−1)."
        },
        {
          "blockId": "p10-162d42f985",
          "page": 10,
          "quote": "The two extra terms are precisely those recorded by the two circuit edges. Thus the augmented\ngraph ramification is zero at every vertex; the identity copies contribute none. All ramification\nindices are prime to p. The resulting tame covering is connected, because each vertex outside\nthe circuit can be joined by a lifted path to a circuit vertex.\nLift these data by the cited theorem. The inverse image of A is exactly the disjoint union of the\nnode annuli above e. For completeness, the inverse-skeleton equality excludes off-skeleton points\nmapping to the skeleton. Each off-skeleton open ball therefore maps into a single off-skeleton\nball downstairs, and continuity identifies the images of their attachment points. Retractions\ncommute on these balls as well as on the skeleton. Since a node annulus is the inverse image of\nits open edge under retraction, this proves the asserted equality for the whole annulus. On each\nannulus above A, the pullback of S has exponent ℓor −ℓin Lemma 4.1. It therefore has an\nanalytic ℓ-th root on each of these disjoint annuli.\nNow suppose that e is separating. Each side of the graph obtained by cutting its interior\ncontains a circuit or a positive-genus vertex. Otherwise that side would be a genus-zero tree\nwith, say, N vertices. Its total valence, including the bridge half-edge, would be 2N −1, contrary\nto stability, which requires at least 3N.\nConstruct a connected unramified degree-ℓcovering of each side, retaining its boundary\nmark for the eventual gluing. If the side contains a circuit, take a connected cyclic graph\ncovering of degree ℓ, with identity residue-curve maps. Such a graph covering is obtained by a\nhomomorphism of its graph fundamental group onto Z/ℓZ. If instead a positive-genus vertex is\nused, take a connected cyclic étale cover of degree ℓof its residue curve, provided by a nonzero\nelement of its Jacobian’s ℓ-torsion [16, Theorem 8.2 and Remark 8.4]. Explicitly, for a line bundle\nL of exact order ℓ, a trivialization L⊗ℓ≃O makes Lℓ−1\nj=0 L⊗j an algebra whose relative spectrum\nis a cyclic étale cover: ℓis invertible, and the exact order of L makes the cover connected. Use ℓ\nidentity copies at the other vertices and match all marked fibres. Paths to the distinguished\nvertex show connectedness of the covered side.\nEach covered side has ℓpreimages of the bridge mark. Match them in pairs to obtain ℓ\ndistinct bridge edges, all with factor one. Every such edge is now nonseparating: after removing\none of them, the two connected covered sides are still joined by the others. The resulting covering\nof the original metrized complex is tame. Indeed the component covers are unramified, all\nfactors are one, and Riemann–Hurwitz again gives zero graph ramification. Lift it to a finite\nétale curve cover. Its triangulation remains stable: at a vertex of local degree d, the quantity\n2g′ −2 + val(v′) is d(2g −2 + val(v)) > 0.\nApply the nonseparating construction to each of these ℓedges. A connected component of\nthe fibre product of the resulting finite étale covers dominates each factor, so all the constructed\nroots persist on a common cover. On each factor-one annulus the pullback of the original S is a\nunit; Lemma 4.1 supplies its root once the new annular coordinate has a root. This proves the\nassertion also for separating edges. Finally, the same finite fibre-product argument treats finitely\nmany original annuli simultaneously."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-10",
      "kind": "lemma",
      "label": "Lemma 4.3",
      "source": {
        "blockId": "p11-878474d913",
        "page": 11,
        "quote": "Lemma 4.3 (Étale covers with roots on collars). Let C/¯k be a smooth proper connected curve\nof genus at least two. Let b1, . . . , bs ∈C(¯k) be distinct, and let Bi be pairwise disjoint sufficiently\nsmall coordinate disks in Can\nΩabout bi, with coordinates ti defined over ¯k and vanishing simply\nat bi. There is a connected finite étale cover Y →CΩand concentric open collars"
      },
      "statement": [
        {
          "blockId": "p11-878474d913",
          "page": 11,
          "quote": "Lemma 4.3 (Étale covers with roots on collars). Let C/¯k be a smooth proper connected curve\nof genus at least two. Let b1, . . . , bs ∈C(¯k) be distinct, and let Bi be pairwise disjoint sufficiently\nsmall coordinate disks in Can\nΩabout bi, with coordinates ti defined over ¯k and vanishing simply\nat bi. There is a connected finite étale cover Y →CΩand concentric open collars"
        },
        {
          "blockId": "p11-cb2d7f4e26",
          "page": 11,
          "quote": "A′\ni = {ri < |ti| < Ri} ⊂Bi,\nri, Ri ∈|Ω×|,\n0 < ri < Ri,"
        },
        {
          "blockId": "p11-e7d0fa5e82",
          "page": 11,
          "quote": "such that ti has an analytic ℓ-th root on the entire inverse image of A′\ni in Y an."
        }
      ],
      "proof": [
        {
          "blockId": "p11-aac65728ac",
          "page": 11,
          "quote": "Proof. All algebraic data descend to a finite extension of k. After a further finite extension,\nchoose a split semistable model with the bi as disjoint smooth sections. Shrinking the coordinate\ndisks causes no problem, since the desired collars need only lie inside the originally specified disks.\nIn an integral local parameter ui at a section, successive blowups at its reductions introduce\nnode annuli\n|π|j+1 < |ui| < |π|j."
        },
        {
          "blockId": "p11-bba5eb485c",
          "page": 11,
          "quote": "On a sufficiently small disk the quotient ti/ui is an analytic unit, since both coordinates have\na simple zero at bi. Its absolute value is constant by Lemma 4.1. Thus these annuli are also\nconcentric annuli for ti, and choosing j large puts them inside Bi. Perform these modifications\nfor all the sections and denote the resulting proper semistable model by C. It need not be stable;\nthe markings can now be forgotten.\nThe RNS theorem, applied with the set of all primes, and its model-domination consequence\n[17, Theorem A and Proposition 2.4(iv)] give, after finite extension of the ground field, a finite\nétale cover whose stable model maps to C. Pass to split reduction and a geometric connected\ncomponent, and write\nf : Y0 −→CΩ,\nY0 −→C"
        },
        {
          "blockId": "p11-eae78e8fc9",
          "page": 11,
          "quote": "for the analytic-base-changed curve cover and the model morphism. No finiteness assertion\nabout the model morphism is needed.\nFix one of the target node annuli Ai ⊂Bi just constructed, and choose an irrational real\nnumber a strictly between its two additive endpoint valuations. Consider the entire inverse\nimage\nf−1{x ∈Ai : −logp |ti(x)| = a}\n."
        },
        {
          "blockId": "p11-d23775987c",
          "page": 11,
          "quote": "Reduction is compatible with the model morphism. A point reducing to a component generic\npoint of Y0 is a model vertex, hence is type 2 and gives ti a value in |Ω×|; its additive valuation\nis rational. It cannot lie above this level. A point reducing to a smooth closed point lies in an\nopen disk tube. If it lies above the chosen level, that smooth point maps to the target node, so\nthe entire disk tube maps into Ai. The pullback of ti is a unit there. By Lemma 4.1 its norm is\nconstant and belongs to |Ω×|, which again excludes the irrational level. All remaining points\nabove the level lie in source node annuli whose nodes map to the target node.\nLet Ni be the union of all source node annuli whose nodes map to the target node defining\nAi; this is a finite union. The complement of Ni in the proper analytic curve Y an\n0\nis compact.\nIntersect it with the inverse image of a closed concentric subannulus of Ai containing the\nchosen level in its interior. On this compact intersection the continuous real function −logp |ti|\nhas compact image avoiding a. Consequently there is an open interval about a with rational\nendpoints whose entire inverse image lies in Ni. It determines a collar A′\ni ⊂Ai.\nEvery source node annulus in Ni maps into Ai by reduction compatibility, so the pullback of\nti is an analytic unit on it. Apply Lemma 4.2 simultaneously to all these source annuli, for all i.\nTheir coordinates acquire ℓ-th roots on a common connected finite étale cover of Y0. Lemma 4.1\nthen gives roots of the pullbacks of the ti on the full inverse images of the A′\ni. The resulting\ncomposite cover of CΩis finite étale."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-11",
      "kind": "proposition",
      "label": "Proposition 4.4",
      "source": {
        "blockId": "p11-81b94308f2",
        "page": 11,
        "quote": "Proposition 4.4 (The first stage). The field Lℓis contained in H."
      },
      "statement": [
        {
          "blockId": "p11-81b94308f2",
          "page": 11,
          "quote": "Proposition 4.4 (The first stage). The field Lℓis contained in H."
        }
      ],
      "proof": [
        {
          "blockId": "p12-a2f25852d0",
          "page": 12,
          "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-12",
      "kind": "lemma",
      "label": "Lemma 5.1",
      "source": {
        "blockId": "p12-8b3352aee6",
        "page": 12,
        "quote": "Lemma 5.1 (Roots after division on a totally degenerate abelian variety). Let J be an abelian\nvariety over Ωwith analytic uniformization q : Tan →Jan, where T is a split torus and the\nkernel Λ ⊂T(Ω) is a free lattice acting by translation. Suppose q is a topological covering and\na local analytic isomorphism. Let A be an analytic space, let a : A →Tan be analytic, and let\nm ≥1 be an integer. Set\nB = A ×q◦a,Jan,[m] Jan."
      },
      "statement": [
        {
          "blockId": "p12-8b3352aee6",
          "page": 12,
          "quote": "Lemma 5.1 (Roots after division on a totally degenerate abelian variety). Let J be an abelian\nvariety over Ωwith analytic uniformization q : Tan →Jan, where T is a split torus and the\nkernel Λ ⊂T(Ω) is a free lattice acting by translation. Suppose q is a topological covering and\na local analytic isomorphism. Let A be an analytic space, let a : A →Tan be analytic, and let\nm ≥1 be an integer. Set\nB = A ×q◦a,Jan,[m] Jan."
        },
        {
          "blockId": "p12-6345d468cf",
          "page": 12,
          "quote": "For every character χ of T, the pullback of χ ◦a to B has an analytic m-th root."
        }
      ],
      "proof": [
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        },
        {
          "blockId": "p12-554fadfdca",
          "page": 12,
          "quote": "To obtain a root of an annular coordinate from the preceding lemma, we arrange that the\npullback of a torus character has slope one. It then differs from the coordinate by a constant\nand an analytic unit close to one. Division on the Jacobian supplies a root of the character on\nthe whole inverse annulus; a sufficiently long annulus leaves room for a middle collar on which\nthe remaining unit also admits a root."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-13",
      "kind": "proposition",
      "label": "Proposition 5.2",
      "source": {
        "blockId": "p13-ad360a6877",
        "page": 13,
        "quote": "Proposition 5.2 (An auxiliary branched cover). Let m ≥1 be an integer, choose an integer"
      },
      "statement": [
        {
          "blockId": "p13-ad360a6877",
          "page": 13,
          "quote": "Proposition 5.2 (An auxiliary branched cover). Let m ≥1 be an integer, choose an integer"
        },
        {
          "blockId": "p13-180905a85c",
          "page": 13,
          "quote": "D > vp(m) +\n1\np −1,"
        },
        {
          "blockId": "p13-2efce9f9ee",
          "page": 13,
          "quote": "and let α ∈k\n× satisfy a = v(α) > 2D. There is a smooth, proper, connected curve Q/k and a\nfinite map Q →P1"
        },
        {
          "blockId": "p13-24a0c55f7f",
          "page": 13,
          "quote": "k whose ramification indices divide ℓ, such that the pullback of the coordinate\nT has an analytic m-th root on the entire inverse image, over Ω, of"
        },
        {
          "blockId": "p13-51845f441b",
          "page": 13,
          "quote": "A◦= {D < v(T) < a −D}."
        }
      ],
      "proof": [
        {
          "blockId": "p13-d8ca492316",
          "page": 13,
          "quote": "Proof. Let Z be the smooth projective normalization of"
        },
        {
          "blockId": "p13-f4696f98c0",
          "page": 13,
          "quote": "yℓ= T(T −1)"
        },
        {
          "blockId": "p13-7b5280585c",
          "page": 13,
          "quote": "T −α ."
        },
        {
          "blockId": "p13-ef8b47c581",
          "page": 13,
          "quote": "The four distinct points 0, α, 1, ∞have orders 1, −1, 1, −1 in the divisor of the displayed rational\nfunction. The cyclic cover is connected, has degree ℓ, and is totally ramified at precisely these\npoints. Riemann–Hurwitz gives"
        },
        {
          "blockId": "p13-9602266c5c",
          "page": 13,
          "quote": "2g(Z) −2 = −2ℓ+ 4(ℓ−1),\ng(Z) = ℓ−1."
        },
        {
          "blockId": "p13-36e42bc923",
          "page": 13,
          "quote": "Here is a split semistable model of Z. Work first over the integers of a sufficiently large finite\nextension of k containing α. The blowup of the model of the line along (T, α) has two rational\ncomponents and one node, with local equation TU = α. The inner component has coordinate\nS = T/α; the four branch sections are disjoint and smooth. The Kummer function reduces\nto T −1 on the outer component and to −S/(S −1) on the inner component. Each of these\ndegree-ℓcovers of a rational component has just two totally ramified points and is rational. Near\nthe node the Kummer function is\nT −1\n1 −U ,"
        },
        {
          "blockId": "p13-3b2bd6a638",
          "page": 13,
          "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
        },
        {
          "blockId": "p13-abfe83435d",
          "page": 13,
          "quote": "q : Tan −→Jan\nΩ,\nker(q) = Λ ⊂T(Ω)."
        },
        {
          "blockId": "p13-c6ed4239e0",
          "page": 13,
          "quote": "Its character lattice is M = H1(Γ, Z); its tropical target is Hom(M, R). The period lattice is\nfree, and q is a topological covering and local analytic isomorphism. These are the Jacobian\nuniformization conventions of [3, Sections 4.1–4.2 and 4.5.2]. The compatibility of the classical\nand tropical Abel–Jacobi maps [3, Proposition 6.1 and Corollary 6.6] identifies the latter with\nintegration on Γ using the edge-length pairing.\nThe open Berkovich annulus Aj strongly deformation retracts to its open skeletal interval\n[2, Section 2.3] and is locally path connected. It is thus simply connected, so i|Aj lifts to an"
        },
        {
          "blockId": "p14-83e6e65311",
          "page": 14,
          "quote": "analytic map bij : Aj →Tan: first lift continuously along the covering q, then use its analytic\nlocal inverses. Orient all edges by increasing v(T). For j′ ̸= j, the cycle cj = ej −ej′ is an\nelement of M. If χj is its character, then uj = χj ◦bij is an analytic unit. The derivative of v(uj)\nalong ej, with respect to v(T), is exactly one: the integral of the harmonic form represented by\ncj over a segment of ej is the length of that segment. Lifts of the tropical map differ only by a\nconstant period, which does not change this derivative.\nBy Lemma 4.1, an analytic unit on an open annulus has a unique dominant Laurent monomial\nat every interior Gauss radius; its exponent is constant and is the slope of its log-norm. It follows\nthat\nuj = c′\njT(1 + hj),\nhj =\nX"
        },
        {
          "blockId": "p14-46e0515936",
          "page": 14,
          "quote": "n̸=0\nhj,nT n,\nc′\nj ∈Ω×."
        },
        {
          "blockId": "p14-c8ba572435",
          "page": 14,
          "quote": "Here c′\njT is the actual coefficient term of degree one. Dominance at every radius r ∈(|α|, 1)\ngives |hj,n|rn < 1. Taking limits at the endpoints yields, for n > 0,"
        },
        {
          "blockId": "p14-9b72f355e7",
          "page": 14,
          "quote": "|hj,n| ≤1,\n|hj,−n| ≤|α|n."
        },
        {
          "blockId": "p14-3a4f58d757",
          "page": 14,
          "quote": "Writing s = v(T), the ultrametric inequality consequently gives"
        },
        {
          "blockId": "p14-dc7170c439",
          "page": 14,
          "quote": "|hj| ≤p−min{s,a−s} < p−D\non Aj ∩{D < s < a −D}."
        },
        {
          "blockId": "p14-ab0531b683",
          "page": 14,
          "quote": "Figure 1 displays the cycle that supplies the slope-one character and the trimming that gives\nthis bound."
        },
        {
          "blockId": "p14-487b788433",
          "page": 14,
          "quote": "Parallel edges of the skeleton"
        },
        {
          "blockId": "p14-f02137f2ac",
          "page": 14,
          "quote": "The middle collar on one edge"
        },
        {
          "blockId": "p14-bcdb9bc4e8",
          "page": 14,
          "quote": "ej"
        },
        {
          "blockId": "p14-51843e9a3f",
          "page": 14,
          "quote": "increasing v(T)"
        },
        {
          "blockId": "p14-abf5f9731d",
          "page": 14,
          "quote": "A◦"
        },
        {
          "blockId": "p14-f786c0231e",
          "page": 14,
          "quote": "ej′"
        },
        {
          "blockId": "p14-58898cb1c4",
          "page": 14,
          "quote": "v(T) = 0\nv(T) = a"
        },
        {
          "blockId": "p14-25d3b270c1",
          "page": 14,
          "quote": "0\nD\na −D\na"
        },
        {
          "blockId": "p14-dd072076ad",
          "page": 14,
          "quote": "..."
        },
        {
          "blockId": "p14-931f971f5f",
          "page": 14,
          "quote": "other edges"
        },
        {
          "blockId": "p14-e027d5a886",
          "page": 14,
          "quote": "uj = c′\njT(1 + hj)\n|hj| < p−D on A◦"
        },
        {
          "blockId": "p14-c66011643c",
          "page": 14,
          "quote": "D > vp(m) +\n1\np −1"
        },
        {
          "blockId": "p14-81229b1e20",
          "page": 14,
          "quote": "ℓedges, each of length a\ncycle cj = ej −ej′"
        },
        {
          "blockId": "p14-e03a60d727",
          "page": 14,
          "quote": "Figure 1: The construction in Proposition 5.2, shown schematically. All edges are oriented\nby increasing v(T); the cycle ej −ej′ gives a character unit uj of slope one on the annulus\ncorresponding to ej. Division by m on the Jacobian supplies a root of uj on the whole inverse\nannulus. Trimming its skeletal interval to D < v(T) < a −D makes hj small enough for 1 + hj\nto have an m-th root, and hence gives a root of T."
        },
        {
          "blockId": "p14-c854a3f922",
          "page": 14,
          "quote": "On every closed subannulus this estimate has a strict uniform margin. The logarithm and\nexponential therefore converge locally uniformly and"
        },
        {
          "blockId": "p14-49024b5738",
          "page": 14,
          "quote": "bj = exp\nm−1 log(1 + hj)\n"
        },
        {
          "blockId": "p14-69e0d744da",
          "page": 14,
          "quote": "is analytic there, with bm\nj\n= 1 + hj. Indeed, since D > 1/(p −1), the logarithm satisfies\n| log(1 + hj)| ≤|hj|, and the valuation of the exponent is greater than D −vp(m) > 1/(p −1).\nFor all ℓannuli at once, form the same algebraic finite étale cover"
        },
        {
          "blockId": "p14-5c842278c6",
          "page": 14,
          "quote": "Z′ = Z ×i,J,[m] J −→Z."
        },
        {
          "blockId": "p14-a396d99e51",
          "page": 14,
          "quote": "It is finite étale for every positive integer m, since the ground field has characteristic zero.\nLemma 5.1 applied to a = bij gives an analytic m-th root of uj on the whole inverse image of Aj."
        },
        {
          "blockId": "p15-9d0c2e8bfe",
          "page": 15,
          "quote": "Dividing it by bj and an m-th root of c′\nj gives an analytic m-th root of T above the middle collar.\nThe annuli Aj are disjoint, so this gives a root on its entire inverse image in Z′. Choose any\nconnected component Q of Z′. Its map to Z is finite étale and surjective, so it is a smooth proper\nconnected curve with the required property. Its composite with Z →P1 has only ramification\nindices 1 or ℓ. All algebraic maps used to define Q are over k; the analytic roots and torus lifts\nare only required over Ω."
        }
      ],
      "guide": []
    },
    {
      "id": "corollary-14",
      "kind": "corollary",
      "label": "Corollary 7.1",
      "source": {
        "blockId": "p16-d357ebb858",
        "page": 16,
        "quote": "Corollary 7.1. Let k/Qp be finite and let X/k be a smooth proper geometrically connected\ncurve of genus at least two. Put K = k(X) and let f\nK ⊂Ksep be the extension corresponding to\nthe full arithmetic étale fundamental group of X. Let w be a real-valued rank-one valuation of\nK extending the p-adic valuation, normalized by w(p) = 1, and fix a prolongation wsep to Ksep.\nRestriction induces an isomorphism of profinite groups"
      },
      "statement": [
        {
          "blockId": "p16-d357ebb858",
          "page": 16,
          "quote": "Corollary 7.1. Let k/Qp be finite and let X/k be a smooth proper geometrically connected\ncurve of genus at least two. Put K = k(X) and let f\nK ⊂Ksep be the extension corresponding to\nthe full arithmetic étale fundamental group of X. Let w be a real-valued rank-one valuation of\nK extending the p-adic valuation, normalized by w(p) = 1, and fix a prolongation wsep to Ksep.\nRestriction induces an isomorphism of profinite groups"
        },
        {
          "blockId": "p16-0a9614cda9",
          "page": 16,
          "quote": "Dwsep(Ksep/K)\n∼\n−−→De\nw(f\nK/K),\new = wsep| e\nK."
        }
      ],
      "proof": [
        {
          "blockId": "p16-45f6d3aaff",
          "page": 16,
          "quote": "Proof. Restriction is surjective. Indeed, lift an automorphism stabilizing ew to an automorphism\nof Ksep/K. Its translate of wsep is another prolongation of ew. Transitivity of Gal(Ksep/f\nK)\non these prolongations allows us to adjust the lift so that it stabilizes wsep. The transitivity\nstatement for an infinite normal extension follows from its finite-level version by compactness.\nAn automorphism stabilizing a rank-one valuation preserves its normalized absolute value.\nTo see why normalization matters, any two order-preserving embeddings of the value group into\nR differ by a positive scale; the value on p fixes that scale. Such an automorphism therefore"
        },
        {
          "blockId": "p16-c72add0b5d",
          "page": 16,
          "quote": "extends as an isometry to E = [\nKsep. If its restriction to f\nK is the identity, it fixes H = c\nf\nK ⊂E\npointwise. Theorem 1.2 gives Ksep ⊂H, so the automorphism is the identity.\nBoth decomposition groups are closed subgroups of their Galois groups.\nFor example,\nstabilization of a valuation ring is the intersection, over its algebraic elements, of conditions on\nfinite Galois orbits, each of which is clopen. Restriction is consequently a continuous bijection\nbetween compact Hausdorff groups, and its inverse is continuous."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-15",
      "kind": "lemma",
      "label": "Lemma 7.2",
      "source": {
        "blockId": "p16-f889b64629",
        "page": 16,
        "quote": "Lemma 7.2. With k, X, K, and f\nK as in Corollary 7.1, let w be any rank-one valuation of K\nextending the p-adic valuation and let ew be any prolongation to f\nK. The projection"
      },
      "statement": [
        {
          "blockId": "p16-f889b64629",
          "page": 16,
          "quote": "Lemma 7.2. With k, X, K, and f\nK as in Corollary 7.1, let w be any rank-one valuation of K\nextending the p-adic valuation and let ew be any prolongation to f\nK. The projection"
        },
        {
          "blockId": "p16-1e032d255e",
          "page": 16,
          "quote": "De\nw(f\nK/K) −→Gk"
        },
        {
          "blockId": "p16-6f906635ba",
          "page": 16,
          "quote": "has no continuous group-homomorphic section."
        }
      ],
      "proof": [
        {
          "blockId": "p16-4611b28177",
          "page": 16,
          "quote": "Proof. Normalize w(p) = 1 and prolong the valuation to Ksep. Suppose that a section exists.\nCorollary 7.1 lifts it to a continuous section"
        },
        {
          "blockId": "p16-cf9338301a",
          "page": 16,
          "quote": "bs : Gk −→GK = Gal(Ksep/K)"
        },
        {
          "blockId": "p16-f8589da2c3",
          "page": 16,
          "quote": "whose image stabilizes this prolongation. The birational section theorem of Koenigsmann places\nthe entire image of bs also in the decomposition group of a prolongation of orda for some a ∈X(k)\n[13, preprint version, Proposition 2.4(2), Definition 1.4, and Observation 1.3]. Here orda is the\nclosed-point valuation of K, trivial on k."
        },
        {
          "blockId": "p17-cd9725f405",
          "page": 17,
          "quote": "Write Σ = bs(Gk) and M = (Ksep)Σ. The subgroup Σ is compact and hence closed, so\nGM = Σ. Let v1 and v2 be the restrictions to M of the two selected prolongations. Each is\nhenselian: its absolute Galois group stabilizes a prolongation, and the transitive action on all\nprolongations therefore has a single point. Both valuations are nontrivial and have rank one.\nIndeed, value groups in an algebraic valued extension differ by a torsion quotient, which preserves\ntheir convex-subgroup chains. Moreover,"
        },
        {
          "blockId": "p17-5c174a043e",
          "page": 17,
          "quote": "v1(p) = 1,\nv2|k× = 0,"
        },
        {
          "blockId": "p17-52809f792f",
          "page": 17,
          "quote": "so these valuations are inequivalent. The resulting contradiction is an instance of the classical\nuniqueness principle for henselian valuations associated with F. K. Schmidt [24]. We give\nexplicitly the two-valuation approximation needed here.\nChoose a nonsquare d ∈k×. It remains a nonsquare in M: otherwise GM = Σ would fix\na square root of d in ¯k, contrary to the surjection Σ →Gk. Let t ∈K be a uniformizer at a.\nChoose a positive integer N with N > v1(t), and put z = pN/t. Then"
        },
        {
          "blockId": "p17-4291df9365",
          "page": 17,
          "quote": "v1(z) > 0,\nv2(z) = −v2(t) < 0."
        },
        {
          "blockId": "p17-45fabab6f0",
          "page": 17,
          "quote": "For positive integers r, set"
        },
        {
          "blockId": "p17-4c55345d29",
          "page": 17,
          "quote": "qr = d + (1 −d)\nzr"
        },
        {
          "blockId": "p17-ff11be89d5",
          "page": 17,
          "quote": "1 + zr ."
        },
        {
          "blockId": "p17-0657feeaca",
          "page": 17,
          "quote": "The denominator is nonzero because v1(zr) > 0. At the first valuation qr tends to d, and at the\nsecond it tends to 1. More precisely,"
        },
        {
          "blockId": "p17-7b97baa8f2",
          "page": 17,
          "quote": "v1(qr/d −1) = v1((1 −d)/d) + rv1(z) −→+∞,"
        },
        {
          "blockId": "p17-3d516e1394",
          "page": 17,
          "quote": "whereas\nv2(qr −1) = v2(d −1) −rv2(z) −→+∞."
        },
        {
          "blockId": "p17-832d186942",
          "page": 17,
          "quote": "For sufficiently large r, Hensel’s lemma applied at 1 gives a square root of qr/d using v1, and a\nsquare root of qr using v2. The sufficient inequalities are"
        },
        {
          "blockId": "p17-444d9d305f",
          "page": 17,
          "quote": "v1(qr/d −1) > 2v1(2),\nv2(qr −1) > 2v2(2)."
        },
        {
          "blockId": "p17-a0e4a91ee4",
          "page": 17,
          "quote": "Their quotient makes d a square in M, a contradiction. This argument includes p = 2 and\narbitrary nondiscrete rank-one value groups."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-16",
      "kind": "proposition",
      "label": "Proposition 8.1",
      "source": {
        "blockId": "p19-07e3bbefa2",
        "page": 19,
        "quote": "Proposition 8.1 (The localization image). Let C/K be a smooth proper geometrically connected\ncurve of genus at least two over a number field. Then"
      },
      "statement": [
        {
          "blockId": "p19-07e3bbefa2",
          "page": 19,
          "quote": "Proposition 8.1 (The localization image). Let C/K be a smooth proper geometrically connected\ncurve of genus at least two over a number field. Then"
        },
        {
          "blockId": "p19-2049c0ef47",
          "page": 19,
          "quote": "loc\nSec(C/K)\n = C(AK)f-cov\n•\n.\n(8.2)"
        }
      ],
      "proof": [
        {
          "blockId": "p19-eb06480d4a",
          "page": 19,
          "quote": "Proof. Let s ∈Sec(C/K). Choose a representative point of the determined component at each\nreal place, and any point at each complex place. Harari–Stix’s Theorem 2.1, with U = 1 and the\nomitted set of places S = ∅, shows that these localizations survive every torsor under a finite\nK-group scheme; their Remark 2.2(1) identifies this as finite-cover descent [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)]. Consequently"
        },
        {
          "blockId": "p19-5c86c998f7",
          "page": 19,
          "quote": "x(s) ∈C(AK)f-cov\n•\n."
        },
        {
          "blockId": "p19-0cb1be5afe",
          "page": 19,
          "quote": "Conversely, let x belong to the finite-cover locus and, by properness, choose an ordinary\nadelic representative P = (Pv) ∈C(AK). Stoll’s cohomological description after Definition 5.2,\ntogether with Definition 5.4, identifies survival with diagonal global H1(K, G)-membership of the"
        },
        {
          "blockId": "p20-9685c75da6",
          "page": 20,
          "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
        }
      ],
      "guide": []
    },
    {
      "id": "theorem-17",
      "kind": "theorem",
      "label": "Theorem 8.2",
      "source": {
        "blockId": "p20-349bbee5b6",
        "page": 20,
        "quote": "Theorem 8.2 (Finite-cover descent criterion). Let C/K be a smooth proper geometrically\nconnected curve of genus at least two over a number field. If"
      },
      "statement": [
        {
          "blockId": "p20-349bbee5b6",
          "page": 20,
          "quote": "Theorem 8.2 (Finite-cover descent criterion). Let C/K be a smooth proper geometrically\nconnected curve of genus at least two over a number field. If"
        },
        {
          "blockId": "p20-13ffcea2da",
          "page": 20,
          "quote": "C(AK)f-cov\n•\n= C(K),\n(8.3)"
        },
        {
          "blockId": "p20-279523de83",
          "page": 20,
          "quote": "then κC : C(K) →Sec(C/K) is bijective."
        }
      ],
      "proof": [
        {
          "blockId": "p20-a963cfc15e",
          "page": 20,
          "quote": "Proof. Let s ∈Sec(C/K). By Proposition 8.1 and (8.3), there is a ∈C(K) such that x(s) is the\ndiagonal localization of a, hence equals the localization of the point section sa.\nAt every finite place the point determined by s is therefore a. At each real place choose a\nitself in the determined component, and choose a also at each complex place. The resulting\nordinary adelic representative of the localization tuple is supported on {a}. Thus the support\nZ(s) in Stix’s terminology, the Zariski-closed support of this tuple, is finite over K. Since C is\nproper of genus at least two, it is hyperbolic and has no cuspidal sections. Stix’s finite-support\nresult therefore makes s a point section [25, arXiv version, Section 3 and Corollary 5]. Finally, if\ntwo points give the same global section, their sections agree after restriction to any one finite\ncompletion. Injectivity in Theorem 1.1 then identifies the two points. This proves injectivity\nand also identifies the point giving s with a."
        },
        {
          "blockId": "p20-e0a64fd793",
          "page": 20,
          "quote": "The equality in (8.3) is supplied by several published criteria. For an abelian variety A/K,\nlet"
        },
        {
          "blockId": "p20-50a0000a2d",
          "page": 20,
          "quote": "!"
        },
        {
          "blockId": "p20-ee36b0409a",
          "page": 20,
          "quote": ","
        },
        {
          "blockId": "p20-eba185748c",
          "page": 20,
          "quote": "Sha(A/K) = ker"
        },
        {
          "blockId": "p20-7ffdf47157",
          "page": 20,
          "quote": "H1(K, A) −→\nY"
        },
        {
          "blockId": "p20-c5713f6b09",
          "page": 20,
          "quote": "v\nH1(Kv, A)"
        },
        {
          "blockId": "p20-6da58faf9e",
          "page": 20,
          "quote": "and write Sha(A/K)div for its maximal divisible subgroup."
        }
      ],
      "guide": []
    },
    {
      "id": "corollary-18",
      "kind": "corollary",
      "label": "Corollary 8.3",
      "source": {
        "blockId": "p20-895f6d0c16",
        "page": 20,
        "quote": "Corollary 8.3 (Abelian targets). Let C/K be a smooth proper geometrically connected curve\nof genus at least two over a number field. The map κC is bijective if either of the following\nconditions holds:"
      },
      "statement": [
        {
          "blockId": "p20-895f6d0c16",
          "page": 20,
          "quote": "Corollary 8.3 (Abelian targets). Let C/K be a smooth proper geometrically connected curve\nof genus at least two over a number field. The map κC is bijective if either of the following\nconditions holds:"
        },
        {
          "blockId": "p20-7899dcad44",
          "page": 20,
          "quote": "(i) there is a nonconstant K-morphism C →A to an abelian variety with A(K) finite and\nSha(A/K)div = 0;"
        },
        {
          "blockId": "p20-38c44a9b7c",
          "page": 20,
          "quote": "(ii) the Jacobian J of C has J(K) finite and Sha(J/K)div = 0."
        }
      ],
      "proof": [
        {
          "blockId": "p20-362ebad260",
          "page": 20,
          "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
        },
        {
          "blockId": "p21-2ac5523931",
          "page": 21,
          "quote": "For an elliptic curve E/Q and a prime p, put"
        },
        {
          "blockId": "p21-86f49c615a",
          "page": 21,
          "quote": "sp(E) = corankZp Selp∞(E/Q),"
        },
        {
          "blockId": "p21-fe267c37c4",
          "page": 21,
          "quote": "where the Selmer group uses the local Kummer conditions at every place."
        }
      ],
      "guide": []
    },
    {
      "id": "corollary-19",
      "kind": "corollary",
      "label": "Corollary 8.4",
      "source": {
        "blockId": "p21-570097c012",
        "page": 21,
        "quote": "Corollary 8.4 (A Selmer-corank-zero elliptic target). Let C/Q be a smooth proper geometrically\nconnected curve of genus at least two. Suppose that there is a nonconstant Q-morphism C →E\nto an elliptic curve and that sp(E) = 0 for at least one prime p. Then κC is bijective."
      },
      "statement": [
        {
          "blockId": "p21-570097c012",
          "page": 21,
          "quote": "Corollary 8.4 (A Selmer-corank-zero elliptic target). Let C/Q be a smooth proper geometrically\nconnected curve of genus at least two. Suppose that there is a nonconstant Q-morphism C →E\nto an elliptic curve and that sp(E) = 0 for at least one prime p. Then κC is bijective."
        }
      ],
      "proof": [
        {
          "blockId": "p21-37f7730139",
          "page": 21,
          "quote": "Proof. The corank-zero case of the companion Selmer converse [18, Theorem 1.1] gives\nrankZ E(Q) = 0 and finiteness of the entire group Sha(E/Q).\nThe Mordell–Weil theorem\ntherefore makes E(Q) finite, while finiteness of Sha(E/Q) makes its maximal divisible subgroup\nzero. Corollary 8.3(i) applies."
        }
      ],
      "guide": []
    },
    {
      "id": "corollary-20",
      "kind": "corollary",
      "label": "Corollary 8.5",
      "source": {
        "blockId": "p21-207584fc3b",
        "page": 21,
        "quote": "Corollary 8.5 (Modular curves). For every positive integer N such that the smooth projective\ngeometrically connected curve X0(N)/Q has genus at least two, the map κX0(N) is bijective. The\nsame assertion holds for the smooth projective geometrically connected curve X1(N)/Q whenever\nits genus is at least two."
      },
      "statement": [
        {
          "blockId": "p21-207584fc3b",
          "page": 21,
          "quote": "Corollary 8.5 (Modular curves). For every positive integer N such that the smooth projective\ngeometrically connected curve X0(N)/Q has genus at least two, the map κX0(N) is bijective. The\nsame assertion holds for the smooth projective geometrically connected curve X1(N)/Q whenever\nits genus is at least two."
        }
      ],
      "proof": [
        {
          "blockId": "p21-b5285c8185",
          "page": 21,
          "quote": "Proof. Stoll’s Corollary 8.8 gives C(AQ)f-ab\n•\n= C(Q) for each of these projective modular curves\nof positive genus [26], with the corrected conventions of [27]. Apply (8.1) and Theorem 8.2."
        },
        {
          "blockId": "p21-eff5224878",
          "page": 21,
          "quote": "The general implication retains the finite-cover equality as a genuine arithmetic hypothesis;\nlocal point-theoreticity alone does not supply it. The elliptic-target specialization uses only\nSelmer corank zero over Q, while the modular-curve cases come directly from Stoll’s descent\nresult and do not depend on that specialization."
        }
      ],
      "guide": []
    },
    {
      "id": "external-21",
      "kind": "external",
      "label": "Reference [1]",
      "source": {
        "blockId": "p22-f749dc35f3",
        "page": 22,
        "quote": "[1] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff. Lifting harmonic\nmorphisms I: metrized complexes and Berkovich skeleta. Research in the Mathematical\nSciences, 2, 2015. Article 7. Preprint version: arXiv:1303.4812v3,\nhttps://arxiv.org/abs/1303.4812v3."
      },
      "statement": [
        {
          "blockId": "p22-f749dc35f3",
          "page": 22,
          "quote": "[1] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff. Lifting harmonic\nmorphisms I: metrized complexes and Berkovich skeleta. Research in the Mathematical\nSciences, 2, 2015. Article 7. Preprint version: arXiv:1303.4812v3,\nhttps://arxiv.org/abs/1303.4812v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-22",
      "kind": "external",
      "label": "Reference [2]",
      "source": {
        "blockId": "p22-4555f88c7b",
        "page": 22,
        "quote": "[2] Matthew Baker, Sam Payne, and Joseph Rabinoff. On the structure of nonarchimedean\nanalytic curves. In Tropical and Non-Archimedean Geometry, volume 605 of Contemporary\nMathematics, pages 93–121. American Mathematical Society, Providence, RI, 2013.\nPreprint version: arXiv:1404.0279v1, https://arxiv.org/abs/1404.0279v1."
      },
      "statement": [
        {
          "blockId": "p22-4555f88c7b",
          "page": 22,
          "quote": "[2] Matthew Baker, Sam Payne, and Joseph Rabinoff. On the structure of nonarchimedean\nanalytic curves. In Tropical and Non-Archimedean Geometry, volume 605 of Contemporary\nMathematics, pages 93–121. American Mathematical Society, Providence, RI, 2013.\nPreprint version: arXiv:1404.0279v1, https://arxiv.org/abs/1404.0279v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-23",
      "kind": "external",
      "label": "Reference [3]",
      "source": {
        "blockId": "p22-1efefd5545",
        "page": 22,
        "quote": "[3] Matthew Baker and Joseph Rabinoff. The skeleton of the Jacobian, the Jacobian of the\nskeleton, and lifting meromorphic functions from tropical to algebraic curves. International\nMathematics Research Notices, 2015(16):7436–7472, 2015. Preprint version:\narXiv:1308.3864v1, https://arxiv.org/abs/1308.3864v1."
      },
      "statement": [
        {
          "blockId": "p22-1efefd5545",
          "page": 22,
          "quote": "[3] Matthew Baker and Joseph Rabinoff. The skeleton of the Jacobian, the Jacobian of the\nskeleton, and lifting meromorphic functions from tropical to algebraic curves. International\nMathematics Research Notices, 2015(16):7436–7472, 2015. Preprint version:\narXiv:1308.3864v1, https://arxiv.org/abs/1308.3864v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-24",
      "kind": "external",
      "label": "Reference [4]",
      "source": {
        "blockId": "p22-f13809174e",
        "page": 22,
        "quote": "[4] L. Alexander Betts and Jakob Stix. Galois sections and p-adic period mappings. Annals of\nMathematics, 201(1):79–166, 2025. Preprint version: arXiv:2204.13674v1,\nhttps://arxiv.org/abs/2204.13674v1."
      },
      "statement": [
        {
          "blockId": "p22-f13809174e",
          "page": 22,
          "quote": "[4] L. Alexander Betts and Jakob Stix. Galois sections and p-adic period mappings. Annals of\nMathematics, 201(1):79–166, 2025. Preprint version: arXiv:2204.13674v1,\nhttps://arxiv.org/abs/2204.13674v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-25",
      "kind": "external",
      "label": "Reference [5]",
      "source": {
        "blockId": "p22-823d36067c",
        "page": 22,
        "quote": "[5] Siegfried Bosch. Stable reduction and rigid analytic uniformization of abelian varieties.\nGroupe d’étude d’analyse ultramétrique, 11:1–4, 1983–1984. Exposé 20.\nhttps://www.numdam.org/item/GAU_1983-1984__11__A12_0/."
      },
      "statement": [
        {
          "blockId": "p22-823d36067c",
          "page": 22,
          "quote": "[5] Siegfried Bosch. Stable reduction and rigid analytic uniformization of abelian varieties.\nGroupe d’étude d’analyse ultramétrique, 11:1–4, 1983–1984. Exposé 20.\nhttps://www.numdam.org/item/GAU_1983-1984__11__A12_0/."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-26",
      "kind": "external",
      "label": "Reference [6]",
      "source": {
        "blockId": "p22-ed2ab4a916",
        "page": 22,
        "quote": "[6] Siegfried Bosch and Werner Lütkebohmert. Stable reduction and uniformization of abelian\nvarieties. II. Inventiones Mathematicae, 78(2):257–297, 1984.\nhttps://doi.org/10.1007/BF01388596."
      },
      "statement": [
        {
          "blockId": "p22-ed2ab4a916",
          "page": 22,
          "quote": "[6] Siegfried Bosch and Werner Lütkebohmert. Stable reduction and uniformization of abelian\nvarieties. II. Inventiones Mathematicae, 78(2):257–297, 1984.\nhttps://doi.org/10.1007/BF01388596."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-27",
      "kind": "external",
      "label": "Reference [7]",
      "source": {
        "blockId": "p22-4850f46dc4",
        "page": 22,
        "quote": "[7] Giulio Bresciani. The section conjecture for the toric fundamental group over p-adic fields.\narXiv:2409.07923v5, 2026. Revision of September 14, 2026.\nhttps://arxiv.org/abs/2409.07923v5."
      },
      "statement": [
        {
          "blockId": "p22-4850f46dc4",
          "page": 22,
          "quote": "[7] Giulio Bresciani. The section conjecture for the toric fundamental group over p-adic fields.\narXiv:2409.07923v5, 2026. Revision of September 14, 2026.\nhttps://arxiv.org/abs/2409.07923v5."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-28",
      "kind": "external",
      "label": "Reference [8]",
      "source": {
        "blockId": "p22-cecc023f16",
        "page": 22,
        "quote": "[8] Giulio Bresciani and Angelo Vistoli. An elementary approach to Stix’s proof of the real\nsection conjecture. arXiv:2012.06278v2, 2020. https://arxiv.org/abs/2012.06278v2."
      },
      "statement": [
        {
          "blockId": "p22-cecc023f16",
          "page": 22,
          "quote": "[8] Giulio Bresciani and Angelo Vistoli. An elementary approach to Stix’s proof of the real\nsection conjecture. arXiv:2012.06278v2, 2020. https://arxiv.org/abs/2012.06278v2."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-29",
      "kind": "external",
      "label": "Reference [9]",
      "source": {
        "blockId": "p22-0bb4058203",
        "page": 22,
        "quote": "[9] Alexander Grothendieck. Letter to G. Faltings (translation into English). In Leila Schneps\nand Pierre Lochak, editors, Geometric Galois Actions, 1, volume 242 of London\nMathematical Society Lecture Note Series, pages 285–293. Cambridge University Press,\n1997. Letter dated June 27, 1983. https://doi.org/10.1017/CBO9780511758874.018."
      },
      "statement": [
        {
          "blockId": "p22-0bb4058203",
          "page": 22,
          "quote": "[9] Alexander Grothendieck. Letter to G. Faltings (translation into English). In Leila Schneps\nand Pierre Lochak, editors, Geometric Galois Actions, 1, volume 242 of London\nMathematical Society Lecture Note Series, pages 285–293. Cambridge University Press,\n1997. Letter dated June 27, 1983. https://doi.org/10.1017/CBO9780511758874.018."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-30",
      "kind": "external",
      "label": "Reference [10]",
      "source": {
        "blockId": "p22-304cc67660",
        "page": 22,
        "quote": "[10] Alexander Grothendieck, editor. Revêtements étales et groupe fondamental (SGA 1),\nvolume 3 of Documents Mathématiques. Société Mathématique de France, Paris, 2003.\nRecomposed and annotated edition of the 1971 original.\nhttps://arxiv.org/abs/math/0206203v2."
      },
      "statement": [
        {
          "blockId": "p22-304cc67660",
          "page": 22,
          "quote": "[10] Alexander Grothendieck, editor. Revêtements étales et groupe fondamental (SGA 1),\nvolume 3 of Documents Mathématiques. Société Mathématique de France, Paris, 2003.\nRecomposed and annotated edition of the 1971 original.\nhttps://arxiv.org/abs/math/0206203v2."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-31",
      "kind": "external",
      "label": "Reference [11]",
      "source": {
        "blockId": "p22-11f3489b82",
        "page": 22,
        "quote": "[11] David Harari and Jakob Stix. Descent obstruction and fundamental exact sequence. In\nJakob Stix, editor, The Arithmetic of Fundamental Groups – PIA 2010, volume 2 of\nContributions in Mathematical and Computational Sciences, pages 147–166. Springer, 2012.\nAuthor preprint dated February 11, 2011. https://www.imo.universite-paris-\nsaclay.fr/~david.harari/articles/finidesc.pdf."
      },
      "statement": [
        {
          "blockId": "p22-11f3489b82",
          "page": 22,
          "quote": "[11] David Harari and Jakob Stix. Descent obstruction and fundamental exact sequence. In\nJakob Stix, editor, The Arithmetic of Fundamental Groups – PIA 2010, volume 2 of\nContributions in Mathematical and Computational Sciences, pages 147–166. Springer, 2012.\nAuthor preprint dated February 11, 2011. https://www.imo.universite-paris-\nsaclay.fr/~david.harari/articles/finidesc.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-32",
      "kind": "external",
      "label": "Reference [12]",
      "source": {
        "blockId": "p22-d1cbf67257",
        "page": 22,
        "quote": "[12] David Harbater. Galois coverings of the arithmetic line. In Number Theory, New York\n1984–1985, volume 1240 of Lecture Notes in Mathematics, pages 165–195. Springer, Berlin,\n1987."
      },
      "statement": [
        {
          "blockId": "p22-d1cbf67257",
          "page": 22,
          "quote": "[12] David Harbater. Galois coverings of the arithmetic line. In Number Theory, New York\n1984–1985, volume 1240 of Lecture Notes in Mathematics, pages 165–195. Springer, Berlin,\n1987."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-33",
      "kind": "external",
      "label": "Reference [13]",
      "source": {
        "blockId": "p23-2d0de417ef",
        "page": 23,
        "quote": "[13] Jochen Koenigsmann. On the ‘section conjecture’ in anabelian geometry. Journal für die\nreine und angewandte Mathematik, 588:221–235, 2005. Preprint version:\narXiv:math/0305226v1, https://arxiv.org/abs/math/0305226v1."
      },
      "statement": [
        {
          "blockId": "p23-2d0de417ef",
          "page": 23,
          "quote": "[13] Jochen Koenigsmann. On the ‘section conjecture’ in anabelian geometry. Journal für die\nreine und angewandte Mathematik, 588:221–235, 2005. Preprint version:\narXiv:math/0305226v1, https://arxiv.org/abs/math/0305226v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-34",
      "kind": "external",
      "label": "Reference [14]",
      "source": {
        "blockId": "p23-88b2132601",
        "page": 23,
        "quote": "[14] Emmanuel Lepage. Resolution of nonsingularities for Mumford curves. Publications of the\nResearch Institute for Mathematical Sciences, 49(4):861–891, 2013.\nhttps://doi.org/10.4171/PRIMS/122."
      },
      "statement": [
        {
          "blockId": "p23-88b2132601",
          "page": 23,
          "quote": "[14] Emmanuel Lepage. Resolution of nonsingularities for Mumford curves. Publications of the\nResearch Institute for Mathematical Sciences, 49(4):861–891, 2013.\nhttps://doi.org/10.4171/PRIMS/122."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-35",
      "kind": "external",
      "label": "Reference [15]",
      "source": {
        "blockId": "p23-87b9aa9eac",
        "page": 23,
        "quote": "[15] Qing Liu. Tout groupe fini est un groupe de Galois sur Qp(T), d’après Harbater. In Recent\nDevelopments in the Inverse Galois Problem, volume 186 of Contemporary Mathematics,\npages 261–265. American Mathematical Society, Providence, RI, 1995. Author version.\nhttps://www.math.u-bordeaux.fr/~qliu/articles/rigidharb.pdf."
      },
      "statement": [
        {
          "blockId": "p23-87b9aa9eac",
          "page": 23,
          "quote": "[15] Qing Liu. Tout groupe fini est un groupe de Galois sur Qp(T), d’après Harbater. In Recent\nDevelopments in the Inverse Galois Problem, volume 186 of Contemporary Mathematics,\npages 261–265. American Mathematical Society, Providence, RI, 1995. Author version.\nhttps://www.math.u-bordeaux.fr/~qliu/articles/rigidharb.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-36",
      "kind": "external",
      "label": "Reference [16]",
      "source": {
        "blockId": "p23-2a525d4b84",
        "page": 23,
        "quote": "[16] James S. Milne. Abelian varieties, 2022. Author version, January 2, 2022, of the chapter in\nArithmetic Geometry (1986), pp. 103–150. Theorem 8.2 and Remark 8.4.\nhttps://www.jmilne.org/math/xnotes/AVs.pdf."
      },
      "statement": [
        {
          "blockId": "p23-2a525d4b84",
          "page": 23,
          "quote": "[16] James S. Milne. Abelian varieties, 2022. Author version, January 2, 2022, of the chapter in\nArithmetic Geometry (1986), pp. 103–150. Theorem 8.2 and Remark 8.4.\nhttps://www.jmilne.org/math/xnotes/AVs.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-37",
      "kind": "external",
      "label": "Reference [17]",
      "source": {
        "blockId": "p23-141acebc6d",
        "page": 23,
        "quote": "[17] Shinichi Mochizuki and Shota Tsujimura. Resolution of nonsingularities, point-theoreticity,\nand metric-admissibility for p-adic hyperbolic curves. RIMS Preprint 1974, 2024. Revision\nof March 30, 2024. Authors’ revised preprint."
      },
      "statement": [
        {
          "blockId": "p23-141acebc6d",
          "page": 23,
          "quote": "[17] Shinichi Mochizuki and Shota Tsujimura. Resolution of nonsingularities, point-theoreticity,\nand metric-admissibility for p-adic hyperbolic curves. RIMS Preprint 1974, 2024. Revision\nof March 30, 2024. Authors’ revised preprint."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-38",
      "kind": "external",
      "label": "Reference [18]",
      "source": {
        "blockId": "p23-75a0e67e86",
        "page": 23,
        "quote": "[18] OpenAI. The Selmer converse for elliptic curves at every prime. OpenAI Math Release\npreprint OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026,\n2026."
      },
      "statement": [
        {
          "blockId": "p23-75a0e67e86",
          "page": 23,
          "quote": "[18] OpenAI. The Selmer converse for elliptic curves at every prime. OpenAI Math Release\npreprint OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026,\n2026."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-39",
      "kind": "external",
      "label": "Reference [19]",
      "source": {
        "blockId": "p23-c0057e72f1",
        "page": 23,
        "quote": "[19] Jesse Pajwani. The valuative section conjecture, étale homotopy, and Berkovich spaces.\nHomology, Homotopy and Applications, 27(2):261–288, 2025. Preprint version:\narXiv:2304.03970v3, https://arxiv.org/abs/2304.03970v3."
      },
      "statement": [
        {
          "blockId": "p23-c0057e72f1",
          "page": 23,
          "quote": "[19] Jesse Pajwani. The valuative section conjecture, étale homotopy, and Berkovich spaces.\nHomology, Homotopy and Applications, 27(2):261–288, 2025. Preprint version:\narXiv:2304.03970v3, https://arxiv.org/abs/2304.03970v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-40",
      "kind": "external",
      "label": "Reference [20]",
      "source": {
        "blockId": "p23-36ff19f210",
        "page": 23,
        "quote": "[20] Jérôme Poineau. Raccord sur les espaces de Berkovich. Algebra & Number Theory,\n4(3):297–334, 2010. Appendix A, Theorem A.1.\nhttps://msp.org/ant/2010/4-3/ant-v4-n3-p03-s.pdf."
      },
      "statement": [
        {
          "blockId": "p23-36ff19f210",
          "page": 23,
          "quote": "[20] Jérôme Poineau. Raccord sur les espaces de Berkovich. Algebra & Number Theory,\n4(3):297–334, 2010. Appendix A, Theorem A.1.\nhttps://msp.org/ant/2010/4-3/ant-v4-n3-p03-s.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-41",
      "kind": "external",
      "label": "Reference [21]",
      "source": {
        "blockId": "p23-db061cda39",
        "page": 23,
        "quote": "[21] Florian Pop. On the birational p-adic section conjecture. Compositio Mathematica,\n146(3):621–637, 2010. https://doi.org/10.1112/S0010437X09004436."
      },
      "statement": [
        {
          "blockId": "p23-db061cda39",
          "page": 23,
          "quote": "[21] Florian Pop. On the birational p-adic section conjecture. Compositio Mathematica,\n146(3):621–637, 2010. https://doi.org/10.1112/S0010437X09004436."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-42",
      "kind": "external",
      "label": "Reference [22]",
      "source": {
        "blockId": "p23-72842ad7d8",
        "page": 23,
        "quote": "[22] Florian Pop and Jakob Stix. Arithmetic in the fundamental group of a p-adic curve: On\nthe p-adic section conjecture for curves. Journal für die reine und angewandte Mathematik,\n725:1–40, 2017. Preprint version: arXiv:1111.1354v1 (2011),\nhttps://arxiv.org/abs/1111.1354v1."
      },
      "statement": [
        {
          "blockId": "p23-72842ad7d8",
          "page": 23,
          "quote": "[22] Florian Pop and Jakob Stix. Arithmetic in the fundamental group of a p-adic curve: On\nthe p-adic section conjecture for curves. Journal für die reine und angewandte Mathematik,\n725:1–40, 2017. Preprint version: arXiv:1111.1354v1 (2011),\nhttps://arxiv.org/abs/1111.1354v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-43",
      "kind": "external",
      "label": "Reference [23]",
      "source": {
        "blockId": "p23-0cffab6f20",
        "page": 23,
        "quote": "[23] Michel Raynaud. Variétés abéliennes et géométrie rigide. In Actes du Congrès International\ndes Mathématiciens (Nice, 1970), volume 1, pages 473–477. Gauthier-Villars, Paris, 1971."
      },
      "statement": [
        {
          "blockId": "p23-0cffab6f20",
          "page": 23,
          "quote": "[23] Michel Raynaud. Variétés abéliennes et géométrie rigide. In Actes du Congrès International\ndes Mathématiciens (Nice, 1970), volume 1, pages 473–477. Gauthier-Villars, Paris, 1971."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-44",
      "kind": "external",
      "label": "Reference [24]",
      "source": {
        "blockId": "p23-1e79982b9d",
        "page": 23,
        "quote": "[24] Friedrich Karl Schmidt. Mehrfach perfekte körper. Mathematische Annalen, 108:1–25, 1933.\nhttps://doi.org/10.1007/BF01452819."
      },
      "statement": [
        {
          "blockId": "p23-1e79982b9d",
          "page": 23,
          "quote": "[24] Friedrich Karl Schmidt. Mehrfach perfekte körper. Mathematische Annalen, 108:1–25, 1933.\nhttps://doi.org/10.1007/BF01452819."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-45",
      "kind": "external",
      "label": "Reference [25]",
      "source": {
        "blockId": "p23-e3f539e7c4",
        "page": 23,
        "quote": "[25] Jakob Stix. On the birational section conjecture with local conditions. Inventiones\nMathematicae, 199(1):239–265, 2015. arXiv version: arXiv:1203.3236v3,\nhttps://arxiv.org/abs/1203.3236v3."
      },
      "statement": [
        {
          "blockId": "p23-e3f539e7c4",
          "page": 23,
          "quote": "[25] Jakob Stix. On the birational section conjecture with local conditions. Inventiones\nMathematicae, 199(1):239–265, 2015. arXiv version: arXiv:1203.3236v3,\nhttps://arxiv.org/abs/1203.3236v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-46",
      "kind": "external",
      "label": "Reference [26]",
      "source": {
        "blockId": "p23-94b55a9e41",
        "page": 23,
        "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
      },
      "statement": [
        {
          "blockId": "p23-94b55a9e41",
          "page": 23,
          "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-47",
      "kind": "external",
      "label": "Reference [27]",
      "source": {
        "blockId": "p24-c2a0cf319b",
        "page": 24,
        "quote": "[27] Michael Stoll. Errata: Finite descent obstructions and rational points on curves, 2017.\nOnline supplement, document dated March 29, 2017.\nhttps://msp.org/ant/2007/1-4/ant-v1-n4-x01-Errata-FiniteDescent.pdf."
      },
      "statement": [
        {
          "blockId": "p24-c2a0cf319b",
          "page": 24,
          "quote": "[27] Michael Stoll. Errata: Finite descent obstructions and rational points on curves, 2017.\nOnline supplement, document dated March 29, 2017.\nhttps://msp.org/ant/2007/1-4/ant-v1-n4-x01-Errata-FiniteDescent.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-48",
      "kind": "external",
      "label": "Reference [28]",
      "source": {
        "blockId": "p24-3a39eed1b5",
        "page": 24,
        "quote": "[28] Akio Tamagawa. Resolution of nonsingularities of families of curves. Publications of the\nResearch Institute for Mathematical Sciences, 40(4):1291–1336, 2004.\nhttps://doi.org/10.2977/PRIMS/1145475448."
      },
      "statement": [
        {
          "blockId": "p24-3a39eed1b5",
          "page": 24,
          "quote": "[28] Akio Tamagawa. Resolution of nonsingularities of families of curves. Publications of the\nResearch Institute for Mathematical Sciences, 40(4):1291–1336, 2004.\nhttps://doi.org/10.2977/PRIMS/1145475448."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-49",
      "kind": "external",
      "label": "Reference [29]",
      "source": {
        "blockId": "p24-5d11e6ebb4",
        "page": 24,
        "quote": "[29] Michael Temkin. Introduction to Berkovich analytic spaces. Lecture notes, Jussieu summer\nschool, 2010. Section 5.1,\nhttps://berkovich-2010.imj-prg.fr/lecture-notes/Temkin-lecture-notes.pdf."
      },
      "statement": [
        {
          "blockId": "p24-5d11e6ebb4",
          "page": 24,
          "quote": "[29] Michael Temkin. Introduction to Berkovich analytic spaces. Lecture notes, Jussieu summer\nschool, 2010. Section 5.1,\nhttps://berkovich-2010.imj-prg.fr/lecture-notes/Temkin-lecture-notes.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "equation-50",
      "kind": "equation",
      "label": "Equation (1.1)",
      "source": {
        "blockId": "p1-356e8bf501",
        "page": 1,
        "quote": "1 −→∆X −→ΠX −→Gk −→1,\n∆X = π1(X¯k),\nΠX = π1(X),\nGk = Gal(¯k/k).\n(1.1)"
      },
      "statement": [
        {
          "blockId": "p1-356e8bf501",
          "page": 1,
          "quote": "1 −→∆X −→ΠX −→Gk −→1,\n∆X = π1(X¯k),\nΠX = π1(X),\nGk = Gal(¯k/k).\n(1.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-51",
      "kind": "equation",
      "label": "Equation (8.1)",
      "source": {
        "blockId": "p19-e53d5cfa01",
        "page": 19,
        "quote": "C(K) ⊆C(AK)f-cov\n•\n⊆C(AK)f-ab\n•\n.\n(8.1)"
      },
      "statement": [
        {
          "blockId": "p19-e53d5cfa01",
          "page": 19,
          "quote": "C(K) ⊆C(AK)f-cov\n•\n⊆C(AK)f-ab\n•\n.\n(8.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-52",
      "kind": "equation",
      "label": "Equation (8.2)",
      "source": {
        "blockId": "p19-2049c0ef47",
        "page": 19,
        "quote": "loc\nSec(C/K)\n = C(AK)f-cov\n•\n.\n(8.2)"
      },
      "statement": [
        {
          "blockId": "p19-2049c0ef47",
          "page": 19,
          "quote": "loc\nSec(C/K)\n = C(AK)f-cov\n•\n.\n(8.2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-53",
      "kind": "equation",
      "label": "Equation (8.3)",
      "source": {
        "blockId": "p20-13ffcea2da",
        "page": 20,
        "quote": "C(AK)f-cov\n•\n= C(K),\n(8.3)"
      },
      "statement": [
        {
          "blockId": "p20-13ffcea2da",
          "page": 20,
          "quote": "C(AK)f-cov\n•\n= C(K),\n(8.3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-54",
      "kind": "equation",
      "label": "Equation (8.4)",
      "source": {
        "blockId": "p21-3cdbd6f10d",
        "page": 21,
        "quote": "ρ−1loc\nSec(C/K)\n ⊆Dlin(C) ∩C(AK)Br.\n(8.4)"
      },
      "statement": [
        {
          "blockId": "p21-3cdbd6f10d",
          "page": 21,
          "quote": "ρ−1loc\nSec(C/K)\n ⊆Dlin(C) ∩C(AK)Br.\n(8.4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "external-55",
      "kind": "external",
      "label": "[7] Theorem 1.5",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "statement": [
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p22-4850f46dc4",
          "page": 22,
          "quote": "[7] Giulio Bresciani. The section conjecture for the toric fundamental group over p-adic fields.\narXiv:2409.07923v5, 2026. Revision of September 14, 2026.\nhttps://arxiv.org/abs/2409.07923v5."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-56",
      "kind": "external",
      "label": "[13] Proposition 2.4",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "statement": [
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p23-2d0de417ef",
          "page": 23,
          "quote": "[13] Jochen Koenigsmann. On the ‘section conjecture’ in anabelian geometry. Journal für die\nreine und angewandte Mathematik, 588:221–235, 2005. Preprint version:\narXiv:math/0305226v1, https://arxiv.org/abs/math/0305226v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-57",
      "kind": "external",
      "label": "[19] Theorem 1.4",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "statement": [
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p23-c0057e72f1",
          "page": 23,
          "quote": "[19] Jesse Pajwani. The valuative section conjecture, étale homotopy, and Berkovich spaces.\nHomology, Homotopy and Applications, 27(2):261–288, 2025. Preprint version:\narXiv:2304.03970v3, https://arxiv.org/abs/2304.03970v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-58",
      "kind": "external",
      "label": "[19] Corollary 1.5",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "statement": [
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p23-c0057e72f1",
          "page": 23,
          "quote": "[19] Jesse Pajwani. The valuative section conjecture, étale homotopy, and Berkovich spaces.\nHomology, Homotopy and Applications, 27(2):261–288, 2025. Preprint version:\narXiv:2304.03970v3, https://arxiv.org/abs/2304.03970v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-59",
      "kind": "external",
      "label": "[7] Definition 3.7",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "statement": [
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p22-4850f46dc4",
          "page": 22,
          "quote": "[7] Giulio Bresciani. The section conjecture for the toric fundamental group over p-adic fields.\narXiv:2409.07923v5, 2026. Revision of September 14, 2026.\nhttps://arxiv.org/abs/2409.07923v5."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-60",
      "kind": "external",
      "label": "[7] Theorem 1.4",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "statement": [
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p22-4850f46dc4",
          "page": 22,
          "quote": "[7] Giulio Bresciani. The section conjecture for the toric fundamental group over p-adic fields.\narXiv:2409.07923v5, 2026. Revision of September 14, 2026.\nhttps://arxiv.org/abs/2409.07923v5."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-61",
      "kind": "external",
      "label": "[7] Theorem 5.10",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "statement": [
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p22-4850f46dc4",
          "page": 22,
          "quote": "[7] Giulio Bresciani. The section conjecture for the toric fundamental group over p-adic fields.\narXiv:2409.07923v5, 2026. Revision of September 14, 2026.\nhttps://arxiv.org/abs/2409.07923v5."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-62",
      "kind": "external",
      "label": "[1] Theorem 7.7",
      "source": {
        "blockId": "p4-563828111d",
        "page": 4,
        "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
      },
      "statement": [
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p22-f749dc35f3",
          "page": 22,
          "quote": "[1] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff. Lifting harmonic\nmorphisms I: metrized complexes and Berkovich skeleta. Research in the Mathematical\nSciences, 2, 2015. Article 7. Preprint version: arXiv:1303.4812v3,\nhttps://arxiv.org/abs/1303.4812v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-63",
      "kind": "external",
      "label": "[17] Proposition 2.4",
      "source": {
        "blockId": "p4-563828111d",
        "page": 4,
        "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
      },
      "statement": [
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p23-141acebc6d",
          "page": 23,
          "quote": "[17] Shinichi Mochizuki and Shota Tsujimura. Resolution of nonsingularities, point-theoreticity,\nand metric-admissibility for p-adic hyperbolic curves. RIMS Preprint 1974, 2024. Revision\nof March 30, 2024. Authors’ revised preprint."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-64",
      "kind": "external",
      "label": "[28] Theorem 0.2",
      "source": {
        "blockId": "p4-563828111d",
        "page": 4,
        "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
      },
      "statement": [
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p24-3a39eed1b5",
          "page": 24,
          "quote": "[28] Akio Tamagawa. Resolution of nonsingularities of families of curves. Publications of the\nResearch Institute for Mathematical Sciences, 40(4):1291–1336, 2004.\nhttps://doi.org/10.2977/PRIMS/1145475448."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-65",
      "kind": "external",
      "label": "[14] Theorem 2.7",
      "source": {
        "blockId": "p4-563828111d",
        "page": 4,
        "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
      },
      "statement": [
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p23-88b2132601",
          "page": 23,
          "quote": "[14] Emmanuel Lepage. Resolution of nonsingularities for Mumford curves. Publications of the\nResearch Institute for Mathematical Sciences, 49(4):861–891, 2013.\nhttps://doi.org/10.4171/PRIMS/122."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-66",
      "kind": "external",
      "label": "[3] Proposition 6.1",
      "source": {
        "blockId": "p4-563828111d",
        "page": 4,
        "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
      },
      "statement": [
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p22-1efefd5545",
          "page": 22,
          "quote": "[3] Matthew Baker and Joseph Rabinoff. The skeleton of the Jacobian, the Jacobian of the\nskeleton, and lifting meromorphic functions from tropical to algebraic curves. International\nMathematics Research Notices, 2015(16):7436–7472, 2015. Preprint version:\narXiv:1308.3864v1, https://arxiv.org/abs/1308.3864v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-67",
      "kind": "external",
      "label": "[3] Corollary 6.6",
      "source": {
        "blockId": "p4-563828111d",
        "page": 4,
        "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
      },
      "statement": [
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p22-1efefd5545",
          "page": 22,
          "quote": "[3] Matthew Baker and Joseph Rabinoff. The skeleton of the Jacobian, the Jacobian of the\nskeleton, and lifting meromorphic functions from tropical to algebraic curves. International\nMathematics Research Notices, 2015(16):7436–7472, 2015. Preprint version:\narXiv:1308.3864v1, https://arxiv.org/abs/1308.3864v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-68",
      "kind": "external",
      "label": "[11] Theorem 2.1",
      "source": {
        "blockId": "p4-526c151cc2",
        "page": 4,
        "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
      },
      "statement": [
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p22-11f3489b82",
          "page": 22,
          "quote": "[11] David Harari and Jakob Stix. Descent obstruction and fundamental exact sequence. In\nJakob Stix, editor, The Arithmetic of Fundamental Groups – PIA 2010, volume 2 of\nContributions in Mathematical and Computational Sciences, pages 147–166. Springer, 2012.\nAuthor preprint dated February 11, 2011. https://www.imo.universite-paris-\nsaclay.fr/~david.harari/articles/finidesc.pdf."
        },
        {
          "blockId": "p19-eb06480d4a",
          "page": 19,
          "quote": "Proof. Let s ∈Sec(C/K). Choose a representative point of the determined component at each\nreal place, and any point at each complex place. Harari–Stix’s Theorem 2.1, with U = 1 and the\nomitted set of places S = ∅, shows that these localizations survive every torsor under a finite\nK-group scheme; their Remark 2.2(1) identifies this as finite-cover descent [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)]. Consequently"
        },
        {
          "blockId": "p20-9685c75da6",
          "page": 20,
          "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-69",
      "kind": "external",
      "label": "[11] Remark 2.2",
      "source": {
        "blockId": "p4-526c151cc2",
        "page": 4,
        "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
      },
      "statement": [
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p22-11f3489b82",
          "page": 22,
          "quote": "[11] David Harari and Jakob Stix. Descent obstruction and fundamental exact sequence. In\nJakob Stix, editor, The Arithmetic of Fundamental Groups – PIA 2010, volume 2 of\nContributions in Mathematical and Computational Sciences, pages 147–166. Springer, 2012.\nAuthor preprint dated February 11, 2011. https://www.imo.universite-paris-\nsaclay.fr/~david.harari/articles/finidesc.pdf."
        },
        {
          "blockId": "p19-eb06480d4a",
          "page": 19,
          "quote": "Proof. Let s ∈Sec(C/K). Choose a representative point of the determined component at each\nreal place, and any point at each complex place. Harari–Stix’s Theorem 2.1, with U = 1 and the\nomitted set of places S = ∅, shows that these localizations survive every torsor under a finite\nK-group scheme; their Remark 2.2(1) identifies this as finite-cover descent [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)]. Consequently"
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-70",
      "kind": "external",
      "label": "[26] Corollary 8.1",
      "source": {
        "blockId": "p4-526c151cc2",
        "page": 4,
        "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
      },
      "statement": [
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p23-94b55a9e41",
          "page": 23,
          "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
        },
        {
          "blockId": "p20-362ebad260",
          "page": 20,
          "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-71",
      "kind": "external",
      "label": "[26] Theorem 8.6",
      "source": {
        "blockId": "p4-526c151cc2",
        "page": 4,
        "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
      },
      "statement": [
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p23-94b55a9e41",
          "page": 23,
          "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
        },
        {
          "blockId": "p20-362ebad260",
          "page": 20,
          "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-72",
      "kind": "external",
      "label": "[26] Corollary 8.8",
      "source": {
        "blockId": "p4-526c151cc2",
        "page": 4,
        "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
      },
      "statement": [
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p23-94b55a9e41",
          "page": 23,
          "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
        },
        {
          "blockId": "p21-b5285c8185",
          "page": 21,
          "quote": "Proof. Stoll’s Corollary 8.8 gives C(AQ)f-ab\n•\n= C(Q) for each of these projective modular curves\nof positive genus [26], with the corrected conventions of [27]. Apply (8.1) and Theorem 8.2."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-73",
      "kind": "external",
      "label": "[18] Theorem 1.1",
      "source": {
        "blockId": "p4-526c151cc2",
        "page": 4,
        "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
      },
      "statement": [
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p23-75a0e67e86",
          "page": 23,
          "quote": "[18] OpenAI. The Selmer converse for elliptic curves at every prime. OpenAI Math Release\npreprint OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026,\n2026."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-74",
      "kind": "external",
      "label": "[10] Corollary 1.8",
      "source": {
        "blockId": "p5-0c4756847d",
        "page": 5,
        "quote": "Every arrow is inside E. In particular a constructed abstract field embedding is not enough by\nitself: for a normal extension F/K0 we will use normality to identify its image with the specified\nsubfield F ⊂E.\nWe shall also use invariance of finite étale covers of a proper variety under extension of\nalgebraically closed fields [10, Exposé X, Corollary 1.8]. Thus a finite étale cover of YΩ, for a\nsmooth proper curve Y/¯k, descends up to isomorphism to a cover of Y . The analytic maps built\non smaller open regions need not descend. Their evaluation at the fixed H-valued point will be\nsufficient."
      },
      "statement": [
        {
          "blockId": "p5-0c4756847d",
          "page": 5,
          "quote": "Every arrow is inside E. In particular a constructed abstract field embedding is not enough by\nitself: for a normal extension F/K0 we will use normality to identify its image with the specified\nsubfield F ⊂E.\nWe shall also use invariance of finite étale covers of a proper variety under extension of\nalgebraically closed fields [10, Exposé X, Corollary 1.8]. Thus a finite étale cover of YΩ, for a\nsmooth proper curve Y/¯k, descends up to isomorphism to a cover of Y . The analytic maps built\non smaller open regions need not descend. Their evaluation at the fixed H-valued point will be\nsufficient."
        },
        {
          "blockId": "p22-304cc67660",
          "page": 22,
          "quote": "[10] Alexander Grothendieck, editor. Revêtements étales et groupe fondamental (SGA 1),\nvolume 3 of Documents Mathématiques. Société Mathématique de France, Paris, 2003.\nRecomposed and annotated edition of the 1971 original.\nhttps://arxiv.org/abs/math/0206203v2."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-75",
      "kind": "external",
      "label": "[20] Theorem A.1",
      "source": {
        "blockId": "p7-15cdd177d5",
        "page": 7,
        "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
      },
      "statement": [
        {
          "blockId": "p7-15cdd177d5",
          "page": 7,
          "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
        },
        {
          "blockId": "p23-36ff19f210",
          "page": 23,
          "quote": "[20] Jérôme Poineau. Raccord sur les espaces de Berkovich. Algebra & Number Theory,\n4(3):297–334, 2010. Appendix A, Theorem A.1.\nhttps://msp.org/ant/2010/4-3/ant-v4-n3-p03-s.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-76",
      "kind": "external",
      "label": "[2] Proposition 2.2",
      "source": {
        "blockId": "p8-6841d631e3",
        "page": 8,
        "quote": "We use the following elementary analytic fact. The exponent in its factorization is the slope\nof the additive valuation of the unit along the annulus skeleton. The unit factorization is the\nstandard annular factorization; compare [2, Proposition 2.2(1)]. We give the argument on the\nentire open annulus, since this is the domain required for gluing."
      },
      "statement": [
        {
          "blockId": "p8-6841d631e3",
          "page": 8,
          "quote": "We use the following elementary analytic fact. The exponent in its factorization is the slope\nof the additive valuation of the unit along the annulus skeleton. The unit factorization is the\nstandard annular factorization; compare [2, Proposition 2.2(1)]. We give the argument on the\nentire open annulus, since this is the domain required for gluing."
        },
        {
          "blockId": "p22-4555f88c7b",
          "page": 22,
          "quote": "[2] Matthew Baker, Sam Payne, and Joseph Rabinoff. On the structure of nonarchimedean\nanalytic curves. In Tropical and Non-Archimedean Geometry, volume 605 of Contemporary\nMathematics, pages 93–121. American Mathematical Society, Providence, RI, 2013.\nPreprint version: arXiv:1404.0279v1, https://arxiv.org/abs/1404.0279v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-77",
      "kind": "external",
      "label": "[1] Definition 2.19",
      "source": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "statement": [
        {
          "blockId": "p9-ef3971a04b",
          "page": 9,
          "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
        },
        {
          "blockId": "p22-f749dc35f3",
          "page": 22,
          "quote": "[1] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff. Lifting harmonic\nmorphisms I: metrized complexes and Berkovich skeleta. Research in the Mathematical\nSciences, 2, 2015. Article 7. Preprint version: arXiv:1303.4812v3,\nhttps://arxiv.org/abs/1303.4812v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-78",
      "kind": "external",
      "label": "[1] Definition 2.21",
      "source": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "statement": [
        {
          "blockId": "p9-ef3971a04b",
          "page": 9,
          "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
        },
        {
          "blockId": "p22-f749dc35f3",
          "page": 22,
          "quote": "[1] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff. Lifting harmonic\nmorphisms I: metrized complexes and Berkovich skeleta. Research in the Mathematical\nSciences, 2, 2015. Article 7. Preprint version: arXiv:1303.4812v3,\nhttps://arxiv.org/abs/1303.4812v3."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-79",
      "kind": "external",
      "label": "[16] Theorem 8.2",
      "source": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "statement": [
        {
          "blockId": "p9-ef3971a04b",
          "page": 9,
          "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
        },
        {
          "blockId": "p23-2a525d4b84",
          "page": 23,
          "quote": "[16] James S. Milne. Abelian varieties, 2022. Author version, January 2, 2022, of the chapter in\nArithmetic Geometry (1986), pp. 103–150. Theorem 8.2 and Remark 8.4.\nhttps://www.jmilne.org/math/xnotes/AVs.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-80",
      "kind": "external",
      "label": "[13] Definition 1.4",
      "source": {
        "blockId": "p16-f8589da2c3",
        "page": 16,
        "quote": "whose image stabilizes this prolongation. The birational section theorem of Koenigsmann places\nthe entire image of bs also in the decomposition group of a prolongation of orda for some a ∈X(k)\n[13, preprint version, Proposition 2.4(2), Definition 1.4, and Observation 1.3]. Here orda is the\nclosed-point valuation of K, trivial on k."
      },
      "statement": [
        {
          "blockId": "p16-f8589da2c3",
          "page": 16,
          "quote": "whose image stabilizes this prolongation. The birational section theorem of Koenigsmann places\nthe entire image of bs also in the decomposition group of a prolongation of orda for some a ∈X(k)\n[13, preprint version, Proposition 2.4(2), Definition 1.4, and Observation 1.3]. Here orda is the\nclosed-point valuation of K, trivial on k."
        },
        {
          "blockId": "p23-2d0de417ef",
          "page": 23,
          "quote": "[13] Jochen Koenigsmann. On the ‘section conjecture’ in anabelian geometry. Journal für die\nreine und angewandte Mathematik, 588:221–235, 2005. Preprint version:\narXiv:math/0305226v1, https://arxiv.org/abs/math/0305226v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-81",
      "kind": "external",
      "label": "[22] Theorem 26",
      "source": {
        "blockId": "p17-6aa1f30eeb",
        "page": 17,
        "quote": "Proof of the surjectivity in Theorem 1.1. Let s : Gk →ΠX be a continuous group-homomorphic\nsection. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26],\nthere is a valuation w of K extending the p-adic valuation, together with a prolongation ew to\nf\nK, such that\ns(Gk) ⊆De\nw(f\nK/K)."
      },
      "statement": [
        {
          "blockId": "p17-6aa1f30eeb",
          "page": 17,
          "quote": "Proof of the surjectivity in Theorem 1.1. Let s : Gk →ΠX be a continuous group-homomorphic\nsection. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26],\nthere is a valuation w of K extending the p-adic valuation, together with a prolongation ew to\nf\nK, such that\ns(Gk) ⊆De\nw(f\nK/K)."
        },
        {
          "blockId": "p23-72842ad7d8",
          "page": 23,
          "quote": "[22] Florian Pop and Jakob Stix. Arithmetic in the fundamental group of a p-adic curve: On\nthe p-adic section conjecture for curves. Journal für die reine und angewandte Mathematik,\n725:1–40, 2017. Preprint version: arXiv:1111.1354v1 (2011),\nhttps://arxiv.org/abs/1111.1354v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-82",
      "kind": "external",
      "label": "[26] Definition 5.2",
      "source": {
        "blockId": "p19-b196bde152",
        "page": 19,
        "quote": "where π0 denotes the set of connected components. Complex places are omitted, as in the\ncorrected convention of Stoll’s erratum [27, Section 1]. For geometrically connected C the\ncomplex component factors in the earlier convention are singletons, but their omission matters\nfor geometrically disconnected torsors. A point of C(AK)• survives a torsor Y →C under a\nfinite K-group scheme G if it lifts to Y ξ(AK)• for some twist Y ξ, where ξ ∈H1(K, G). Write\nC(AK)f-cov\n•\nfor the points surviving every such torsor, and C(AK)f-ab\n•\nfor those surviving every\nsuch torsor with G abelian [26, Definitions 5.2 and 5.4]. We identify C(K) with its diagonal\nimage. The definitions give"
      },
      "statement": [
        {
          "blockId": "p19-b196bde152",
          "page": 19,
          "quote": "where π0 denotes the set of connected components. Complex places are omitted, as in the\ncorrected convention of Stoll’s erratum [27, Section 1]. For geometrically connected C the\ncomplex component factors in the earlier convention are singletons, but their omission matters\nfor geometrically disconnected torsors. A point of C(AK)• survives a torsor Y →C under a\nfinite K-group scheme G if it lifts to Y ξ(AK)• for some twist Y ξ, where ξ ∈H1(K, G). Write\nC(AK)f-cov\n•\nfor the points surviving every such torsor, and C(AK)f-ab\n•\nfor those surviving every\nsuch torsor with G abelian [26, Definitions 5.2 and 5.4]. We identify C(K) with its diagonal\nimage. The definitions give"
        },
        {
          "blockId": "p23-94b55a9e41",
          "page": 23,
          "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
        },
        {
          "blockId": "p19-0cb1be5afe",
          "page": 19,
          "quote": "Conversely, let x belong to the finite-cover locus and, by properness, choose an ordinary\nadelic representative P = (Pv) ∈C(AK). Stoll’s cohomological description after Definition 5.2,\ntogether with Definition 5.4, identifies survival with diagonal global H1(K, G)-membership of the"
        },
        {
          "blockId": "p20-9685c75da6",
          "page": 20,
          "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-83",
      "kind": "external",
      "label": "[26] Definition 5.4",
      "source": {
        "blockId": "p19-b196bde152",
        "page": 19,
        "quote": "where π0 denotes the set of connected components. Complex places are omitted, as in the\ncorrected convention of Stoll’s erratum [27, Section 1]. For geometrically connected C the\ncomplex component factors in the earlier convention are singletons, but their omission matters\nfor geometrically disconnected torsors. A point of C(AK)• survives a torsor Y →C under a\nfinite K-group scheme G if it lifts to Y ξ(AK)• for some twist Y ξ, where ξ ∈H1(K, G). Write\nC(AK)f-cov\n•\nfor the points surviving every such torsor, and C(AK)f-ab\n•\nfor those surviving every\nsuch torsor with G abelian [26, Definitions 5.2 and 5.4]. We identify C(K) with its diagonal\nimage. The definitions give"
      },
      "statement": [
        {
          "blockId": "p19-b196bde152",
          "page": 19,
          "quote": "where π0 denotes the set of connected components. Complex places are omitted, as in the\ncorrected convention of Stoll’s erratum [27, Section 1]. For geometrically connected C the\ncomplex component factors in the earlier convention are singletons, but their omission matters\nfor geometrically disconnected torsors. A point of C(AK)• survives a torsor Y →C under a\nfinite K-group scheme G if it lifts to Y ξ(AK)• for some twist Y ξ, where ξ ∈H1(K, G). Write\nC(AK)f-cov\n•\nfor the points surviving every such torsor, and C(AK)f-ab\n•\nfor those surviving every\nsuch torsor with G abelian [26, Definitions 5.2 and 5.4]. We identify C(K) with its diagonal\nimage. The definitions give"
        },
        {
          "blockId": "p23-94b55a9e41",
          "page": 23,
          "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
        },
        {
          "blockId": "p19-0cb1be5afe",
          "page": 19,
          "quote": "Conversely, let x belong to the finite-cover locus and, by properness, choose an ordinary\nadelic representative P = (Pv) ∈C(AK). Stoll’s cohomological description after Definition 5.2,\ntogether with Definition 5.4, identifies survival with diagonal global H1(K, G)-membership of the"
        },
        {
          "blockId": "p20-9685c75da6",
          "page": 20,
          "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-84",
      "kind": "external",
      "label": "[26] Proposition 8.5",
      "source": {
        "blockId": "p20-362ebad260",
        "page": 20,
        "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
      },
      "statement": [
        {
          "blockId": "p20-362ebad260",
          "page": 20,
          "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
        },
        {
          "blockId": "p23-94b55a9e41",
          "page": 23,
          "quote": "[26] Michael Stoll. Finite descent obstructions and rational points on curves. Algebra & Number\nTheory, 1(4):349–391, 2007. Revised preprint:\nhttps://mathe2.uni-bayreuth.de/stoll/papers/covers-revised.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-85",
      "kind": "external",
      "label": "[11] Theorem 4.1",
      "source": {
        "blockId": "p21-a353547873",
        "page": 21,
        "quote": "We return to an arbitrary smooth proper geometrically connected curve C/K of genus at least\ntwo over a number field, without assuming (8.3). The localization image in Proposition 8.1\nsatisfies additional descent conditions and, under a restriction on K, has finite image at each\nfinite place.\nLet ρ : C(AK) →C(AK)• be the natural projection. Write Dlin(C) for the ordinary adelic\ndescent set: P belongs to it if, for every C-torsor T →C under a linear algebraic K-group\nG, it lifts to T ξ(AK) for some global twist ξ ∈H1(K, G). Harari–Stix’s Theorem 4.1 sends\nthe finite-torsor condition established in Proposition 8.1 to this linear descent condition, while\ntheir Corollary 3.1, applied to the geometric abelianization of the interpolating section, gives\nBrauer–Manin membership [11, preprint version, Corollary 3.1 and Theorem 4.1]. Consequently"
      },
      "statement": [
        {
          "blockId": "p21-a353547873",
          "page": 21,
          "quote": "We return to an arbitrary smooth proper geometrically connected curve C/K of genus at least\ntwo over a number field, without assuming (8.3). The localization image in Proposition 8.1\nsatisfies additional descent conditions and, under a restriction on K, has finite image at each\nfinite place.\nLet ρ : C(AK) →C(AK)• be the natural projection. Write Dlin(C) for the ordinary adelic\ndescent set: P belongs to it if, for every C-torsor T →C under a linear algebraic K-group\nG, it lifts to T ξ(AK) for some global twist ξ ∈H1(K, G). Harari–Stix’s Theorem 4.1 sends\nthe finite-torsor condition established in Proposition 8.1 to this linear descent condition, while\ntheir Corollary 3.1, applied to the geometric abelianization of the interpolating section, gives\nBrauer–Manin membership [11, preprint version, Corollary 3.1 and Theorem 4.1]. Consequently"
        },
        {
          "blockId": "p22-11f3489b82",
          "page": 22,
          "quote": "[11] David Harari and Jakob Stix. Descent obstruction and fundamental exact sequence. In\nJakob Stix, editor, The Arithmetic of Fundamental Groups – PIA 2010, volume 2 of\nContributions in Mathematical and Computational Sciences, pages 147–166. Springer, 2012.\nAuthor preprint dated February 11, 2011. https://www.imo.universite-paris-\nsaclay.fr/~david.harari/articles/finidesc.pdf."
        },
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p5-d4b9ec4b22",
          "page": 5,
          "quote": "preprint version, Corollary 3.1 and Theorem 4.1]. If F contains no CM subfield, Betts–Stix’s\ntheorem makes the projection of the localization image to C(Fv) finite at every finite place v [4,\npreprint version, Theorem A]. These statements constrain the localization image; the finite-cover\nequality remains a separate arithmetic hypothesis in the global section-conjecture criterion."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-86",
      "kind": "external",
      "label": "[11] Corollary 3.1",
      "source": {
        "blockId": "p21-a353547873",
        "page": 21,
        "quote": "We return to an arbitrary smooth proper geometrically connected curve C/K of genus at least\ntwo over a number field, without assuming (8.3). The localization image in Proposition 8.1\nsatisfies additional descent conditions and, under a restriction on K, has finite image at each\nfinite place.\nLet ρ : C(AK) →C(AK)• be the natural projection. Write Dlin(C) for the ordinary adelic\ndescent set: P belongs to it if, for every C-torsor T →C under a linear algebraic K-group\nG, it lifts to T ξ(AK) for some global twist ξ ∈H1(K, G). Harari–Stix’s Theorem 4.1 sends\nthe finite-torsor condition established in Proposition 8.1 to this linear descent condition, while\ntheir Corollary 3.1, applied to the geometric abelianization of the interpolating section, gives\nBrauer–Manin membership [11, preprint version, Corollary 3.1 and Theorem 4.1]. Consequently"
      },
      "statement": [
        {
          "blockId": "p21-a353547873",
          "page": 21,
          "quote": "We return to an arbitrary smooth proper geometrically connected curve C/K of genus at least\ntwo over a number field, without assuming (8.3). The localization image in Proposition 8.1\nsatisfies additional descent conditions and, under a restriction on K, has finite image at each\nfinite place.\nLet ρ : C(AK) →C(AK)• be the natural projection. Write Dlin(C) for the ordinary adelic\ndescent set: P belongs to it if, for every C-torsor T →C under a linear algebraic K-group\nG, it lifts to T ξ(AK) for some global twist ξ ∈H1(K, G). Harari–Stix’s Theorem 4.1 sends\nthe finite-torsor condition established in Proposition 8.1 to this linear descent condition, while\ntheir Corollary 3.1, applied to the geometric abelianization of the interpolating section, gives\nBrauer–Manin membership [11, preprint version, Corollary 3.1 and Theorem 4.1]. Consequently"
        },
        {
          "blockId": "p22-11f3489b82",
          "page": 22,
          "quote": "[11] David Harari and Jakob Stix. Descent obstruction and fundamental exact sequence. In\nJakob Stix, editor, The Arithmetic of Fundamental Groups – PIA 2010, volume 2 of\nContributions in Mathematical and Computational Sciences, pages 147–166. Springer, 2012.\nAuthor preprint dated February 11, 2011. https://www.imo.universite-paris-\nsaclay.fr/~david.harari/articles/finidesc.pdf."
        },
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p5-d4b9ec4b22",
          "page": 5,
          "quote": "preprint version, Corollary 3.1 and Theorem 4.1]. If F contains no CM subfield, Betts–Stix’s\ntheorem makes the projection of the localization image to C(Fv) finite at every finite place v [4,\npreprint version, Theorem A]. These statements constrain the localization image; the finite-cover\nequality remains a separate arithmetic hypothesis in the global section-conjecture criterion."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-87",
      "kind": "external",
      "label": "[4] Definition 1.1",
      "source": {
        "blockId": "p21-c8382ca489",
        "page": 21,
        "quote": "for every finite place v, with image in C(Kv) [4, arXiv version, Definition 1.1 and Theorem A].\nThis asserts neither injectivity of localization nor finiteness of the global section set, and\nsupplies no effective or uniform bound."
      },
      "statement": [
        {
          "blockId": "p21-c8382ca489",
          "page": 21,
          "quote": "for every finite place v, with image in C(Kv) [4, arXiv version, Definition 1.1 and Theorem A].\nThis asserts neither injectivity of localization nor finiteness of the global section set, and\nsupplies no effective or uniform bound."
        },
        {
          "blockId": "p22-f13809174e",
          "page": 22,
          "quote": "[4] L. Alexander Betts and Jakob Stix. Galois sections and p-adic period mappings. Annals of\nMathematics, 201(1):79–166, 2025. Preprint version: arXiv:2204.13674v1,\nhttps://arxiv.org/abs/2204.13674v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "curated-88",
      "kind": "definition",
      "label": "Geometric sections and the fundamental group sequence",
      "source": {
        "blockId": "p1-02c6e2932a",
        "page": 1,
        "quote": "Let p be a prime, let k/Qp be a finite extension, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Choose an algebraic closure ¯k and a geometric base point\nof X. The full arithmetic étale fundamental group fits into"
      },
      "statement": [
        {
          "blockId": "p1-02c6e2932a",
          "page": 1,
          "quote": "Let p be a prime, let k/Qp be a finite extension, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Choose an algebraic closure ¯k and a geometric base point\nof X. The full arithmetic étale fundamental group fits into"
        },
        {
          "blockId": "p1-356e8bf501",
          "page": 1,
          "quote": "1 −→∆X −→ΠX −→Gk −→1,\n∆X = π1(X¯k),\nΠX = π1(X),\nGk = Gal(¯k/k).\n(1.1)"
        },
        {
          "blockId": "p2-fa6c63fa9e",
          "page": 2,
          "quote": "A rational point a ∈X(k) induces a continuous group-homomorphic section sa : Gk →ΠX. Its\nclass under conjugation by ∆X is independent of the path used to identify base points. The local\np-adic section conjecture asserts that this construction accounts for all sections, and distinguishes\nall rational points.\nA section whose class comes from a rational point is called geometric. The conjecture thus\nasks whether rational points can be recovered from the splittings of the arithmetic fundamental\ngroup. It is a local analogue of the section conjecture in Grothendieck’s letter to Faltings of 27\nJune 1983, formulated over fields finitely generated over the prime field [9, equation (7) and the\nsubsequent properness correction]. The local question retains the full arithmetic and geometric\ngroups in (1.1), while bringing the geometry of p-adic valuations and reduction into the problem."
        }
      ],
      "proof": [],
      "guide": []
    }
  ],
  "edges": [
    {
      "id": "edge-1",
      "from": "equation-52",
      "to": "proposition-16",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p19-2049c0ef47",
        "page": 19,
        "quote": "loc\nSec(C/K)\n = C(AK)f-cov\n•\n.\n(8.2)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-2",
      "from": "equation-53",
      "to": "theorem-17",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p20-13ffcea2da",
        "page": 20,
        "quote": "C(AK)f-cov\n•\n= C(K),\n(8.3)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-3",
      "from": "corollary-14",
      "to": "theorem-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p3-fe1bba6732",
        "page": 3,
        "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
      },
      "note": ""
    },
    {
      "id": "edge-4",
      "from": "external-44",
      "to": "theorem-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p3-fe1bba6732",
        "page": 3,
        "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-5",
      "from": "lemma-3",
      "to": "proposition-4",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p6-bba07b773a",
        "page": 6,
        "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
      },
      "note": ""
    },
    {
      "id": "edge-6",
      "from": "lemma-3",
      "to": "proposition-4",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-15cdd177d5",
        "page": 7,
        "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
      },
      "note": ""
    },
    {
      "id": "edge-7",
      "from": "external-75",
      "to": "proposition-4",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-15cdd177d5",
        "page": 7,
        "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-8",
      "from": "proposition-4",
      "to": "lemma-5",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-7113083a50",
        "page": 7,
        "quote": "Lemma 3.3. In Proposition 3.2, suppose that the composite VΩ→XΩis isomorphic, over XΩ,\nto the base change of a finite morphism V →X¯k from a smooth proper connected curve, and\nassume that ξ ∈U. If ¯k(V ) admits a K0-embedding into H, then ¯k(C) admits a K0-embedding\ninto H."
      },
      "note": ""
    },
    {
      "id": "edge-9",
      "from": "proposition-4",
      "to": "lemma-5",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-a80d8ea63e",
        "page": 7,
        "quote": "Proof. The given embedding determines an H-valued point of V . Together with the fixed\ninclusion Ω⊂H, it determines an H-valued point of V ׯk Ω, and the chosen isomorphism\ntransports this point to VΩ. Its image in XΩ(H) is exactly the fixed point induced by K0 ,→H\nand Ω,→H. In particular its underlying analytic image is ξ ∈U.\nAn H-valued point of an algebraic variety over Ωis equivalently an H-valued point of its\nanalytification [29, Fact 5.1.3]. More explicitly, it gives an analytic point v together with an\nisometric embedding of the completed residue field H(v) into H. The analytic map over U in\nProposition 3.2 induces a compatible map on completed residue fields, and hence an H-valued\npoint of CΩover the same point of XΩ(H). Project this point to C. Its image on X¯k is the\ngeneric point with the specified map K0 →H. Since C →X¯k is finite, its support on C is also\ngeneric. The point therefore induces the required field embedding ¯k(C) ,→H over K0.\nNeither the analytic map over U nor a chosen local trivialization has been asserted to descend\nto ¯k."
      },
      "note": ""
    },
    {
      "id": "edge-10",
      "from": "lemma-6",
      "to": "remark-7",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p8-1aa850b304",
        "page": 8,
        "quote": "Remark 3.5. Lemma 3.4 makes no assertion about tame reduction in residue characteristic p, nor\nabout composing two arbitrarily ramified covers. Its last assertion about composition requires\nthe upper cover to be étale."
      },
      "note": ""
    },
    {
      "id": "edge-11",
      "from": "lemma-8",
      "to": "lemma-9",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "note": ""
    },
    {
      "id": "edge-12",
      "from": "external-62",
      "to": "lemma-9",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-13",
      "from": "external-77",
      "to": "lemma-9",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-14",
      "from": "external-78",
      "to": "lemma-9",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-15",
      "from": "external-79",
      "to": "lemma-9",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p9-ef3971a04b",
        "page": 9,
        "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-16",
      "from": "lemma-8",
      "to": "lemma-9",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p10-162d42f985",
        "page": 10,
        "quote": "The two extra terms are precisely those recorded by the two circuit edges. Thus the augmented\ngraph ramification is zero at every vertex; the identity copies contribute none. All ramification\nindices are prime to p. The resulting tame covering is connected, because each vertex outside\nthe circuit can be joined by a lifted path to a circuit vertex.\nLift these data by the cited theorem. The inverse image of A is exactly the disjoint union of the\nnode annuli above e. For completeness, the inverse-skeleton equality excludes off-skeleton points\nmapping to the skeleton. Each off-skeleton open ball therefore maps into a single off-skeleton\nball downstairs, and continuity identifies the images of their attachment points. Retractions\ncommute on these balls as well as on the skeleton. Since a node annulus is the inverse image of\nits open edge under retraction, this proves the asserted equality for the whole annulus. On each\nannulus above A, the pullback of S has exponent ℓor −ℓin Lemma 4.1. It therefore has an\nanalytic ℓ-th root on each of these disjoint annuli.\nNow suppose that e is separating. Each side of the graph obtained by cutting its interior\ncontains a circuit or a positive-genus vertex. Otherwise that side would be a genus-zero tree\nwith, say, N vertices. Its total valence, including the bridge half-edge, would be 2N −1, contrary\nto stability, which requires at least 3N.\nConstruct a connected unramified degree-ℓcovering of each side, retaining its boundary\nmark for the eventual gluing. If the side contains a circuit, take a connected cyclic graph\ncovering of degree ℓ, with identity residue-curve maps. Such a graph covering is obtained by a\nhomomorphism of its graph fundamental group onto Z/ℓZ. If instead a positive-genus vertex is\nused, take a connected cyclic étale cover of degree ℓof its residue curve, provided by a nonzero\nelement of its Jacobian’s ℓ-torsion [16, Theorem 8.2 and Remark 8.4]. Explicitly, for a line bundle\nL of exact order ℓ, a trivialization L⊗ℓ≃O makes Lℓ−1\nj=0 L⊗j an algebra whose relative spectrum\nis a cyclic étale cover: ℓis invertible, and the exact order of L makes the cover connected. Use ℓ\nidentity copies at the other vertices and match all marked fibres. Paths to the distinguished\nvertex show connectedness of the covered side.\nEach covered side has ℓpreimages of the bridge mark. Match them in pairs to obtain ℓ\ndistinct bridge edges, all with factor one. Every such edge is now nonseparating: after removing\none of them, the two connected covered sides are still joined by the others. The resulting covering\nof the original metrized complex is tame. Indeed the component covers are unramified, all\nfactors are one, and Riemann–Hurwitz again gives zero graph ramification. Lift it to a finite\nétale curve cover. Its triangulation remains stable: at a vertex of local degree d, the quantity\n2g′ −2 + val(v′) is d(2g −2 + val(v)) > 0.\nApply the nonseparating construction to each of these ℓedges. A connected component of\nthe fibre product of the resulting finite étale covers dominates each factor, so all the constructed\nroots persist on a common cover. On each factor-one annulus the pullback of the original S is a\nunit; Lemma 4.1 supplies its root once the new annular coordinate has a root. This proves the\nassertion also for separating edges. Finally, the same finite fibre-product argument treats finitely\nmany original annuli simultaneously."
      },
      "note": ""
    },
    {
      "id": "edge-17",
      "from": "lemma-8",
      "to": "lemma-10",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p11-bba5eb485c",
        "page": 11,
        "quote": "On a sufficiently small disk the quotient ti/ui is an analytic unit, since both coordinates have\na simple zero at bi. Its absolute value is constant by Lemma 4.1. Thus these annuli are also\nconcentric annuli for ti, and choosing j large puts them inside Bi. Perform these modifications\nfor all the sections and denote the resulting proper semistable model by C. It need not be stable;\nthe markings can now be forgotten.\nThe RNS theorem, applied with the set of all primes, and its model-domination consequence\n[17, Theorem A and Proposition 2.4(iv)] give, after finite extension of the ground field, a finite\nétale cover whose stable model maps to C. Pass to split reduction and a geometric connected\ncomponent, and write\nf : Y0 −→CΩ,\nY0 −→C"
      },
      "note": ""
    },
    {
      "id": "edge-18",
      "from": "external-63",
      "to": "lemma-10",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p11-bba5eb485c",
        "page": 11,
        "quote": "On a sufficiently small disk the quotient ti/ui is an analytic unit, since both coordinates have\na simple zero at bi. Its absolute value is constant by Lemma 4.1. Thus these annuli are also\nconcentric annuli for ti, and choosing j large puts them inside Bi. Perform these modifications\nfor all the sections and denote the resulting proper semistable model by C. It need not be stable;\nthe markings can now be forgotten.\nThe RNS theorem, applied with the set of all primes, and its model-domination consequence\n[17, Theorem A and Proposition 2.4(iv)] give, after finite extension of the ground field, a finite\nétale cover whose stable model maps to C. Pass to split reduction and a geometric connected\ncomponent, and write\nf : Y0 −→CΩ,\nY0 −→C"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-19",
      "from": "lemma-8",
      "to": "lemma-10",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p11-d23775987c",
        "page": 11,
        "quote": "Reduction is compatible with the model morphism. A point reducing to a component generic\npoint of Y0 is a model vertex, hence is type 2 and gives ti a value in |Ω×|; its additive valuation\nis rational. It cannot lie above this level. A point reducing to a smooth closed point lies in an\nopen disk tube. If it lies above the chosen level, that smooth point maps to the target node, so\nthe entire disk tube maps into Ai. The pullback of ti is a unit there. By Lemma 4.1 its norm is\nconstant and belongs to |Ω×|, which again excludes the irrational level. All remaining points\nabove the level lie in source node annuli whose nodes map to the target node.\nLet Ni be the union of all source node annuli whose nodes map to the target node defining\nAi; this is a finite union. The complement of Ni in the proper analytic curve Y an\n0\nis compact.\nIntersect it with the inverse image of a closed concentric subannulus of Ai containing the\nchosen level in its interior. On this compact intersection the continuous real function −logp |ti|\nhas compact image avoiding a. Consequently there is an open interval about a with rational\nendpoints whose entire inverse image lies in Ni. It determines a collar A′\ni ⊂Ai.\nEvery source node annulus in Ni maps into Ai by reduction compatibility, so the pullback of\nti is an analytic unit on it. Apply Lemma 4.2 simultaneously to all these source annuli, for all i.\nTheir coordinates acquire ℓ-th roots on a common connected finite étale cover of Y0. Lemma 4.1\nthen gives roots of the pullbacks of the ti on the full inverse images of the A′\ni. The resulting\ncomposite cover of CΩis finite étale."
      },
      "note": ""
    },
    {
      "id": "edge-20",
      "from": "lemma-9",
      "to": "lemma-10",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p11-d23775987c",
        "page": 11,
        "quote": "Reduction is compatible with the model morphism. A point reducing to a component generic\npoint of Y0 is a model vertex, hence is type 2 and gives ti a value in |Ω×|; its additive valuation\nis rational. It cannot lie above this level. A point reducing to a smooth closed point lies in an\nopen disk tube. If it lies above the chosen level, that smooth point maps to the target node, so\nthe entire disk tube maps into Ai. The pullback of ti is a unit there. By Lemma 4.1 its norm is\nconstant and belongs to |Ω×|, which again excludes the irrational level. All remaining points\nabove the level lie in source node annuli whose nodes map to the target node.\nLet Ni be the union of all source node annuli whose nodes map to the target node defining\nAi; this is a finite union. The complement of Ni in the proper analytic curve Y an\n0\nis compact.\nIntersect it with the inverse image of a closed concentric subannulus of Ai containing the\nchosen level in its interior. On this compact intersection the continuous real function −logp |ti|\nhas compact image avoiding a. Consequently there is an open interval about a with rational\nendpoints whose entire inverse image lies in Ni. It determines a collar A′\ni ⊂Ai.\nEvery source node annulus in Ni maps into Ai by reduction compatibility, so the pullback of\nti is an analytic unit on it. Apply Lemma 4.2 simultaneously to all these source annuli, for all i.\nTheir coordinates acquire ℓ-th roots on a common connected finite étale cover of Y0. Lemma 4.1\nthen gives roots of the pullbacks of the ti on the full inverse images of the A′\ni. The resulting\ncomposite cover of CΩis finite étale."
      },
      "note": ""
    },
    {
      "id": "edge-21",
      "from": "lemma-3",
      "to": "proposition-11",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-a2f25852d0",
        "page": 12,
        "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
      },
      "note": ""
    },
    {
      "id": "edge-22",
      "from": "lemma-10",
      "to": "proposition-11",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-a2f25852d0",
        "page": 12,
        "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
      },
      "note": ""
    },
    {
      "id": "edge-23",
      "from": "proposition-4",
      "to": "proposition-11",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-a2f25852d0",
        "page": 12,
        "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
      },
      "note": ""
    },
    {
      "id": "edge-24",
      "from": "lemma-5",
      "to": "proposition-11",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-a2f25852d0",
        "page": 12,
        "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
      },
      "note": ""
    },
    {
      "id": "edge-25",
      "from": "external-74",
      "to": "proposition-11",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-a2f25852d0",
        "page": 12,
        "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-26",
      "from": "external-43",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-3b2bd6a638",
        "page": 13,
        "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-27",
      "from": "external-26",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-3b2bd6a638",
        "page": 13,
        "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-28",
      "from": "external-25",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-3b2bd6a638",
        "page": 13,
        "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-29",
      "from": "external-23",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-c6ed4239e0",
        "page": 13,
        "quote": "Its character lattice is M = H1(Γ, Z); its tropical target is Hom(M, R). The period lattice is\nfree, and q is a topological covering and local analytic isomorphism. These are the Jacobian\nuniformization conventions of [3, Sections 4.1–4.2 and 4.5.2]. The compatibility of the classical\nand tropical Abel–Jacobi maps [3, Proposition 6.1 and Corollary 6.6] identifies the latter with\nintegration on Γ using the edge-length pairing.\nThe open Berkovich annulus Aj strongly deformation retracts to its open skeletal interval\n[2, Section 2.3] and is locally path connected. It is thus simply connected, so i|Aj lifts to an"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-30",
      "from": "external-66",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-c6ed4239e0",
        "page": 13,
        "quote": "Its character lattice is M = H1(Γ, Z); its tropical target is Hom(M, R). The period lattice is\nfree, and q is a topological covering and local analytic isomorphism. These are the Jacobian\nuniformization conventions of [3, Sections 4.1–4.2 and 4.5.2]. The compatibility of the classical\nand tropical Abel–Jacobi maps [3, Proposition 6.1 and Corollary 6.6] identifies the latter with\nintegration on Γ using the edge-length pairing.\nThe open Berkovich annulus Aj strongly deformation retracts to its open skeletal interval\n[2, Section 2.3] and is locally path connected. It is thus simply connected, so i|Aj lifts to an"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-31",
      "from": "external-67",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-c6ed4239e0",
        "page": 13,
        "quote": "Its character lattice is M = H1(Γ, Z); its tropical target is Hom(M, R). The period lattice is\nfree, and q is a topological covering and local analytic isomorphism. These are the Jacobian\nuniformization conventions of [3, Sections 4.1–4.2 and 4.5.2]. The compatibility of the classical\nand tropical Abel–Jacobi maps [3, Proposition 6.1 and Corollary 6.6] identifies the latter with\nintegration on Γ using the edge-length pairing.\nThe open Berkovich annulus Aj strongly deformation retracts to its open skeletal interval\n[2, Section 2.3] and is locally path connected. It is thus simply connected, so i|Aj lifts to an"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-32",
      "from": "lemma-8",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p14-83e6e65311",
        "page": 14,
        "quote": "analytic map bij : Aj →Tan: first lift continuously along the covering q, then use its analytic\nlocal inverses. Orient all edges by increasing v(T). For j′ ̸= j, the cycle cj = ej −ej′ is an\nelement of M. If χj is its character, then uj = χj ◦bij is an analytic unit. The derivative of v(uj)\nalong ej, with respect to v(T), is exactly one: the integral of the harmonic form represented by\ncj over a segment of ej is the length of that segment. Lifts of the tropical map differ only by a\nconstant period, which does not change this derivative.\nBy Lemma 4.1, an analytic unit on an open annulus has a unique dominant Laurent monomial\nat every interior Gauss radius; its exponent is constant and is the slope of its log-norm. It follows\nthat\nuj = c′\njT(1 + hj),\nhj =\nX"
      },
      "note": ""
    },
    {
      "id": "edge-33",
      "from": "lemma-12",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p14-a396d99e51",
        "page": 14,
        "quote": "It is finite étale for every positive integer m, since the ground field has characteristic zero.\nLemma 5.1 applied to a = bij gives an analytic m-th root of uj on the whole inverse image of Aj."
      },
      "note": ""
    },
    {
      "id": "edge-34",
      "from": "lemma-3",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-0ad83e953c",
        "page": 15,
        "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
      },
      "note": ""
    },
    {
      "id": "edge-35",
      "from": "proposition-13",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-0ad83e953c",
        "page": 15,
        "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
      },
      "note": ""
    },
    {
      "id": "edge-36",
      "from": "lemma-6",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-acbc14cc1c",
        "page": 15,
        "quote": "the coordinate T has an analytic mb-th root. Pull this cover back by the nonconstant morphism\nX¯k →P1\n¯k given by T = tb/βb, and take a normalized connected component. Denote the resulting\ncover by Wb →X¯k.\nLemma 3.4 shows that its horizontal indices still divide ℓ. Roots pull back to every point of\nthe normalized component, so tb has an analytic mb-th root on the entire inverse image in W an\nb,Ω\nof the collar\n|βb|p−(ab−Db) < |tb| < |βb|p−Db"
      },
      "note": ""
    },
    {
      "id": "edge-37",
      "from": "lemma-6",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-8ac499fdb2",
        "page": 15,
        "quote": "inside Bb. Here we multiplied the pulled-back root of T by an mb-th root of βb in Ω.\nEmbed the finitely many function fields ¯k(Wb) into the fixed Ksep over K0 and form their\ncompositum. Let W →X¯k be the associated smooth proper connected curve. It dominates\nevery Wb; its horizontal indices divide ℓby Lemma 3.4. All collar roots therefore pull back to\nthe full inverse images in W an\nΩ.\nProposition 3.2, with Y = WΩ, produces a connected finite étale cover VΩ→WΩand an\nanalytic map from its exterior region to Can\nΩover Xan\nΩ. Since W is proper over the algebraically\nclosed field ¯k, invariance of finite étale covers under the algebraically closed extension ¯k ⊂Ω[10]\ngives a connected finite étale cover V →W and an isomorphism"
      },
      "note": ""
    },
    {
      "id": "edge-38",
      "from": "proposition-4",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-8ac499fdb2",
        "page": 15,
        "quote": "inside Bb. Here we multiplied the pulled-back root of T by an mb-th root of βb in Ω.\nEmbed the finitely many function fields ¯k(Wb) into the fixed Ksep over K0 and form their\ncompositum. Let W →X¯k be the associated smooth proper connected curve. It dominates\nevery Wb; its horizontal indices divide ℓby Lemma 3.4. All collar roots therefore pull back to\nthe full inverse images in W an\nΩ.\nProposition 3.2, with Y = WΩ, produces a connected finite étale cover VΩ→WΩand an\nanalytic map from its exterior region to Can\nΩover Xan\nΩ. Since W is proper over the algebraically\nclosed field ¯k, invariance of finite étale covers under the algebraically closed extension ¯k ⊂Ω[10]\ngives a connected finite étale cover V →W and an isomorphism"
      },
      "note": ""
    },
    {
      "id": "edge-39",
      "from": "external-30",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-8ac499fdb2",
        "page": 15,
        "quote": "inside Bb. Here we multiplied the pulled-back root of T by an mb-th root of βb in Ω.\nEmbed the finitely many function fields ¯k(Wb) into the fixed Ksep over K0 and form their\ncompositum. Let W →X¯k be the associated smooth proper connected curve. It dominates\nevery Wb; its horizontal indices divide ℓby Lemma 3.4. All collar roots therefore pull back to\nthe full inverse images in W an\nΩ.\nProposition 3.2, with Y = WΩ, produces a connected finite étale cover VΩ→WΩand an\nanalytic map from its exterior region to Can\nΩover Xan\nΩ. Since W is proper over the algebraically\nclosed field ¯k, invariance of finite étale covers under the algebraically closed extension ¯k ⊂Ω[10]\ngives a connected finite étale cover V →W and an isomorphism"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-40",
      "from": "lemma-6",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-e981bdf0e0",
        "page": 15,
        "quote": "No descent assertion is needed for the exterior analytic map.\nAlthough no such bound was imposed on C →X¯k, the composite V →X¯k still has horizontal\nindices dividing ℓ. By Lemma 3.4, its function field embeds over K0 in Lℓ. This embedding\ncan, if desired, be chosen to extend the already chosen embedding of ¯k(W): extend that\nembedding to ¯k(V ) inside Ksep and then take its Galois closure, which still has indices dividing\nℓ. Proposition 4.4 places its image in H."
      },
      "note": ""
    },
    {
      "id": "edge-41",
      "from": "proposition-11",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-e981bdf0e0",
        "page": 15,
        "quote": "No descent assertion is needed for the exterior analytic map.\nAlthough no such bound was imposed on C →X¯k, the composite V →X¯k still has horizontal\nindices dividing ℓ. By Lemma 3.4, its function field embeds over K0 in Lℓ. This embedding\ncan, if desired, be chosen to extend the already chosen embedding of ¯k(W): extend that\nembedding to ¯k(V ) inside Ksep and then take its Galois closure, which still has indices dividing\nℓ. Proposition 4.4 places its image in H."
      },
      "note": ""
    },
    {
      "id": "edge-42",
      "from": "lemma-5",
      "to": "theorem-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p16-7e7543d8a3",
        "page": 16,
        "quote": "Lemma 3.3 now gives a K0-embedding F = ¯k(C) ,→H ⊂E. Because F/K0 is normal, this\nembedding has image the chosen subfield F of E. Indeed a primitive element of F is sent to a\nroot of its minimal polynomial, all of whose roots already belong to F; the same holds for every\nelement. The image, having degree [F : K0], is therefore F itself. Consequently F ⊂H.\nEvery element of Ksep lies in a finite Galois extension of K0 inside Ksep: this is an algebraic\nclosure of the characteristic-zero field K0.\nTaking the union over those extensions proves\nKsep ⊂H inside E."
      },
      "note": ""
    },
    {
      "id": "edge-43",
      "from": "theorem-2",
      "to": "corollary-14",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p16-c72add0b5d",
        "page": 16,
        "quote": "extends as an isometry to E = [\nKsep. If its restriction to f\nK is the identity, it fixes H = c\nf\nK ⊂E\npointwise. Theorem 1.2 gives Ksep ⊂H, so the automorphism is the identity.\nBoth decomposition groups are closed subgroups of their Galois groups.\nFor example,\nstabilization of a valuation ring is the intersection, over its algebraic elements, of conditions on\nfinite Galois orbits, each of which is clopen. Restriction is consequently a continuous bijection\nbetween compact Hausdorff groups, and its inverse is continuous."
      },
      "note": ""
    },
    {
      "id": "edge-44",
      "from": "corollary-14",
      "to": "lemma-15",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p16-f889b64629",
        "page": 16,
        "quote": "Lemma 7.2. With k, X, K, and f\nK as in Corollary 7.1, let w be any rank-one valuation of K\nextending the p-adic valuation and let ew be any prolongation to f\nK. The projection"
      },
      "note": ""
    },
    {
      "id": "edge-45",
      "from": "corollary-14",
      "to": "lemma-15",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p16-4611b28177",
        "page": 16,
        "quote": "Proof. Normalize w(p) = 1 and prolong the valuation to Ksep. Suppose that a section exists.\nCorollary 7.1 lifts it to a continuous section"
      },
      "note": ""
    },
    {
      "id": "edge-46",
      "from": "external-56",
      "to": "lemma-15",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p16-f8589da2c3",
        "page": 16,
        "quote": "whose image stabilizes this prolongation. The birational section theorem of Koenigsmann places\nthe entire image of bs also in the decomposition group of a prolongation of orda for some a ∈X(k)\n[13, preprint version, Proposition 2.4(2), Definition 1.4, and Observation 1.3]. Here orda is the\nclosed-point valuation of K, trivial on k."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-47",
      "from": "external-80",
      "to": "lemma-15",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p16-f8589da2c3",
        "page": 16,
        "quote": "whose image stabilizes this prolongation. The birational section theorem of Koenigsmann places\nthe entire image of bs also in the decomposition group of a prolongation of orda for some a ∈X(k)\n[13, preprint version, Proposition 2.4(2), Definition 1.4, and Observation 1.3]. Here orda is the\nclosed-point valuation of K, trivial on k."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-48",
      "from": "external-44",
      "to": "lemma-15",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p17-52809f792f",
        "page": 17,
        "quote": "so these valuations are inequivalent. The resulting contradiction is an instance of the classical\nuniqueness principle for henselian valuations associated with F. K. Schmidt [24]. We give\nexplicitly the two-valuation approximation needed here.\nChoose a nonsquare d ∈k×. It remains a nonsquare in M: otherwise GM = Σ would fix\na square root of d in ¯k, contrary to the surjection Σ →Gk. Let t ∈K be a uniformizer at a.\nChoose a positive integer N with N > v1(t), and put z = pN/t. Then"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-49",
      "from": "external-81",
      "to": "theorem-1",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p17-6aa1f30eeb",
        "page": 17,
        "quote": "Proof of the surjectivity in Theorem 1.1. Let s : Gk →ΠX be a continuous group-homomorphic\nsection. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26],\nthere is a valuation w of K extending the p-adic valuation, together with a prolongation ew to\nf\nK, such that\ns(Gk) ⊆De\nw(f\nK/K)."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-50",
      "from": "lemma-15",
      "to": "theorem-1",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p17-7b7f5a8a8a",
        "page": 17,
        "quote": "Since Γk has rational rank one and ordered rank is at most rational rank, w has rank at most\ntwo. Rank one is excluded by Lemma 7.2. If w has rank two, let u be its rank-one coarsening.\nThe nonzero proper convex subgroup of the rank-two value group has a unique extension to\nthe value group of ew: it consists of those values some positive multiple of which lies in the\noriginal subgroup. This follows because an algebraic valued extension has torsion quotient of\nvalue groups. Every automorphism stabilizing ew preserves this subgroup, so s(Gk) stabilizes the\nresulting prolongation of u. If u is nontrivial on k, it can be normalized to extend the p-adic\nvaluation and is excluded by the same lemma."
      },
      "note": ""
    },
    {
      "id": "edge-51",
      "from": "external-42",
      "to": "theorem-1",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p18-9a49bae2ba",
        "page": 18,
        "quote": "It remains that u is trivial on k. Properness gives it a center on X. This center cannot be\ngeneric because u is nontrivial, so it is a closed point a. The valuation ring of u dominates the\ndiscrete valuation ring OX,a. Every element of K× is a power of a uniformizer times a unit of\nthis local ring; hence u is equivalent to orda. Its residue field is the finite extension k(a)/k. The\nsecond valuation in the composition w is the unique prolongation of the p-adic valuation to k(a).\nThus w is of type 2h.\nThe chosen horizontal prolongation to K¯k selects a geometric point above a. Its stabilizer\nacts on constants through Gk(a) ⊆Gk, with the embedding determined by that point. Since\ns(Gk) projects onto Gk, necessarily k(a) = k. For the full étale tower the horizontal inertia\nis trivial, and the horizontal decomposition group is the point section at a. Equivalently, the\ntype-2h conclusion of [22, preprint version, Section 6.1] gives s = sa after the corresponding\ngeometric choice of base point. This proves surjectivity.\nNotice that a type-2h valuation can lie over a non-rational closed point. Rationality here\nfollows from the surjection to Gk; no unrestricted equivalence between “not type 2h” and “not\noriginating from a rational point” is used."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-52",
      "from": "external-68",
      "to": "proposition-16",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p19-eb06480d4a",
        "page": 19,
        "quote": "Proof. Let s ∈Sec(C/K). Choose a representative point of the determined component at each\nreal place, and any point at each complex place. Harari–Stix’s Theorem 2.1, with U = 1 and the\nomitted set of places S = ∅, shows that these localizations survive every torsor under a finite\nK-group scheme; their Remark 2.2(1) identifies this as finite-cover descent [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)]. Consequently"
      },
      "note": "Reviewed external reference: The prose names Harari–Stix and repeats the result inside citation [11]. External source not retrieved or verified."
    },
    {
      "id": "edge-53",
      "from": "external-69",
      "to": "proposition-16",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p19-eb06480d4a",
        "page": 19,
        "quote": "Proof. Let s ∈Sec(C/K). Choose a representative point of the determined component at each\nreal place, and any point at each complex place. Harari–Stix’s Theorem 2.1, with U = 1 and the\nomitted set of places S = ∅, shows that these localizations survive every torsor under a finite\nK-group scheme; their Remark 2.2(1) identifies this as finite-cover descent [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)]. Consequently"
      },
      "note": "Reviewed external reference: The possessive their refers to Harari–Stix, and [11] names the same remark. External source not retrieved or verified."
    },
    {
      "id": "edge-54",
      "from": "external-82",
      "to": "proposition-16",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p19-0cb1be5afe",
        "page": 19,
        "quote": "Conversely, let x belong to the finite-cover locus and, by properness, choose an ordinary\nadelic representative P = (Pv) ∈C(AK). Stoll’s cohomological description after Definition 5.2,\ntogether with Definition 5.4, identifies survival with diagonal global H1(K, G)-membership of the"
      },
      "note": "Reviewed external reference: Stoll’s cohomological description continues onto page 20, where the citation is [26]. External source not retrieved or verified."
    },
    {
      "id": "edge-55",
      "from": "external-83",
      "to": "proposition-16",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p19-0cb1be5afe",
        "page": 19,
        "quote": "Conversely, let x belong to the finite-cover locus and, by properness, choose an ordinary\nadelic representative P = (Pv) ∈C(AK). Stoll’s cohomological description after Definition 5.2,\ntogether with Definition 5.4, identifies survival with diagonal global H1(K, G)-membership of the"
      },
      "note": "Reviewed external reference: Stoll’s cohomological description continues onto page 20, where the citation is [26]. External source not retrieved or verified."
    },
    {
      "id": "edge-56",
      "from": "external-68",
      "to": "proposition-16",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p20-9685c75da6",
        "page": 20,
        "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
      },
      "note": "Reviewed external reference: The prose names Harari–Stix and repeats the result inside citation [11]. External source not retrieved or verified."
    },
    {
      "id": "edge-57",
      "from": "external-46",
      "to": "proposition-16",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-9685c75da6",
        "page": 20,
        "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-58",
      "from": "external-69",
      "to": "proposition-16",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-9685c75da6",
        "page": 20,
        "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-59",
      "from": "proposition-16",
      "to": "theorem-17",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-a963cfc15e",
        "page": 20,
        "quote": "Proof. Let s ∈Sec(C/K). By Proposition 8.1 and (8.3), there is a ∈C(K) such that x(s) is the\ndiagonal localization of a, hence equals the localization of the point section sa.\nAt every finite place the point determined by s is therefore a. At each real place choose a\nitself in the determined component, and choose a also at each complex place. The resulting\nordinary adelic representative of the localization tuple is supported on {a}. Thus the support\nZ(s) in Stix’s terminology, the Zariski-closed support of this tuple, is finite over K. Since C is\nproper of genus at least two, it is hyperbolic and has no cuspidal sections. Stix’s finite-support\nresult therefore makes s a point section [25, arXiv version, Section 3 and Corollary 5]. Finally, if\ntwo points give the same global section, their sections agree after restriction to any one finite\ncompletion. Injectivity in Theorem 1.1 then identifies the two points. This proves injectivity\nand also identifies the point giving s with a."
      },
      "note": ""
    },
    {
      "id": "edge-60",
      "from": "theorem-1",
      "to": "theorem-17",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-a963cfc15e",
        "page": 20,
        "quote": "Proof. Let s ∈Sec(C/K). By Proposition 8.1 and (8.3), there is a ∈C(K) such that x(s) is the\ndiagonal localization of a, hence equals the localization of the point section sa.\nAt every finite place the point determined by s is therefore a. At each real place choose a\nitself in the determined component, and choose a also at each complex place. The resulting\nordinary adelic representative of the localization tuple is supported on {a}. Thus the support\nZ(s) in Stix’s terminology, the Zariski-closed support of this tuple, is finite over K. Since C is\nproper of genus at least two, it is hyperbolic and has no cuspidal sections. Stix’s finite-support\nresult therefore makes s a point section [25, arXiv version, Section 3 and Corollary 5]. Finally, if\ntwo points give the same global section, their sections agree after restriction to any one finite\ncompletion. Injectivity in Theorem 1.1 then identifies the two points. This proves injectivity\nand also identifies the point giving s with a."
      },
      "note": ""
    },
    {
      "id": "edge-61",
      "from": "external-71",
      "to": "corollary-18",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p20-362ebad260",
        "page": 20,
        "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
      },
      "note": "Reviewed external reference: Both occurrences refer to Stoll’s theorem from [26]; [27] supplies corrected covering arguments. External source not retrieved or verified."
    },
    {
      "id": "edge-62",
      "from": "external-70",
      "to": "corollary-18",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p20-362ebad260",
        "page": 20,
        "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
      },
      "note": "Reviewed external reference: The possessive his refers to Stoll, cited as [26] in the same sentence. External source not retrieved or verified."
    },
    {
      "id": "edge-63",
      "from": "external-84",
      "to": "corollary-18",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p20-362ebad260",
        "page": 20,
        "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
      },
      "note": "Reviewed external reference: The prose describes the repair in [27] of the proposition used by Stoll’s theorem from [26], not local Corollary 8.5. External source not retrieved or verified."
    },
    {
      "id": "edge-64",
      "from": "theorem-17",
      "to": "corollary-18",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-362ebad260",
        "page": 20,
        "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
      },
      "note": ""
    },
    {
      "id": "edge-65",
      "from": "external-46",
      "to": "corollary-18",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-362ebad260",
        "page": 20,
        "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-66",
      "from": "external-47",
      "to": "corollary-18",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-362ebad260",
        "page": 20,
        "quote": "Proof. Stoll’s Theorem 8.6 gives C(AK)f-ab\n•\n= C(K) under the first condition, and his Corol-\nlary 8.1 gives the same equality under the second, without requiring a K-rational base point on\nC [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6\nare in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3),\nso Theorem 8.2 applies."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-67",
      "from": "corollary-18",
      "to": "corollary-19",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p21-37f7730139",
        "page": 21,
        "quote": "Proof. The corank-zero case of the companion Selmer converse [18, Theorem 1.1] gives\nrankZ E(Q) = 0 and finiteness of the entire group Sha(E/Q).\nThe Mordell–Weil theorem\ntherefore makes E(Q) finite, while finiteness of Sha(E/Q) makes its maximal divisible subgroup\nzero. Corollary 8.3(i) applies."
      },
      "note": ""
    },
    {
      "id": "edge-68",
      "from": "external-73",
      "to": "corollary-19",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p21-37f7730139",
        "page": 21,
        "quote": "Proof. The corank-zero case of the companion Selmer converse [18, Theorem 1.1] gives\nrankZ E(Q) = 0 and finiteness of the entire group Sha(E/Q).\nThe Mordell–Weil theorem\ntherefore makes E(Q) finite, while finiteness of Sha(E/Q) makes its maximal divisible subgroup\nzero. Corollary 8.3(i) applies."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-69",
      "from": "external-72",
      "to": "corollary-20",
      "relation": "cited-in-proof",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p21-b5285c8185",
        "page": 21,
        "quote": "Proof. Stoll’s Corollary 8.8 gives C(AQ)f-ab\n•\n= C(Q) for each of these projective modular curves\nof positive genus [26], with the corrected conventions of [27]. Apply (8.1) and Theorem 8.2."
      },
      "note": "Reviewed external reference: Stoll is named and [26] is the cited source, with corrected conventions in [27]. External source not retrieved or verified."
    },
    {
      "id": "edge-70",
      "from": "theorem-17",
      "to": "corollary-20",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p21-b5285c8185",
        "page": 21,
        "quote": "Proof. Stoll’s Corollary 8.8 gives C(AQ)f-ab\n•\n= C(Q) for each of these projective modular curves\nof positive genus [26], with the corrected conventions of [27]. Apply (8.1) and Theorem 8.2."
      },
      "note": ""
    },
    {
      "id": "edge-71",
      "from": "external-46",
      "to": "corollary-20",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p21-b5285c8185",
        "page": 21,
        "quote": "Proof. Stoll’s Corollary 8.8 gives C(AQ)f-ab\n•\n= C(Q) for each of these projective modular curves\nof positive genus [26], with the corrected conventions of [27]. Apply (8.1) and Theorem 8.2."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-72",
      "from": "external-47",
      "to": "corollary-20",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p21-b5285c8185",
        "page": 21,
        "quote": "Proof. Stoll’s Corollary 8.8 gives C(AQ)f-ab\n•\n= C(Q) for each of these projective modular curves\nof positive genus [26], with the corrected conventions of [27]. Apply (8.1) and Theorem 8.2."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-73",
      "from": "curated-88",
      "to": "theorem-1",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-02c6e2932a",
        "page": 1,
        "quote": "Let p be a prime, let k/Qp be a finite extension, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Choose an algebraic closure ¯k and a geometric base point\nof X. The full arithmetic étale fundamental group fits into"
      },
      "note": "Agent-proposed background relation from the cited definition; not proof verification."
    }
  ],
  "notation": [
    {
      "symbol": "A",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "occurrences": [
        {
          "blockId": "p2-fa6c63fa9e",
          "page": 2,
          "quote": "A rational point a ∈X(k) induces a continuous group-homomorphic section sa : Gk →ΠX. Its\nclass under conjugation by ∆X is independent of the path used to identify base points. The local\np-adic section conjecture asserts that this construction accounts for all sections, and distinguishes\nall rational points.\nA section whose class comes from a rational point is called geometric. The conjecture thus\nasks whether rational points can be recovered from the splittings of the arithmetic fundamental\ngroup. It is a local analogue of the section conjecture in Grothendieck’s letter to Faltings of 27\nJune 1983, formulated over fields finitely generated over the prime field [9, equation (7) and the\nsubsequent properness correction]. The local question retains the full arithmetic and geometric\ngroups in (1.1), while bringing the geometry of p-adic valuations and reduction into the problem."
        },
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p3-23cd8374c1",
          "page": 3,
          "quote": "Put K = k(X) and choose a separable closure Ksep containing ¯k.\nLet f\nK ⊂Ksep be the\ncompositum of the function fields of all connected finite étale covers of X, with all their K-\nembeddings in Ksep. Then Gal(f\nK/K) = ΠX after the chosen base point identification. A\nreal-valued rank-one norm on K means a nonarchimedean field absolute value with values in R≥0;\nin particular its support is zero. We normalize an extension of the p-adic norm by |p| = p−1."
        },
        {
          "blockId": "p3-fe1bba6732",
          "page": 3,
          "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
        },
        {
          "blockId": "p3-1d11e06d43",
          "page": 3,
          "quote": "The geometric content lies in Theorem 1.2. A finite extension of ¯k(X) is the function field of a\nfinite curve cover, which may have branch points. Near such a point, with local coordinate t, a\nbranch has the form t = ze. If another cover supplies a root of t of order divisible by all these\nbranch indices, the pulled-back cover splits over an annulus around the branch point. One can\nthen replace the part over the inner disk by a trivial cover and glue to obtain an étale cover.\nThe roots must exist on the entire inverse image of the annulus: a root on only one sheet does\nnot trivialize the full cover needed for gluing. We use this operation twice, with different sources\nof collar roots.\nThis gluing and algebraization method belongs to the formal and rigid analytic patching\ntradition of Harbater and Liu [12, 15]. Poineau treats the corresponding Berkovich framework\n[20, Section 1 and Appendix A]. Section 3 proves the precise replacement and field-embedding\nstatements used here. In particular, evaluating the resulting exterior analytic map retains the\nspecified valued field embedding; the map itself need not descend to ¯k.\nFix an odd prime ℓ̸= p. In the first stage, coordinates on the node annuli of a stable model\nacquire ℓ-th roots on suitable finite étale covers. The construction uses the lifting theorem"
        },
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p5-d4b9ec4b22",
          "page": 5,
          "quote": "preprint version, Corollary 3.1 and Theorem 4.1]. If F contains no CM subfield, Betts–Stix’s\ntheorem makes the projection of the localization image to C(Fv) finite at every finite place v [4,\npreprint version, Theorem A]. These statements constrain the localization image; the finite-cover\nequality remains a separate arithmetic hypothesis in the global section-conjecture criterion."
        },
        {
          "blockId": "p5-7f0c749ae4",
          "page": 5,
          "quote": "The inclusion of the completion H into E is isometric: it identifies H with the closure of f\nK in\nE. Since k is complete, the restriction of the chosen norm to ¯k is its unique prolongation. Thus\nΩ:= b¯k is a specified subfield of H. It is algebraically closed and has additive value group Q,\nwith the normalization v = −logp | · |. The equality of value groups before and after completion\nfollows because every nonzero limit has the same norm as sufficiently close approximants.\nWe use Berkovich analytifications over Ω. A field embedding K0 ,→H determines a point of\nX¯k(H) supported at the generic point of X¯k. Together with the specified embedding Ω,→H,\nit determines a point of XΩ(H), whose analytic image we denote by ξ. Its projection to X¯k is\ngeneric; its support on XΩneed not be generic. In particular, ξ can be an Ω-rational point. We\nretain the whole H-valued point, including the embedding of its completed residue field into H,\nwhenever we use ξ.\nIf F/K0 is finite, its normalization gives a smooth proper connected curve over ¯k with a finite\nmap to X¯k. We call the ramification indices of this map at closed points horizontal ramification\nindices. The corresponding discrete valuations are trivial on ¯k and have residue characteristic\nzero. These indices are unrelated to possible wild ramification on a characteristic-p reduction\nmodel.\nFix an odd prime ℓ̸= p. Inside Ksep, let Lℓbe the compositum of all finite Galois extensions\nF/K0 whose horizontal ramification indices divide ℓ. The unramified extensions are included.\nIn particular f\nK ⊂Lℓ: the universal étale field over K0 is the same field f\nK after constants are\nincluded. One may see this by descending any finite étale cover of X¯k to a finite extension of k\nand then composing with the finite étale constant-field cover of X.\nThe field-theoretic goal can be displayed as follows. The top row consists of the defining\ninclusions; the two right vertical arrows are the containments proved in the successive stages."
        },
        {
          "blockId": "p7-15cdd177d5",
          "page": 7,
          "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
        },
        {
          "blockId": "p8-c97559e01e",
          "page": 8,
          "quote": "Proof. All valuations in this statement are the horizontal, discrete closed-point valuations,\nwhich are trivial on ¯k. Their residue characteristic is zero. The completed base field at such\na point is A = ¯k((t)). Every finite extension of A has residue field ¯k and is of the form ¯k((u))\nfor a uniformizer u: successive subtraction of residue representatives gives the power-series\nexpansion in its complete valuation ring. If its ramification index is e, then t = uea(u) for a unit\na(u) ∈¯k[[u]]×. The unit has an e-th root in ¯k[[u]], by Hensel’s lemma in residue characteristic\nzero. Changing the parameter gives t = ze. Thus, inside a fixed algebraic closure, the local field\nis A(t1/e), which is cyclic over A since ¯k contains all roots of unity.\nFor e | n, this is a subfield of A(t1/n). All local factors of a finite compositum, or of a normal\nclosure formed from conjugates, therefore remain subfields of this same cyclic extension. This\nproves the assertions about subextensions, composita, and Galois closures.\nFor pullback, write the local base map as t = sra(s), where r ≥1 and a(s) is a unit.\nExtracting its e-th root in ¯k[[s]] shows that every normalized local factor after pullback has\nramification index e/ gcd(e, r), which divides e. Finally, ramification indices multiply in a tower,\nand an étale upper cover contributes index 1. Apply the Galois-closure assertion to a cover\nwith indices dividing ℓand embed that closure into the fixed Ksep. Its field lies in the defining\ncompositum Lℓ."
        },
        {
          "blockId": "p8-36813d5eeb",
          "page": 8,
          "quote": "Lemma 4.1 (Units on disks and annuli). An analytic unit on an open disk over Ωhas constant\nabsolute value in |Ω×|. If A = {r < |S| < R} is an open annulus and u ∈O(A)×, then"
        },
        {
          "blockId": "p8-6b360aa50e",
          "page": 8,
          "quote": "u = cSd(1 + h),\nc ∈Ω×,\nd ∈Z,\nh ∈O(A),\n|h(x)| < 1\n(x ∈A),"
        },
        {
          "blockId": "p8-fcb18b4841",
          "page": 8,
          "quote": "where cSd is a term of the Laurent expansion of u. The function 1 + h has an analytic ℓ-th root\non A. Consequently, on any analytic cover of A on which S has an ℓ-th root, so does u."
        },
        {
          "blockId": "p9-7df61f8099",
          "page": 9,
          "quote": "the reduction of u is a monomial. Thus exactly one Laurent term dominates at that radius. Its\nexponent cannot change inside the radius interval. Indeed, convergence on compact subannuli\nreduces any proposed change to finitely many competing terms; a tie between two terms occurs\nat a radius in |Ω×|, since the additive value group is Q. Such a tie contradicts the preceding\nreduction argument. A single term cSd therefore dominates throughout A, giving the asserted\nfactorization. The same argument on closed subdisks uses that the only units in eΩ[s] are\nconstants and proves the disk assertion. Its constant norm belongs to |Ω×| by evaluation at a\nrational point of the disk.\nSince ℓis a unit in the valuation ring, the binomial series for (1 + h)1/ℓconverges on every\ncompact subannulus; the resulting functions agree on overlaps. A root of c lies in Ω, and a root\nof Sd follows from a root of S, also when d < 0."
        },
        {
          "blockId": "p9-ee00c7509d",
          "page": 9,
          "quote": "Recall the data attached to a split stable model of a proper curve. Its skeleton is a finite\nmetric graph. Each vertex carries the smooth normalization of the corresponding special-fibre\ncomponent, with one marked point for each incident edge end. A loop has two distinct marks\non the same normalization. The tube of a node is an open annulus, and the tube of a smooth\nspecial-fibre point is an open disk. The graph with these marked residue curves is its metrized\ncomplex. Stability means\n2g(v) −2 + val(v) > 0"
        },
        {
          "blockId": "p9-a7590a88c3",
          "page": 9,
          "quote": "Lemma 4.2 (Roots on stable node annuli). Let C/Ωbe a smooth proper connected curve of\ngenus at least two, equipped with its stable skeleton. Let A be a node annulus and S an annular\ncoordinate on A. There is a connected finite étale cover C′ →C such that S has an analytic ℓ-th\nroot on the entire inverse image of A. A single such cover can be chosen for any finite collection\nof node annuli."
        },
        {
          "blockId": "p9-ef3971a04b",
          "page": 9,
          "quote": "Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along\nedges, divisible by ℓabove the chosen edge. On an inverse annulus this makes the Laurent\nexponent of the pulled-back coordinate divisible by ℓ; its constant factor and the factor 1 + h\nalso have ℓ-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a\ncircuit. For a separating edge, we first pass to an unramified cover in which every edge above it\nlies on a circuit.\nWe use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A\ncovering to which it applies consists of finite residue-curve maps and a finite harmonic graph\nmap: every fibre of every marked point is represented by edge ends, their ramification indices\nequal the edge expansion factors, and the component degree is the sum of these factors over\neach base direction. The residue maps must be tamely ramified, and the augmented graph\nramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no\npunctures. The lift is then finite étale, and has exactly the prescribed inverse images of the\nvertex set and skeleton [1, Definitions 4.25 and 4.31].\nFirst suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit\ncontaining e. At each vertex on this circuit, let P, Q be the two circuit marks on its residue\ncurve D. Divisibility of Pic0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that\nP −Q + ℓE is principal. Extract an ℓ-th root of a function with this divisor and take the smooth\nnormalization. The resulting cyclic cover of D has degree ℓ, is connected because the order at P\nis prime to ℓ, and is totally ramified at P, Q and nowhere else. Each other marked point has ℓ\ndistinct preimages. At every vertex outside the circuit take ℓidentity copies of its residue curve.\nAbove each circuit edge use one edge of expansion factor ℓand of length equal to the\noriginal length divided by ℓ. Above each remaining edge use ℓfactor-one edges, matching all the\nunramified marked fibres at its ends. These rules also handle loops and chords between circuit"
        },
        {
          "blockId": "p10-162d42f985",
          "page": 10,
          "quote": "The two extra terms are precisely those recorded by the two circuit edges. Thus the augmented\ngraph ramification is zero at every vertex; the identity copies contribute none. All ramification\nindices are prime to p. The resulting tame covering is connected, because each vertex outside\nthe circuit can be joined by a lifted path to a circuit vertex.\nLift these data by the cited theorem. The inverse image of A is exactly the disjoint union of the\nnode annuli above e. For completeness, the inverse-skeleton equality excludes off-skeleton points\nmapping to the skeleton. Each off-skeleton open ball therefore maps into a single off-skeleton\nball downstairs, and continuity identifies the images of their attachment points. Retractions\ncommute on these balls as well as on the skeleton. Since a node annulus is the inverse image of\nits open edge under retraction, this proves the asserted equality for the whole annulus. On each\nannulus above A, the pullback of S has exponent ℓor −ℓin Lemma 4.1. It therefore has an\nanalytic ℓ-th root on each of these disjoint annuli.\nNow suppose that e is separating. Each side of the graph obtained by cutting its interior\ncontains a circuit or a positive-genus vertex. Otherwise that side would be a genus-zero tree\nwith, say, N vertices. Its total valence, including the bridge half-edge, would be 2N −1, contrary\nto stability, which requires at least 3N.\nConstruct a connected unramified degree-ℓcovering of each side, retaining its boundary\nmark for the eventual gluing. If the side contains a circuit, take a connected cyclic graph\ncovering of degree ℓ, with identity residue-curve maps. Such a graph covering is obtained by a\nhomomorphism of its graph fundamental group onto Z/ℓZ. If instead a positive-genus vertex is\nused, take a connected cyclic étale cover of degree ℓof its residue curve, provided by a nonzero\nelement of its Jacobian’s ℓ-torsion [16, Theorem 8.2 and Remark 8.4]. Explicitly, for a line bundle\nL of exact order ℓ, a trivialization L⊗ℓ≃O makes Lℓ−1\nj=0 L⊗j an algebra whose relative spectrum\nis a cyclic étale cover: ℓis invertible, and the exact order of L makes the cover connected. Use ℓ\nidentity copies at the other vertices and match all marked fibres. Paths to the distinguished\nvertex show connectedness of the covered side.\nEach covered side has ℓpreimages of the bridge mark. Match them in pairs to obtain ℓ\ndistinct bridge edges, all with factor one. Every such edge is now nonseparating: after removing\none of them, the two connected covered sides are still joined by the others. The resulting covering\nof the original metrized complex is tame. Indeed the component covers are unramified, all\nfactors are one, and Riemann–Hurwitz again gives zero graph ramification. Lift it to a finite\nétale curve cover. Its triangulation remains stable: at a vertex of local degree d, the quantity\n2g′ −2 + val(v′) is d(2g −2 + val(v)) > 0.\nApply the nonseparating construction to each of these ℓedges. A connected component of\nthe fibre product of the resulting finite étale covers dominates each factor, so all the constructed\nroots persist on a common cover. On each factor-one annulus the pullback of the original S is a\nunit; Lemma 4.1 supplies its root once the new annular coordinate has a root. This proves the\nassertion also for separating edges. Finally, the same finite fibre-product argument treats finitely\nmany original annuli simultaneously."
        },
        {
          "blockId": "p11-cb2d7f4e26",
          "page": 11,
          "quote": "A′\ni = {ri < |ti| < Ri} ⊂Bi,\nri, Ri ∈|Ω×|,\n0 < ri < Ri,"
        }
      ],
      "occurrenceCount": 38
    },
    {
      "symbol": "B",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p6-bba07b773a",
        "page": 6,
        "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
      },
      "occurrences": [
        {
          "blockId": "p6-bba07b773a",
          "page": 6,
          "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
        },
        {
          "blockId": "p6-cc1c5411e6",
          "page": 6,
          "quote": "b∈B\n{|tb| ≤rb},"
        },
        {
          "blockId": "p7-15cdd177d5",
          "page": 7,
          "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
        },
        {
          "blockId": "p12-8b3352aee6",
          "page": 12,
          "quote": "Lemma 5.1 (Roots after division on a totally degenerate abelian variety). Let J be an abelian\nvariety over Ωwith analytic uniformization q : Tan →Jan, where T is a split torus and the\nkernel Λ ⊂T(Ω) is a free lattice acting by translation. Suppose q is a topological covering and\na local analytic isomorphism. Let A be an analytic space, let a : A →Tan be analytic, and let\nm ≥1 be an integer. Set\nB = A ×q◦a,Jan,[m] Jan."
        },
        {
          "blockId": "p12-6345d468cf",
          "page": 12,
          "quote": "For every character χ of T, the pullback of χ ◦a to B has an analytic m-th root."
        },
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        },
        {
          "blockId": "p15-0ad83e953c",
          "page": 15,
          "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
        }
      ],
      "occurrenceCount": 7
    },
    {
      "symbol": "Bi",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p11-878474d913",
        "page": 11,
        "quote": "Lemma 4.3 (Étale covers with roots on collars). Let C/¯k be a smooth proper connected curve\nof genus at least two. Let b1, . . . , bs ∈C(¯k) be distinct, and let Bi be pairwise disjoint sufficiently\nsmall coordinate disks in Can\nΩabout bi, with coordinates ti defined over ¯k and vanishing simply\nat bi. There is a connected finite étale cover Y →CΩand concentric open collars"
      },
      "occurrences": [
        {
          "blockId": "p11-878474d913",
          "page": 11,
          "quote": "Lemma 4.3 (Étale covers with roots on collars). Let C/¯k be a smooth proper connected curve\nof genus at least two. Let b1, . . . , bs ∈C(¯k) be distinct, and let Bi be pairwise disjoint sufficiently\nsmall coordinate disks in Can\nΩabout bi, with coordinates ti defined over ¯k and vanishing simply\nat bi. There is a connected finite étale cover Y →CΩand concentric open collars"
        },
        {
          "blockId": "p11-cb2d7f4e26",
          "page": 11,
          "quote": "A′\ni = {ri < |ti| < Ri} ⊂Bi,\nri, Ri ∈|Ω×|,\n0 < ri < Ri,"
        },
        {
          "blockId": "p11-bba5eb485c",
          "page": 11,
          "quote": "On a sufficiently small disk the quotient ti/ui is an analytic unit, since both coordinates have\na simple zero at bi. Its absolute value is constant by Lemma 4.1. Thus these annuli are also\nconcentric annuli for ti, and choosing j large puts them inside Bi. Perform these modifications\nfor all the sections and denote the resulting proper semistable model by C. It need not be stable;\nthe markings can now be forgotten.\nThe RNS theorem, applied with the set of all primes, and its model-domination consequence\n[17, Theorem A and Proposition 2.4(iv)] give, after finite extension of the ground field, a finite\nétale cover whose stable model maps to C. Pass to split reduction and a geometric connected\ncomponent, and write\nf : Y0 −→CΩ,\nY0 −→C"
        },
        {
          "blockId": "p11-eae78e8fc9",
          "page": 11,
          "quote": "for the analytic-base-changed curve cover and the model morphism. No finiteness assertion\nabout the model morphism is needed.\nFix one of the target node annuli Ai ⊂Bi just constructed, and choose an irrational real\nnumber a strictly between its two additive endpoint valuations. Consider the entire inverse\nimage\nf−1{x ∈Ai : −logp |ti(x)| = a}\n."
        }
      ],
      "occurrenceCount": 4,
      "latex": "B_{i}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "J",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p12-8b3352aee6",
        "page": 12,
        "quote": "Lemma 5.1 (Roots after division on a totally degenerate abelian variety). Let J be an abelian\nvariety over Ωwith analytic uniformization q : Tan →Jan, where T is a split torus and the\nkernel Λ ⊂T(Ω) is a free lattice acting by translation. Suppose q is a topological covering and\na local analytic isomorphism. Let A be an analytic space, let a : A →Tan be analytic, and let\nm ≥1 be an integer. Set\nB = A ×q◦a,Jan,[m] Jan."
      },
      "occurrences": [
        {
          "blockId": "p12-8b3352aee6",
          "page": 12,
          "quote": "Lemma 5.1 (Roots after division on a totally degenerate abelian variety). Let J be an abelian\nvariety over Ωwith analytic uniformization q : Tan →Jan, where T is a split torus and the\nkernel Λ ⊂T(Ω) is a free lattice acting by translation. Suppose q is a topological covering and\na local analytic isomorphism. Let A be an analytic space, let a : A →Tan be analytic, and let\nm ≥1 be an integer. Set\nB = A ×q◦a,Jan,[m] Jan."
        },
        {
          "blockId": "p13-3b2bd6a638",
          "page": 13,
          "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
        },
        {
          "blockId": "p14-5c842278c6",
          "page": 14,
          "quote": "Z′ = Z ×i,J,[m] J −→Z."
        },
        {
          "blockId": "p18-796f538cd7",
          "page": 18,
          "quote": "Proof of the injectivity in Theorem 1.1. Suppose a, b ∈X(k) have the same section class modulo\nthe full geometric fundamental group. Let J be the Jacobian of X, and use a as base point for\nthe Abel–Jacobi map\nia : X −→J,\nx 7−→[x −a]."
        },
        {
          "blockId": "p18-d2c2d7a7c2",
          "page": 18,
          "quote": "Their induced sections on J are again geometrically conjugate. For every positive integer n,\nmultiplication [n] : J →J is a finite étale torsor under J[n], also when p | n since k has\ncharacteristic zero. Conjugate sections give isomorphic Gk-actions on its geometric fiber, so the\nexistence of a fixed point is unchanged by conjugation. The section at ia(a) = 0 has the fixed\nlift 0. Therefore the fiber over ia(b) has a Gk-fixed point, which is a k-rational point of that\nfiber. Thus\nia(b) ∈\n\\"
        },
        {
          "blockId": "p18-bb146811cd",
          "page": 18,
          "quote": "This is also the usual Kummer argument: the class of the multiplication torsor pulled back along\na section is unchanged by geometric conjugation.\nThe compact p-adic analytic group J(k) has an open subgroup U ∼= Zg(X)[k:Qp]\np\nby the local\nlogarithm. If q = [J(k) : U], then qJ(k) ⊆U. For each r ≥0, take n = qpr in the displayed\nintersection. Any element of that intersection can then be written as pr(qy) with y ∈J(k), and\nhence belongs to prU. As U is a finite free Zp-module,"
        },
        {
          "blockId": "p20-38c44a9b7c",
          "page": 20,
          "quote": "(ii) the Jacobian J of C has J(K) finite and Sha(J/K)div = 0."
        }
      ],
      "occurrenceCount": 7
    },
    {
      "symbol": "K",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p3-23cd8374c1",
        "page": 3,
        "quote": "Put K = k(X) and choose a separable closure Ksep containing ¯k.\nLet f\nK ⊂Ksep be the\ncompositum of the function fields of all connected finite étale covers of X, with all their K-\nembeddings in Ksep. Then Gal(f\nK/K) = ΠX after the chosen base point identification. A\nreal-valued rank-one norm on K means a nonarchimedean field absolute value with values in R≥0;\nin particular its support is zero. We normalize an extension of the p-adic norm by |p| = p−1."
      },
      "occurrences": [
        {
          "blockId": "p3-23cd8374c1",
          "page": 3,
          "quote": "Put K = k(X) and choose a separable closure Ksep containing ¯k.\nLet f\nK ⊂Ksep be the\ncompositum of the function fields of all connected finite étale covers of X, with all their K-\nembeddings in Ksep. Then Gal(f\nK/K) = ΠX after the chosen base point identification. A\nreal-valued rank-one norm on K means a nonarchimedean field absolute value with values in R≥0;\nin particular its support is zero. We normalize an extension of the p-adic norm by |p| = p−1."
        },
        {
          "blockId": "p3-de0986e624",
          "page": 3,
          "quote": "Theorem 1.2. Let p be a prime, let k/Qp be finite, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Put K = k(X), choose a separable closure Ksep, and let\nf\nK ⊂Ksep be the compositum of the function fields of all connected finite étale covers of X, with\nall their K-embeddings. For every real-valued rank-one norm on K extending the normalized\np-adic norm on k, and every prolongation to Ksep, set"
        },
        {
          "blockId": "p3-5c3066fd86",
          "page": 3,
          "quote": "E = [\nKsep,\nH = c\nf\nK ⊂E,"
        },
        {
          "blockId": "p3-fe1bba6732",
          "page": 3,
          "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
        },
        {
          "blockId": "p5-b47be4ae89",
          "page": 5,
          "quote": "K ⊂K0 := ¯k(X) ⊂f\nK ⊂Ksep ⊂E,\nH = c\nf\nK ⊂E."
        },
        {
          "blockId": "p5-7f0c749ae4",
          "page": 5,
          "quote": "The inclusion of the completion H into E is isometric: it identifies H with the closure of f\nK in\nE. Since k is complete, the restriction of the chosen norm to ¯k is its unique prolongation. Thus\nΩ:= b¯k is a specified subfield of H. It is algebraically closed and has additive value group Q,\nwith the normalization v = −logp | · |. The equality of value groups before and after completion\nfollows because every nonzero limit has the same norm as sufficiently close approximants.\nWe use Berkovich analytifications over Ω. A field embedding K0 ,→H determines a point of\nX¯k(H) supported at the generic point of X¯k. Together with the specified embedding Ω,→H,\nit determines a point of XΩ(H), whose analytic image we denote by ξ. Its projection to X¯k is\ngeneric; its support on XΩneed not be generic. In particular, ξ can be an Ω-rational point. We\nretain the whole H-valued point, including the embedding of its completed residue field into H,\nwhenever we use ξ.\nIf F/K0 is finite, its normalization gives a smooth proper connected curve over ¯k with a finite\nmap to X¯k. We call the ramification indices of this map at closed points horizontal ramification\nindices. The corresponding discrete valuations are trivial on ¯k and have residue characteristic\nzero. These indices are unrelated to possible wild ramification on a characteristic-p reduction\nmodel.\nFix an odd prime ℓ̸= p. Inside Ksep, let Lℓbe the compositum of all finite Galois extensions\nF/K0 whose horizontal ramification indices divide ℓ. The unramified extensions are included.\nIn particular f\nK ⊂Lℓ: the universal étale field over K0 is the same field f\nK after constants are\nincluded. One may see this by descending any finite étale cover of X¯k to a finite extension of k\nand then composing with the finite étale constant-field cover of X.\nThe field-theoretic goal can be displayed as follows. The top row consists of the defining\ninclusions; the two right vertical arrows are the containments proved in the successive stages."
        },
        {
          "blockId": "p5-928dc44837",
          "page": 5,
          "quote": "f\nK −−−−→Lℓ−−−−→Ksep\ny\nyfirst stage\nysecond stage"
        },
        {
          "blockId": "p5-0c4756847d",
          "page": 5,
          "quote": "Every arrow is inside E. In particular a constructed abstract field embedding is not enough by\nitself: for a normal extension F/K0 we will use normality to identify its image with the specified\nsubfield F ⊂E.\nWe shall also use invariance of finite étale covers of a proper variety under extension of\nalgebraically closed fields [10, Exposé X, Corollary 1.8]. Thus a finite étale cover of YΩ, for a\nsmooth proper curve Y/¯k, descends up to isomorphism to a cover of Y . The analytic maps built\non smaller open regions need not descend. Their evaluation at the fixed H-valued point will be\nsufficient."
        },
        {
          "blockId": "p7-7113083a50",
          "page": 7,
          "quote": "Lemma 3.3. In Proposition 3.2, suppose that the composite VΩ→XΩis isomorphic, over XΩ,\nto the base change of a finite morphism V →X¯k from a smooth proper connected curve, and\nassume that ξ ∈U. If ¯k(V ) admits a K0-embedding into H, then ¯k(C) admits a K0-embedding\ninto H."
        },
        {
          "blockId": "p7-a80d8ea63e",
          "page": 7,
          "quote": "Proof. The given embedding determines an H-valued point of V . Together with the fixed\ninclusion Ω⊂H, it determines an H-valued point of V ׯk Ω, and the chosen isomorphism\ntransports this point to VΩ. Its image in XΩ(H) is exactly the fixed point induced by K0 ,→H\nand Ω,→H. In particular its underlying analytic image is ξ ∈U.\nAn H-valued point of an algebraic variety over Ωis equivalently an H-valued point of its\nanalytification [29, Fact 5.1.3]. More explicitly, it gives an analytic point v together with an\nisometric embedding of the completed residue field H(v) into H. The analytic map over U in\nProposition 3.2 induces a compatible map on completed residue fields, and hence an H-valued\npoint of CΩover the same point of XΩ(H). Project this point to C. Its image on X¯k is the\ngeneric point with the specified map K0 →H. Since C →X¯k is finite, its support on C is also\ngeneric. The point therefore induces the required field embedding ¯k(C) ,→H over K0.\nNeither the analytic map over U nor a chosen local trivialization has been asserted to descend\nto ¯k."
        },
        {
          "blockId": "p8-c7ae99a541",
          "page": 8,
          "quote": "Lemma 3.4. Fix a positive integer n. Among finite extensions of K0, the condition that every\nramification index at a closed point of X¯k divide n is preserved by taking subextensions, finite\ncomposita, and Galois closures. It is also preserved by normalized pullback along a finite morphism\nof smooth proper curves and by composition on top with a finite étale cover. Consequently every\nfinite cover of X¯k with indices dividing ℓhas a K0-embedding of its function field into Lℓ."
        },
        {
          "blockId": "p12-a2f25852d0",
          "page": 12,
          "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
        },
        {
          "blockId": "p15-388037fca7",
          "page": 15,
          "quote": "The first stage, Proposition 4.4, places the field Lℓinside H. We now use the auxiliary covers\nfrom Proposition 5.2 to realize every finite Galois extension of K0 in that same completion. The\ncovers used to create the collars may be branched; their horizontal ramification indices divide ℓ,\nso the first stage is sufficient to accommodate them."
        },
        {
          "blockId": "p15-0ad83e953c",
          "page": 15,
          "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
        },
        {
          "blockId": "p15-8ac499fdb2",
          "page": 15,
          "quote": "inside Bb. Here we multiplied the pulled-back root of T by an mb-th root of βb in Ω.\nEmbed the finitely many function fields ¯k(Wb) into the fixed Ksep over K0 and form their\ncompositum. Let W →X¯k be the associated smooth proper connected curve. It dominates\nevery Wb; its horizontal indices divide ℓby Lemma 3.4. All collar roots therefore pull back to\nthe full inverse images in W an\nΩ.\nProposition 3.2, with Y = WΩ, produces a connected finite étale cover VΩ→WΩand an\nanalytic map from its exterior region to Can\nΩover Xan\nΩ. Since W is proper over the algebraically\nclosed field ¯k, invariance of finite étale covers under the algebraically closed extension ¯k ⊂Ω[10]\ngives a connected finite étale cover V →W and an isomorphism"
        },
        {
          "blockId": "p15-e981bdf0e0",
          "page": 15,
          "quote": "No descent assertion is needed for the exterior analytic map.\nAlthough no such bound was imposed on C →X¯k, the composite V →X¯k still has horizontal\nindices dividing ℓ. By Lemma 3.4, its function field embeds over K0 in Lℓ. This embedding\ncan, if desired, be chosen to extend the already chosen embedding of ¯k(W): extend that\nembedding to ¯k(V ) inside Ksep and then take its Galois closure, which still has indices dividing\nℓ. Proposition 4.4 places its image in H."
        },
        {
          "blockId": "p16-7e7543d8a3",
          "page": 16,
          "quote": "Lemma 3.3 now gives a K0-embedding F = ¯k(C) ,→H ⊂E. Because F/K0 is normal, this\nembedding has image the chosen subfield F of E. Indeed a primitive element of F is sent to a\nroot of its minimal polynomial, all of whose roots already belong to F; the same holds for every\nelement. The image, having degree [F : K0], is therefore F itself. Consequently F ⊂H.\nEvery element of Ksep lies in a finite Galois extension of K0 inside Ksep: this is an algebraic\nclosure of the characteristic-zero field K0.\nTaking the union over those extensions proves\nKsep ⊂H inside E."
        },
        {
          "blockId": "p16-d357ebb858",
          "page": 16,
          "quote": "Corollary 7.1. Let k/Qp be finite and let X/k be a smooth proper geometrically connected\ncurve of genus at least two. Put K = k(X) and let f\nK ⊂Ksep be the extension corresponding to\nthe full arithmetic étale fundamental group of X. Let w be a real-valued rank-one valuation of\nK extending the p-adic valuation, normalized by w(p) = 1, and fix a prolongation wsep to Ksep.\nRestriction induces an isomorphism of profinite groups"
        },
        {
          "blockId": "p16-0a9614cda9",
          "page": 16,
          "quote": "Dwsep(Ksep/K)\n∼\n−−→De\nw(f\nK/K),\new = wsep| e\nK."
        },
        {
          "blockId": "p16-45f6d3aaff",
          "page": 16,
          "quote": "Proof. Restriction is surjective. Indeed, lift an automorphism stabilizing ew to an automorphism\nof Ksep/K. Its translate of wsep is another prolongation of ew. Transitivity of Gal(Ksep/f\nK)\non these prolongations allows us to adjust the lift so that it stabilizes wsep. The transitivity\nstatement for an infinite normal extension follows from its finite-level version by compactness.\nAn automorphism stabilizing a rank-one valuation preserves its normalized absolute value.\nTo see why normalization matters, any two order-preserving embeddings of the value group into\nR differ by a positive scale; the value on p fixes that scale. Such an automorphism therefore"
        }
      ],
      "occurrenceCount": 56
    },
    {
      "symbol": "Ni",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p11-d23775987c",
        "page": 11,
        "quote": "Reduction is compatible with the model morphism. A point reducing to a component generic\npoint of Y0 is a model vertex, hence is type 2 and gives ti a value in |Ω×|; its additive valuation\nis rational. It cannot lie above this level. A point reducing to a smooth closed point lies in an\nopen disk tube. If it lies above the chosen level, that smooth point maps to the target node, so\nthe entire disk tube maps into Ai. The pullback of ti is a unit there. By Lemma 4.1 its norm is\nconstant and belongs to |Ω×|, which again excludes the irrational level. All remaining points\nabove the level lie in source node annuli whose nodes map to the target node.\nLet Ni be the union of all source node annuli whose nodes map to the target node defining\nAi; this is a finite union. The complement of Ni in the proper analytic curve Y an\n0\nis compact.\nIntersect it with the inverse image of a closed concentric subannulus of Ai containing the\nchosen level in its interior. On this compact intersection the continuous real function −logp |ti|\nhas compact image avoiding a. Consequently there is an open interval about a with rational\nendpoints whose entire inverse image lies in Ni. It determines a collar A′\ni ⊂Ai.\nEvery source node annulus in Ni maps into Ai by reduction compatibility, so the pullback of\nti is an analytic unit on it. Apply Lemma 4.2 simultaneously to all these source annuli, for all i.\nTheir coordinates acquire ℓ-th roots on a common connected finite étale cover of Y0. Lemma 4.1\nthen gives roots of the pullbacks of the ti on the full inverse images of the A′\ni. The resulting\ncomposite cover of CΩis finite étale."
      },
      "occurrences": [
        {
          "blockId": "p11-d23775987c",
          "page": 11,
          "quote": "Reduction is compatible with the model morphism. A point reducing to a component generic\npoint of Y0 is a model vertex, hence is type 2 and gives ti a value in |Ω×|; its additive valuation\nis rational. It cannot lie above this level. A point reducing to a smooth closed point lies in an\nopen disk tube. If it lies above the chosen level, that smooth point maps to the target node, so\nthe entire disk tube maps into Ai. The pullback of ti is a unit there. By Lemma 4.1 its norm is\nconstant and belongs to |Ω×|, which again excludes the irrational level. All remaining points\nabove the level lie in source node annuli whose nodes map to the target node.\nLet Ni be the union of all source node annuli whose nodes map to the target node defining\nAi; this is a finite union. The complement of Ni in the proper analytic curve Y an\n0\nis compact.\nIntersect it with the inverse image of a closed concentric subannulus of Ai containing the\nchosen level in its interior. On this compact intersection the continuous real function −logp |ti|\nhas compact image avoiding a. Consequently there is an open interval about a with rational\nendpoints whose entire inverse image lies in Ni. It determines a collar A′\ni ⊂Ai.\nEvery source node annulus in Ni maps into Ai by reduction compatibility, so the pullback of\nti is an analytic unit on it. Apply Lemma 4.2 simultaneously to all these source annuli, for all i.\nTheir coordinates acquire ℓ-th roots on a common connected finite étale cover of Y0. Lemma 4.1\nthen gives roots of the pullbacks of the ti on the full inverse images of the A′\ni. The resulting\ncomposite cover of CΩis finite étale."
        }
      ],
      "occurrenceCount": 1,
      "latex": "N_{i}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "U",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p6-b993750202",
        "page": 6,
        "quote": "Set\nU = Xan\nΩ\\\n["
      },
      "occurrences": [
        {
          "blockId": "p6-b993750202",
          "page": 6,
          "quote": "Set\nU = Xan\nΩ\\\n["
        },
        {
          "blockId": "p6-6863f2684b",
          "page": 6,
          "quote": "(V an\nΩ)|U −→Can\nΩ\nover Xan\nΩ."
        },
        {
          "blockId": "p6-966efaa943",
          "page": 6,
          "quote": "The notation on the left denotes the inverse image of U under the composite map to Xan\nΩ."
        },
        {
          "blockId": "p7-51fd4b0c57",
          "page": 7,
          "quote": "Proof. The inverse image YU of U and the inverse images Yb of the disks {|tb| < Rb} form an\nopen cover of Y an. Distinct Yb are disjoint, and YU ∩Yb = g−1(Ab).\nOn YU, take the pullback of Can\nΩ→Xan\nΩ. It is finite étale of degree d, since U contains no\nbranch point. On each Yb, take the trivial degree-d cover. These two covers are isomorphic\nover their overlap. Indeed, write qmb\nb\n= g∗tb for the assumed analytic root. In the local factor\ncorresponding to c ∈f−1(b), put ac = qmb/ec\nb\n. The polynomial defining that factor splits as"
        },
        {
          "blockId": "p7-7113083a50",
          "page": 7,
          "quote": "Lemma 3.3. In Proposition 3.2, suppose that the composite VΩ→XΩis isomorphic, over XΩ,\nto the base change of a finite morphism V →X¯k from a smooth proper connected curve, and\nassume that ξ ∈U. If ¯k(V ) admits a K0-embedding into H, then ¯k(C) admits a K0-embedding\ninto H."
        },
        {
          "blockId": "p7-a80d8ea63e",
          "page": 7,
          "quote": "Proof. The given embedding determines an H-valued point of V . Together with the fixed\ninclusion Ω⊂H, it determines an H-valued point of V ׯk Ω, and the chosen isomorphism\ntransports this point to VΩ. Its image in XΩ(H) is exactly the fixed point induced by K0 ,→H\nand Ω,→H. In particular its underlying analytic image is ξ ∈U.\nAn H-valued point of an algebraic variety over Ωis equivalently an H-valued point of its\nanalytification [29, Fact 5.1.3]. More explicitly, it gives an analytic point v together with an\nisometric embedding of the completed residue field H(v) into H. The analytic map over U in\nProposition 3.2 induces a compatible map on completed residue fields, and hence an H-valued\npoint of CΩover the same point of XΩ(H). Project this point to C. Its image on X¯k is the\ngeneric point with the specified map K0 →H. Since C →X¯k is finite, its support on C is also\ngeneric. The point therefore induces the required field embedding ¯k(C) ,→H over K0.\nNeither the analytic map over U nor a chosen local trivialization has been asserted to descend\nto ¯k."
        },
        {
          "blockId": "p13-36e42bc923",
          "page": 13,
          "quote": "Here is a split semistable model of Z. Work first over the integers of a sufficiently large finite\nextension of k containing α. The blowup of the model of the line along (T, α) has two rational\ncomponents and one node, with local equation TU = α. The inner component has coordinate\nS = T/α; the four branch sections are disjoint and smooth. The Kummer function reduces\nto T −1 on the outer component and to −S/(S −1) on the inner component. Each of these\ndegree-ℓcovers of a rational component has just two totally ramified points and is rational. Near\nthe node the Kummer function is\nT −1\n1 −U ,"
        },
        {
          "blockId": "p18-bb146811cd",
          "page": 18,
          "quote": "This is also the usual Kummer argument: the class of the multiplication torsor pulled back along\na section is unchanged by geometric conjugation.\nThe compact p-adic analytic group J(k) has an open subgroup U ∼= Zg(X)[k:Qp]\np\nby the local\nlogarithm. If q = [J(k) : U], then qJ(k) ⊆U. For each r ≥0, take n = qpr in the displayed\nintersection. Any element of that intersection can then be written as pr(qy) with y ∈J(k), and\nhence belongs to prU. As U is a finite free Zp-module,"
        },
        {
          "blockId": "p19-eb06480d4a",
          "page": 19,
          "quote": "Proof. Let s ∈Sec(C/K). Choose a representative point of the determined component at each\nreal place, and any point at each complex place. Harari–Stix’s Theorem 2.1, with U = 1 and the\nomitted set of places S = ∅, shows that these localizations survive every torsor under a finite\nK-group scheme; their Remark 2.2(1) identifies this as finite-cover descent [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)]. Consequently"
        },
        {
          "blockId": "p20-9685c75da6",
          "page": 20,
          "quote": "local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor\nthrough the real components, and complex places impose no lifting condition, in the corrected\nconvention already used above. Thus every such P satisfies the ordinary finite-torsor condition.\nHarari–Stix’s Theorem 2.1 with U = 1 and S = ∅, using implication (i)⇒(iii), gives a global\nsection whose restrictions are the point sections of Pv, up to conjugation by the full geometric\nfundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and\nreal-component uniqueness used above identify its localization with x, proving (8.2)."
        }
      ],
      "occurrenceCount": 10
    },
    {
      "symbol": "Z",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p13-d8ca492316",
        "page": 13,
        "quote": "Proof. Let Z be the smooth projective normalization of"
      },
      "occurrences": [
        {
          "blockId": "p7-15d8e7fa2f",
          "page": 7,
          "quote": "ζ∈µec\n(Z −ζac)."
        },
        {
          "blockId": "p8-6b360aa50e",
          "page": 8,
          "quote": "u = cSd(1 + h),\nc ∈Ω×,\nd ∈Z,\nh ∈O(A),\n|h(x)| < 1\n(x ∈A),"
        },
        {
          "blockId": "p10-162d42f985",
          "page": 10,
          "quote": "The two extra terms are precisely those recorded by the two circuit edges. Thus the augmented\ngraph ramification is zero at every vertex; the identity copies contribute none. All ramification\nindices are prime to p. The resulting tame covering is connected, because each vertex outside\nthe circuit can be joined by a lifted path to a circuit vertex.\nLift these data by the cited theorem. The inverse image of A is exactly the disjoint union of the\nnode annuli above e. For completeness, the inverse-skeleton equality excludes off-skeleton points\nmapping to the skeleton. Each off-skeleton open ball therefore maps into a single off-skeleton\nball downstairs, and continuity identifies the images of their attachment points. Retractions\ncommute on these balls as well as on the skeleton. Since a node annulus is the inverse image of\nits open edge under retraction, this proves the asserted equality for the whole annulus. On each\nannulus above A, the pullback of S has exponent ℓor −ℓin Lemma 4.1. It therefore has an\nanalytic ℓ-th root on each of these disjoint annuli.\nNow suppose that e is separating. Each side of the graph obtained by cutting its interior\ncontains a circuit or a positive-genus vertex. Otherwise that side would be a genus-zero tree\nwith, say, N vertices. Its total valence, including the bridge half-edge, would be 2N −1, contrary\nto stability, which requires at least 3N.\nConstruct a connected unramified degree-ℓcovering of each side, retaining its boundary\nmark for the eventual gluing. If the side contains a circuit, take a connected cyclic graph\ncovering of degree ℓ, with identity residue-curve maps. Such a graph covering is obtained by a\nhomomorphism of its graph fundamental group onto Z/ℓZ. If instead a positive-genus vertex is\nused, take a connected cyclic étale cover of degree ℓof its residue curve, provided by a nonzero\nelement of its Jacobian’s ℓ-torsion [16, Theorem 8.2 and Remark 8.4]. Explicitly, for a line bundle\nL of exact order ℓ, a trivialization L⊗ℓ≃O makes Lℓ−1\nj=0 L⊗j an algebra whose relative spectrum\nis a cyclic étale cover: ℓis invertible, and the exact order of L makes the cover connected. Use ℓ\nidentity copies at the other vertices and match all marked fibres. Paths to the distinguished\nvertex show connectedness of the covered side.\nEach covered side has ℓpreimages of the bridge mark. Match them in pairs to obtain ℓ\ndistinct bridge edges, all with factor one. Every such edge is now nonseparating: after removing\none of them, the two connected covered sides are still joined by the others. The resulting covering\nof the original metrized complex is tame. Indeed the component covers are unramified, all\nfactors are one, and Riemann–Hurwitz again gives zero graph ramification. Lift it to a finite\nétale curve cover. Its triangulation remains stable: at a vertex of local degree d, the quantity\n2g′ −2 + val(v′) is d(2g −2 + val(v)) > 0.\nApply the nonseparating construction to each of these ℓedges. A connected component of\nthe fibre product of the resulting finite étale covers dominates each factor, so all the constructed\nroots persist on a common cover. On each factor-one annulus the pullback of the original S is a\nunit; Lemma 4.1 supplies its root once the new annular coordinate has a root. This proves the\nassertion also for separating edges. Finally, the same finite fibre-product argument treats finitely\nmany original annuli simultaneously."
        },
        {
          "blockId": "p13-d8ca492316",
          "page": 13,
          "quote": "Proof. Let Z be the smooth projective normalization of"
        },
        {
          "blockId": "p13-9602266c5c",
          "page": 13,
          "quote": "2g(Z) −2 = −2ℓ+ 4(ℓ−1),\ng(Z) = ℓ−1."
        },
        {
          "blockId": "p13-36e42bc923",
          "page": 13,
          "quote": "Here is a split semistable model of Z. Work first over the integers of a sufficiently large finite\nextension of k containing α. The blowup of the model of the line along (T, α) has two rational\ncomponents and one node, with local equation TU = α. The inner component has coordinate\nS = T/α; the four branch sections are disjoint and smooth. The Kummer function reduces\nto T −1 on the outer component and to −S/(S −1) on the inner component. Each of these\ndegree-ℓcovers of a rational component has just two totally ramified points and is rational. Near\nthe node the Kummer function is\nT −1\n1 −U ,"
        },
        {
          "blockId": "p13-3b2bd6a638",
          "page": 13,
          "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
        },
        {
          "blockId": "p13-c6ed4239e0",
          "page": 13,
          "quote": "Its character lattice is M = H1(Γ, Z); its tropical target is Hom(M, R). The period lattice is\nfree, and q is a topological covering and local analytic isomorphism. These are the Jacobian\nuniformization conventions of [3, Sections 4.1–4.2 and 4.5.2]. The compatibility of the classical\nand tropical Abel–Jacobi maps [3, Proposition 6.1 and Corollary 6.6] identifies the latter with\nintegration on Γ using the edge-length pairing.\nThe open Berkovich annulus Aj strongly deformation retracts to its open skeletal interval\n[2, Section 2.3] and is locally path connected. It is thus simply connected, so i|Aj lifts to an"
        },
        {
          "blockId": "p14-5c842278c6",
          "page": 14,
          "quote": "Z′ = Z ×i,J,[m] J −→Z."
        },
        {
          "blockId": "p15-9d0c2e8bfe",
          "page": 15,
          "quote": "Dividing it by bj and an m-th root of c′\nj gives an analytic m-th root of T above the middle collar.\nThe annuli Aj are disjoint, so this gives a root on its entire inverse image in Z′. Choose any\nconnected component Q of Z′. Its map to Z is finite étale and surjective, so it is a smooth proper\nconnected curve with the required property. Its composite with Z →P1 has only ramification\nindices 1 or ℓ. All algebraic maps used to define Q are over k; the analytic roots and torus lifts\nare only required over Ω."
        },
        {
          "blockId": "p20-a963cfc15e",
          "page": 20,
          "quote": "Proof. Let s ∈Sec(C/K). By Proposition 8.1 and (8.3), there is a ∈C(K) such that x(s) is the\ndiagonal localization of a, hence equals the localization of the point section sa.\nAt every finite place the point determined by s is therefore a. At each real place choose a\nitself in the determined component, and choose a also at each complex place. The resulting\nordinary adelic representative of the localization tuple is supported on {a}. Thus the support\nZ(s) in Stix’s terminology, the Zariski-closed support of this tuple, is finite over K. Since C is\nproper of genus at least two, it is hyperbolic and has no cuspidal sections. Stix’s finite-support\nresult therefore makes s a point section [25, arXiv version, Section 3 and Corollary 5]. Finally, if\ntwo points give the same global section, their sections agree after restriction to any one finite\ncompletion. Injectivity in Theorem 1.1 then identifies the two points. This proves injectivity\nand also identifies the point giving s with a."
        }
      ],
      "occurrenceCount": 11
    },
    {
      "symbol": "aB",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p12-08e5a73bba",
        "page": 12,
        "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
      },
      "occurrences": [
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        }
      ],
      "occurrenceCount": 1,
      "latex": "a_{B}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "ac",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p7-51fd4b0c57",
        "page": 7,
        "quote": "Proof. The inverse image YU of U and the inverse images Yb of the disks {|tb| < Rb} form an\nopen cover of Y an. Distinct Yb are disjoint, and YU ∩Yb = g−1(Ab).\nOn YU, take the pullback of Can\nΩ→Xan\nΩ. It is finite étale of degree d, since U contains no\nbranch point. On each Yb, take the trivial degree-d cover. These two covers are isomorphic\nover their overlap. Indeed, write qmb\nb\n= g∗tb for the assumed analytic root. In the local factor\ncorresponding to c ∈f−1(b), put ac = qmb/ec\nb\n. The polynomial defining that factor splits as"
      },
      "occurrences": [
        {
          "blockId": "p7-51fd4b0c57",
          "page": 7,
          "quote": "Proof. The inverse image YU of U and the inverse images Yb of the disks {|tb| < Rb} form an\nopen cover of Y an. Distinct Yb are disjoint, and YU ∩Yb = g−1(Ab).\nOn YU, take the pullback of Can\nΩ→Xan\nΩ. It is finite étale of degree d, since U contains no\nbranch point. On each Yb, take the trivial degree-d cover. These two covers are isomorphic\nover their overlap. Indeed, write qmb\nb\n= g∗tb for the assumed analytic root. In the local factor\ncorresponding to c ∈f−1(b), put ac = qmb/ec\nb\n. The polynomial defining that factor splits as"
        },
        {
          "blockId": "p7-15d8e7fa2f",
          "page": 7,
          "quote": "ζ∈µec\n(Z −ζac)."
        },
        {
          "blockId": "p7-15cdd177d5",
          "page": 7,
          "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
        }
      ],
      "occurrenceCount": 3,
      "latex": "a_{c}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "b",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-7f0c749ae4",
        "page": 5,
        "quote": "The inclusion of the completion H into E is isometric: it identifies H with the closure of f\nK in\nE. Since k is complete, the restriction of the chosen norm to ¯k is its unique prolongation. Thus\nΩ:= b¯k is a specified subfield of H. It is algebraically closed and has additive value group Q,\nwith the normalization v = −logp | · |. The equality of value groups before and after completion\nfollows because every nonzero limit has the same norm as sufficiently close approximants.\nWe use Berkovich analytifications over Ω. A field embedding K0 ,→H determines a point of\nX¯k(H) supported at the generic point of X¯k. Together with the specified embedding Ω,→H,\nit determines a point of XΩ(H), whose analytic image we denote by ξ. Its projection to X¯k is\ngeneric; its support on XΩneed not be generic. In particular, ξ can be an Ω-rational point. We\nretain the whole H-valued point, including the embedding of its completed residue field into H,\nwhenever we use ξ.\nIf F/K0 is finite, its normalization gives a smooth proper connected curve over ¯k with a finite\nmap to X¯k. We call the ramification indices of this map at closed points horizontal ramification\nindices. The corresponding discrete valuations are trivial on ¯k and have residue characteristic\nzero. These indices are unrelated to possible wild ramification on a characteristic-p reduction\nmodel.\nFix an odd prime ℓ̸= p. Inside Ksep, let Lℓbe the compositum of all finite Galois extensions\nF/K0 whose horizontal ramification indices divide ℓ. The unramified extensions are included.\nIn particular f\nK ⊂Lℓ: the universal étale field over K0 is the same field f\nK after constants are\nincluded. One may see this by descending any finite étale cover of X¯k to a finite extension of k\nand then composing with the finite étale constant-field cover of X.\nThe field-theoretic goal can be displayed as follows. The top row consists of the defining\ninclusions; the two right vertical arrows are the containments proved in the successive stages."
      },
      "occurrences": [
        {
          "blockId": "p5-7f0c749ae4",
          "page": 5,
          "quote": "The inclusion of the completion H into E is isometric: it identifies H with the closure of f\nK in\nE. Since k is complete, the restriction of the chosen norm to ¯k is its unique prolongation. Thus\nΩ:= b¯k is a specified subfield of H. It is algebraically closed and has additive value group Q,\nwith the normalization v = −logp | · |. The equality of value groups before and after completion\nfollows because every nonzero limit has the same norm as sufficiently close approximants.\nWe use Berkovich analytifications over Ω. A field embedding K0 ,→H determines a point of\nX¯k(H) supported at the generic point of X¯k. Together with the specified embedding Ω,→H,\nit determines a point of XΩ(H), whose analytic image we denote by ξ. Its projection to X¯k is\ngeneric; its support on XΩneed not be generic. In particular, ξ can be an Ω-rational point. We\nretain the whole H-valued point, including the embedding of its completed residue field into H,\nwhenever we use ξ.\nIf F/K0 is finite, its normalization gives a smooth proper connected curve over ¯k with a finite\nmap to X¯k. We call the ramification indices of this map at closed points horizontal ramification\nindices. The corresponding discrete valuations are trivial on ¯k and have residue characteristic\nzero. These indices are unrelated to possible wild ramification on a characteristic-p reduction\nmodel.\nFix an odd prime ℓ̸= p. Inside Ksep, let Lℓbe the compositum of all finite Galois extensions\nF/K0 whose horizontal ramification indices divide ℓ. The unramified extensions are included.\nIn particular f\nK ⊂Lℓ: the universal étale field over K0 is the same field f\nK after constants are\nincluded. One may see this by descending any finite étale cover of X¯k to a finite extension of k\nand then composing with the finite étale constant-field cover of X.\nThe field-theoretic goal can be displayed as follows. The top row consists of the defining\ninclusions; the two right vertical arrows are the containments proved in the successive stages."
        },
        {
          "blockId": "p6-4a7d2c52b5",
          "page": 6,
          "quote": "Lemma 3.1. Let f : C →X¯k be a finite surjective morphism of smooth proper connected curves,\nand let b ∈X(¯k). Fix a rational function tb which is a local parameter at b. There is an open\ndisk Bb about b in Xan\nΩ, with coordinate tb, such that its full inverse image in Can\nΩis a disjoint\nunion of open disks. On the disk about c ∈f−1(b) there is a coordinate zc for which"
        },
        {
          "blockId": "p6-c0041351b3",
          "page": 6,
          "quote": "where ec is the ramification index at c. No condition that ec be prime to p is required. For finitely\nmany distinct points b, the disks can be chosen disjoint and disjoint from the fixed point ξ."
        },
        {
          "blockId": "p6-0f4a412be2",
          "page": 6,
          "quote": "Here it is enough to make the valuation of a(u)/a(0) −1 greater than vp(ec) + 1/(p −1). Thus\nzc = u a(u)1/ec is analytic with a nonzero derivative at the center. Analytic inversion gives a\nsmaller disk on which it is a coordinate and tb = zec\nc .\nDo this in disjoint neighborhoods of all the finitely many points above b. The complement of\nthese neighborhoods in the proper analytic curve Can\nΩis compact. Its image under f is compact,\nclosed, and does not contain b. Consequently a sufficiently small disk about b has its entire\ninverse image in the chosen neighborhoods. Shrink it once more so that, in every zc-chart, that\ninverse image is the disk defined by the corresponding bound on |zc|ec.\nThe point ξ does not equal any point of X(¯k), since its projection to X¯k is generic. The\nshrinking disks at b form a neighborhood basis there. Hausdorffness therefore permits all disks\nto avoid ξ and one another. We may choose their radii in |Ω×|. The argument also applies when\nξ is an Ω-rational point whose projection to X¯k is generic."
        },
        {
          "blockId": "p6-bba07b773a",
          "page": 6,
          "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
        },
        {
          "blockId": "p6-cc1c5411e6",
          "page": 6,
          "quote": "b∈B\n{|tb| ≤rb},"
        },
        {
          "blockId": "p7-51fd4b0c57",
          "page": 7,
          "quote": "Proof. The inverse image YU of U and the inverse images Yb of the disks {|tb| < Rb} form an\nopen cover of Y an. Distinct Yb are disjoint, and YU ∩Yb = g−1(Ab).\nOn YU, take the pullback of Can\nΩ→Xan\nΩ. It is finite étale of degree d, since U contains no\nbranch point. On each Yb, take the trivial degree-d cover. These two covers are isomorphic\nover their overlap. Indeed, write qmb\nb\n= g∗tb for the assumed analytic root. In the local factor\ncorresponding to c ∈f−1(b), put ac = qmb/ec\nb\n. The polynomial defining that factor splits as"
        },
        {
          "blockId": "p11-878474d913",
          "page": 11,
          "quote": "Lemma 4.3 (Étale covers with roots on collars). Let C/¯k be a smooth proper connected curve\nof genus at least two. Let b1, . . . , bs ∈C(¯k) be distinct, and let Bi be pairwise disjoint sufficiently\nsmall coordinate disks in Can\nΩabout bi, with coordinates ti defined over ¯k and vanishing simply\nat bi. There is a connected finite étale cover Y →CΩand concentric open collars"
        },
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        },
        {
          "blockId": "p15-0ad83e953c",
          "page": 15,
          "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
        },
        {
          "blockId": "p15-acbc14cc1c",
          "page": 15,
          "quote": "the coordinate T has an analytic mb-th root. Pull this cover back by the nonconstant morphism\nX¯k →P1\n¯k given by T = tb/βb, and take a normalized connected component. Denote the resulting\ncover by Wb →X¯k.\nLemma 3.4 shows that its horizontal indices still divide ℓ. Roots pull back to every point of\nthe normalized component, so tb has an analytic mb-th root on the entire inverse image in W an\nb,Ω\nof the collar\n|βb|p−(ab−Db) < |tb| < |βb|p−Db"
        },
        {
          "blockId": "p15-8ac499fdb2",
          "page": 15,
          "quote": "inside Bb. Here we multiplied the pulled-back root of T by an mb-th root of βb in Ω.\nEmbed the finitely many function fields ¯k(Wb) into the fixed Ksep over K0 and form their\ncompositum. Let W →X¯k be the associated smooth proper connected curve. It dominates\nevery Wb; its horizontal indices divide ℓby Lemma 3.4. All collar roots therefore pull back to\nthe full inverse images in W an\nΩ.\nProposition 3.2, with Y = WΩ, produces a connected finite étale cover VΩ→WΩand an\nanalytic map from its exterior region to Can\nΩover Xan\nΩ. Since W is proper over the algebraically\nclosed field ¯k, invariance of finite étale covers under the algebraically closed extension ¯k ⊂Ω[10]\ngives a connected finite étale cover V →W and an isomorphism"
        },
        {
          "blockId": "p18-796f538cd7",
          "page": 18,
          "quote": "Proof of the injectivity in Theorem 1.1. Suppose a, b ∈X(k) have the same section class modulo\nthe full geometric fundamental group. Let J be the Jacobian of X, and use a as base point for\nthe Abel–Jacobi map\nia : X −→J,\nx 7−→[x −a]."
        },
        {
          "blockId": "p18-d2c2d7a7c2",
          "page": 18,
          "quote": "Their induced sections on J are again geometrically conjugate. For every positive integer n,\nmultiplication [n] : J →J is a finite étale torsor under J[n], also when p | n since k has\ncharacteristic zero. Conjugate sections give isomorphic Gk-actions on its geometric fiber, so the\nexistence of a fixed point is unchanged by conjugation. The section at ia(a) = 0 has the fixed\nlift 0. Therefore the fiber over ia(b) has a Gk-fixed point, which is a k-rational point of that\nfiber. Thus\nia(b) ∈\n\\"
        },
        {
          "blockId": "p18-f9740ddbdc",
          "page": 18,
          "quote": "and hence ia(b) = 0. If a ̸= b, the linear equivalence of the degree-one divisors a and b would\ngive a degree-one morphism X →P1, forcing genus zero. Hence a = b, as required."
        }
      ],
      "occurrenceCount": 15
    },
    {
      "symbol": "ew",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p16-f889b64629",
        "page": 16,
        "quote": "Lemma 7.2. With k, X, K, and f\nK as in Corollary 7.1, let w be any rank-one valuation of K\nextending the p-adic valuation and let ew be any prolongation to f\nK. The projection"
      },
      "occurrences": [
        {
          "blockId": "p16-0a9614cda9",
          "page": 16,
          "quote": "Dwsep(Ksep/K)\n∼\n−−→De\nw(f\nK/K),\new = wsep| e\nK."
        },
        {
          "blockId": "p16-45f6d3aaff",
          "page": 16,
          "quote": "Proof. Restriction is surjective. Indeed, lift an automorphism stabilizing ew to an automorphism\nof Ksep/K. Its translate of wsep is another prolongation of ew. Transitivity of Gal(Ksep/f\nK)\non these prolongations allows us to adjust the lift so that it stabilizes wsep. The transitivity\nstatement for an infinite normal extension follows from its finite-level version by compactness.\nAn automorphism stabilizing a rank-one valuation preserves its normalized absolute value.\nTo see why normalization matters, any two order-preserving embeddings of the value group into\nR differ by a positive scale; the value on p fixes that scale. Such an automorphism therefore"
        },
        {
          "blockId": "p16-f889b64629",
          "page": 16,
          "quote": "Lemma 7.2. With k, X, K, and f\nK as in Corollary 7.1, let w be any rank-one valuation of K\nextending the p-adic valuation and let ew be any prolongation to f\nK. The projection"
        },
        {
          "blockId": "p17-6aa1f30eeb",
          "page": 17,
          "quote": "Proof of the surjectivity in Theorem 1.1. Let s : Gk →ΠX be a continuous group-homomorphic\nsection. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26],\nthere is a valuation w of K extending the p-adic valuation, together with a prolongation ew to\nf\nK, such that\ns(Gk) ⊆De\nw(f\nK/K)."
        },
        {
          "blockId": "p17-7b7f5a8a8a",
          "page": 17,
          "quote": "Since Γk has rational rank one and ordered rank is at most rational rank, w has rank at most\ntwo. Rank one is excluded by Lemma 7.2. If w has rank two, let u be its rank-one coarsening.\nThe nonzero proper convex subgroup of the rank-two value group has a unique extension to\nthe value group of ew: it consists of those values some positive multiple of which lies in the\noriginal subgroup. This follows because an algebraic valued extension has torsion quotient of\nvalue groups. Every automorphism stabilizing ew preserves this subgroup, so s(Gk) stabilizes the\nresulting prolongation of u. If u is nontrivial on k, it can be normalized to extend the p-adic\nvaluation and is excluded by the same lemma."
        }
      ],
      "occurrenceCount": 5
    },
    {
      "symbol": "mb",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p6-bba07b773a",
        "page": 6,
        "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
      },
      "occurrences": [
        {
          "blockId": "p6-bba07b773a",
          "page": 6,
          "quote": "Proposition 3.2. Let f : C →X¯k be as in Lemma 3.1, of degree d, with branch set B. Choose\npairwise disjoint disks Bb as in that lemma. For each b ∈B, let mb be a positive multiple of all\nramification indices above b. Suppose that g : Y →XΩis a finite surjective morphism from a\nsmooth proper connected curve and that there are radii 0 < rb < Rb in |Ω×|, strictly inside Bb,\nsuch that g∗tb has an analytic mb-th root on the whole inverse image of"
        },
        {
          "blockId": "p15-0ad83e953c",
          "page": 15,
          "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
        },
        {
          "blockId": "p15-acbc14cc1c",
          "page": 15,
          "quote": "the coordinate T has an analytic mb-th root. Pull this cover back by the nonconstant morphism\nX¯k →P1\n¯k given by T = tb/βb, and take a normalized connected component. Denote the resulting\ncover by Wb →X¯k.\nLemma 3.4 shows that its horizontal indices still divide ℓ. Roots pull back to every point of\nthe normalized component, so tb has an analytic mb-th root on the entire inverse image in W an\nb,Ω\nof the collar\n|βb|p−(ab−Db) < |tb| < |βb|p−Db"
        },
        {
          "blockId": "p15-8ac499fdb2",
          "page": 15,
          "quote": "inside Bb. Here we multiplied the pulled-back root of T by an mb-th root of βb in Ω.\nEmbed the finitely many function fields ¯k(Wb) into the fixed Ksep over K0 and form their\ncompositum. Let W →X¯k be the associated smooth proper connected curve. It dominates\nevery Wb; its horizontal indices divide ℓby Lemma 3.4. All collar roots therefore pull back to\nthe full inverse images in W an\nΩ.\nProposition 3.2, with Y = WΩ, produces a connected finite étale cover VΩ→WΩand an\nanalytic map from its exterior region to Can\nΩover Xan\nΩ. Since W is proper over the algebraically\nclosed field ¯k, invariance of finite étale covers under the algebraically closed extension ¯k ⊂Ω[10]\ngives a connected finite étale cover V →W and an isomorphism"
        }
      ],
      "occurrenceCount": 4,
      "latex": "m_{b}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "p",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p1-02c6e2932a",
        "page": 1,
        "quote": "Let p be a prime, let k/Qp be a finite extension, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Choose an algebraic closure ¯k and a geometric base point\nof X. The full arithmetic étale fundamental group fits into"
      },
      "occurrences": [
        {
          "blockId": "p1-428b594b2d",
          "page": 1,
          "quote": "The p-adic section conjecture"
        },
        {
          "blockId": "p1-d25e2968bf",
          "page": 1,
          "quote": "We prove the local p-adic section conjecture for smooth proper geometrically connected\ncurves of genus at least two, over every finite extension of Qp. Combined with established\nfinite-descent theorems, this also yields the global section conjecture for smooth proper\ngeometrically connected curves of genus at least two over number fields when their finite-\ncover descent locus equals their rational points; this includes X0(N) and X1(N) of genus at\nleast two over Q."
        },
        {
          "blockId": "p1-bd26276fa6",
          "page": 1,
          "quote": "4\nPrime-to-p roots from stable reduction\n8\n4.1\nUnits and roots on node annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n8\n4.2\nMoving disk collars into the stable skeleton\n. . . . . . . . . . . . . . . . . . . . .\n10"
        },
        {
          "blockId": "p1-02c6e2932a",
          "page": 1,
          "quote": "Let p be a prime, let k/Qp be a finite extension, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Choose an algebraic closure ¯k and a geometric base point\nof X. The full arithmetic étale fundamental group fits into"
        },
        {
          "blockId": "p2-fa6c63fa9e",
          "page": 2,
          "quote": "A rational point a ∈X(k) induces a continuous group-homomorphic section sa : Gk →ΠX. Its\nclass under conjugation by ∆X is independent of the path used to identify base points. The local\np-adic section conjecture asserts that this construction accounts for all sections, and distinguishes\nall rational points.\nA section whose class comes from a rational point is called geometric. The conjecture thus\nasks whether rational points can be recovered from the splittings of the arithmetic fundamental\ngroup. It is a local analogue of the section conjecture in Grothendieck’s letter to Faltings of 27\nJune 1983, formulated over fields finitely generated over the prime field [9, equation (7) and the\nsubsequent properness correction]. The local question retains the full arithmetic and geometric\ngroups in (1.1), while bringing the geometry of p-adic valuations and reduction into the problem."
        },
        {
          "blockId": "p2-14f3098072",
          "page": 2,
          "quote": "Theorem 1.1. For every prime p, every finite extension k/Qp, and every smooth proper\ngeometrically connected curve X/k of genus at least two, the map"
        },
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p3-23cd8374c1",
          "page": 3,
          "quote": "Put K = k(X) and choose a separable closure Ksep containing ¯k.\nLet f\nK ⊂Ksep be the\ncompositum of the function fields of all connected finite étale covers of X, with all their K-\nembeddings in Ksep. Then Gal(f\nK/K) = ΠX after the chosen base point identification. A\nreal-valued rank-one norm on K means a nonarchimedean field absolute value with values in R≥0;\nin particular its support is zero. We normalize an extension of the p-adic norm by |p| = p−1."
        },
        {
          "blockId": "p3-de0986e624",
          "page": 3,
          "quote": "Theorem 1.2. Let p be a prime, let k/Qp be finite, and let X/k be a smooth proper geometrically\nconnected curve of genus at least two. Put K = k(X), choose a separable closure Ksep, and let\nf\nK ⊂Ksep be the compositum of the function fields of all connected finite étale covers of X, with\nall their K-embeddings. For every real-valued rank-one norm on K extending the normalized\np-adic norm on k, and every prolongation to Ksep, set"
        },
        {
          "blockId": "p3-fe1bba6732",
          "page": 3,
          "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
        },
        {
          "blockId": "p3-1d11e06d43",
          "page": 3,
          "quote": "The geometric content lies in Theorem 1.2. A finite extension of ¯k(X) is the function field of a\nfinite curve cover, which may have branch points. Near such a point, with local coordinate t, a\nbranch has the form t = ze. If another cover supplies a root of t of order divisible by all these\nbranch indices, the pulled-back cover splits over an annulus around the branch point. One can\nthen replace the part over the inner disk by a trivial cover and glue to obtain an étale cover.\nThe roots must exist on the entire inverse image of the annulus: a root on only one sheet does\nnot trivialize the full cover needed for gluing. We use this operation twice, with different sources\nof collar roots.\nThis gluing and algebraization method belongs to the formal and rigid analytic patching\ntradition of Harbater and Liu [12, 15]. Poineau treats the corresponding Berkovich framework\n[20, Section 1 and Appendix A]. Section 3 proves the precise replacement and field-embedding\nstatements used here. In particular, evaluating the resulting exterior analytic map retains the\nspecified valued field embedding; the map itself need not descend to ¯k.\nFix an odd prime ℓ̸= p. In the first stage, coordinates on the node annuli of a stable model\nacquire ℓ-th roots on suitable finite étale covers. The construction uses the lifting theorem"
        },
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p5-7f0c749ae4",
          "page": 5,
          "quote": "The inclusion of the completion H into E is isometric: it identifies H with the closure of f\nK in\nE. Since k is complete, the restriction of the chosen norm to ¯k is its unique prolongation. Thus\nΩ:= b¯k is a specified subfield of H. It is algebraically closed and has additive value group Q,\nwith the normalization v = −logp | · |. The equality of value groups before and after completion\nfollows because every nonzero limit has the same norm as sufficiently close approximants.\nWe use Berkovich analytifications over Ω. A field embedding K0 ,→H determines a point of\nX¯k(H) supported at the generic point of X¯k. Together with the specified embedding Ω,→H,\nit determines a point of XΩ(H), whose analytic image we denote by ξ. Its projection to X¯k is\ngeneric; its support on XΩneed not be generic. In particular, ξ can be an Ω-rational point. We\nretain the whole H-valued point, including the embedding of its completed residue field into H,\nwhenever we use ξ.\nIf F/K0 is finite, its normalization gives a smooth proper connected curve over ¯k with a finite\nmap to X¯k. We call the ramification indices of this map at closed points horizontal ramification\nindices. The corresponding discrete valuations are trivial on ¯k and have residue characteristic\nzero. These indices are unrelated to possible wild ramification on a characteristic-p reduction\nmodel.\nFix an odd prime ℓ̸= p. Inside Ksep, let Lℓbe the compositum of all finite Galois extensions\nF/K0 whose horizontal ramification indices divide ℓ. The unramified extensions are included.\nIn particular f\nK ⊂Lℓ: the universal étale field over K0 is the same field f\nK after constants are\nincluded. One may see this by descending any finite étale cover of X¯k to a finite extension of k\nand then composing with the finite étale constant-field cover of X.\nThe field-theoretic goal can be displayed as follows. The top row consists of the defining\ninclusions; the two right vertical arrows are the containments proved in the successive stages."
        },
        {
          "blockId": "p6-c0041351b3",
          "page": 6,
          "quote": "where ec is the ramification index at c. No condition that ec be prime to p is required. For finitely\nmany distinct points b, the disks can be chosen disjoint and disjoint from the fixed point ξ."
        },
        {
          "blockId": "p6-0f4a412be2",
          "page": 6,
          "quote": "Here it is enough to make the valuation of a(u)/a(0) −1 greater than vp(ec) + 1/(p −1). Thus\nzc = u a(u)1/ec is analytic with a nonzero derivative at the center. Analytic inversion gives a\nsmaller disk on which it is a coordinate and tb = zec\nc .\nDo this in disjoint neighborhoods of all the finitely many points above b. The complement of\nthese neighborhoods in the proper analytic curve Can\nΩis compact. Its image under f is compact,\nclosed, and does not contain b. Consequently a sufficiently small disk about b has its entire\ninverse image in the chosen neighborhoods. Shrink it once more so that, in every zc-chart, that\ninverse image is the disk defined by the corresponding bound on |zc|ec.\nThe point ξ does not equal any point of X(¯k), since its projection to X¯k is generic. The\nshrinking disks at b form a neighborhood basis there. Hausdorffness therefore permits all disks\nto avoid ξ and one another. We may choose their radii in |Ω×|. The argument also applies when\nξ is an Ω-rational point whose projection to X¯k is generic."
        },
        {
          "blockId": "p7-15cdd177d5",
          "page": 7,
          "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
        },
        {
          "blockId": "p8-1aa850b304",
          "page": 8,
          "quote": "Remark 3.5. Lemma 3.4 makes no assertion about tame reduction in residue characteristic p, nor\nabout composing two arbitrarily ramified covers. Its last assertion about composition requires\nthe upper cover to be étale."
        },
        {
          "blockId": "p8-6016f20943",
          "page": 8,
          "quote": "4\nPrime-to-p roots from stable reduction"
        },
        {
          "blockId": "p8-44bd051e25",
          "page": 8,
          "quote": "We first treat the extensions whose horizontal ramification indices divide the fixed prime ℓ̸= p.\nThe patching construction reduces this task to extracting ℓ-th roots of coordinates on collars\naround the branch points. We shall find these roots in finite étale covers of XΩ. There are two\nsteps: extract roots on stable node annuli, and use resolution of nonsingularities to reach such\nannuli from arbitrary small disks."
        }
      ],
      "occurrenceCount": 43
    },
    {
      "symbol": "s",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p2-2b25b1a064",
        "page": 2,
        "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
      },
      "occurrences": [
        {
          "blockId": "p2-fa6c63fa9e",
          "page": 2,
          "quote": "A rational point a ∈X(k) induces a continuous group-homomorphic section sa : Gk →ΠX. Its\nclass under conjugation by ∆X is independent of the path used to identify base points. The local\np-adic section conjecture asserts that this construction accounts for all sections, and distinguishes\nall rational points.\nA section whose class comes from a rational point is called geometric. The conjecture thus\nasks whether rational points can be recovered from the splittings of the arithmetic fundamental\ngroup. It is a local analogue of the section conjecture in Grothendieck’s letter to Faltings of 27\nJune 1983, formulated over fields finitely generated over the prime field [9, equation (7) and the\nsubsequent properness correction]. The local question retains the full arithmetic and geometric\ngroups in (1.1), while bringing the geometry of p-adic valuations and reduction into the problem."
        },
        {
          "blockId": "p2-3127881a48",
          "page": 2,
          "quote": "(\ns : Gk →π1(X) a continuous group homomorphism"
        },
        {
          "blockId": "p2-b83772449f",
          "page": 2,
          "quote": "such that (π1(X) →Gk) ◦s = idGk"
        },
        {
          "blockId": "p2-2b25b1a064",
          "page": 2,
          "quote": "The birational analogue replaces the arithmetic étale fundamental group by the absolute Galois\ngroup of the curve’s function field. For a Galois extension with a chosen valuation, the decompo-\nsition group consists of the automorphisms preserving its valuation ring. Koenigsmann proves\nthat every birational section over a finite extension of Qp is contained in the decomposition\ngroup of a rational point [13, preprint version, Proposition 2.4(2)]. Pop subsequently obtains a\nbirational criterion from sections of an elementary abelian quotient that lift to the corresponding\nelementary metabelian quotient [21, Theorems A and A′]. These results make the passage from\nan étale section to a birational section a natural objective. Such a lift is additional information:\na section of the quotient group in (1.1) does not by itself supply a splitting of the absolute Galois\nsequence.\nPop–Stix localize every étale section at a valuation extending the p-adic valuation [22, preprint\nversion, Theorem 26]. This reduces the problem to understanding which valuations can support\na section. Pajwani develops this valuative approach through étale homotopy and fixed points\nin Berkovich spaces [19, preprint version, Theorem 1.4 and Corollary 1.5]. The passage from a\nfixed valuation to a rational point remains the geometric issue addressed here.\nRecent results also constrain the possible nongeometric sections. Bresciani introduces toric\nsections and proves the section conjecture for them; they map injectively to the set of étale\nsections [7, Definition 3.7 and Theorems 1.4 and 5.10]. For an arbitrary nongeometric étale\nsection, his Theorem 1.5 produces a finite étale neighbourhood whose index is divisible by p.\nHere a neighbourhood is a connected finite étale cover to which the section lifts, and the index\nof a curve is the greatest common divisor of the degrees of its closed points. In a different\ndirection, Betts–Stix prove finiteness of the local image of global Selmer sections for smooth\nproper geometrically connected curves of genus at least two over number fields containing no\nCM subfield [4, preprint version, Theorem A]. Their Selmer sections already arise from points at\nevery completion; the finiteness assertion concerns the local images of this globally constrained\nset of sections.\nOur lifting step is a statement about completions of the universal étale function field. It\ncompares the absolute and étale decomposition groups at every rank-one valuation extending the\np-adic norm. This supplies the birational lift at precisely the valuations that must be excluded\nfrom the Pop–Stix localization."
        },
        {
          "blockId": "p3-fe1bba6732",
          "page": 3,
          "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
        },
        {
          "blockId": "p4-563828111d",
          "page": 4,
          "quote": "of Amini–Baker–Brugallé–Rabinoff, which realizes tame coverings of metrized complexes as\ncurve covers [1, Theorem 7.7]. A metrized complex records the skeleton together with the\nresidue curve at each vertex and the marked points corresponding to edge ends. Small collars\naround arbitrary branch points need not themselves be stable node annuli. The resolution\nof nonsingularities theorem of Mochizuki–Tsujimura supplies a finite étale cover whose stable\nmodel maps to a prescribed semistable model [17, Theorem A and Proposition 2.4(iv)]. This\nextends Tamagawa’s result for closed points on a stable model [28, Theorem 0.2(v)] and Lepage’s\ncorresponding resolution theorem for Mumford curves [14, Theorem 2.7]. The general theorem of\nMochizuki–Tsujimura allows us to move these collars into the stable-node construction. Patching\nthen places in H every finite Galois extension whose horizontal ramification indices divide ℓ.\nHere horizontal indices refer to ramification at closed points over ¯k. Section 4 proves this first\nstage.\nThe second stage produces roots of arbitrary positive order m while keeping the auxiliary\ncover’s horizontal indices dividing ℓ. We construct a cyclic cover of the projective line whose\nskeleton has two vertices and ℓparallel edges. A cycle using two distinct edges gives a character\nof the torus uniformizing its Jacobian. Along a chosen annulus that character has slope one,\nso it is a constant times the annular coordinate times an analytic unit close to one. Figure 1\nshows the geometry. This uses the nonarchimedean uniformization theory of Raynaud and\nBosch–Lütkebohmert [23, 6] and the compatibility of classical and tropical Abel–Jacobi maps\nproved by Baker–Rabinoff [3, Proposition 6.1 and Corollary 6.6].\nPullback by multiplication by m on the Jacobian supplies a root of the character on the\nwhole inverse annulus. On a sufficiently long middle collar, the remaining unit has an m-th root\nby the p-adic logarithm and exponential. This yields a root of the coordinate itself. The auxiliary\ncovers still have horizontal indices dividing ℓ, so the first stage places their function fields in\nH. Section 5 constructs these roots, and a second application of patching in Section 6 realizes\narbitrary finite Galois extensions in the same completion. The common requirement of roots on\nevery inverse-image sheet is what allows both stages to use the same patching proposition."
        },
        {
          "blockId": "p4-526c151cc2",
          "page": 4,
          "quote": "The local theorem also identifies the possible localizations of global sections. Let C/F be a\nsmooth proper geometrically connected curve of genus at least two over a number field. A global\nsection determines a point of C(Fv) at every finite place by Theorem 1.1, and a connected\ncomponent of C(Fv) at every real place by the real section theorem [8, Introduction, Theorem].\nTogether these form a modified adelic point; complex places contribute no factor. The finite-cover\ndescent locus consists of the tuples that lift to some global twist of each torsor under a finite\nF-group scheme. Proposition 8.1 shows that this locus is exactly the image of global sections\nunder localization. Both inclusions use the descent bridge of Harari–Stix [11, preprint version,\nTheorem 2.1 and Remark 2.2(1)].\nWhen this finite-cover locus consists exactly of the diagonal rational points, Stix’s finite-\nsupport theorem identifies every global section with a point section [25, arXiv version, Corollary 5].\nThe resulting bijection is Theorem 8.2. Stoll’s finite-descent criteria supply the required equality\nwhen the curve has a nonconstant morphism over its number field to an abelian variety with\nfinitely many rational points and trivial divisible Tate–Shafarevich subgroup, or when its Jacobian\nhas these properties. They also cover the smooth projective modular curves X0(N) and X1(N)\nof genus at least two over Q [26, Corollary 8.1, Theorem 8.6, and Corollary 8.8], with the\ncorrections in [27]. The Selmer converse in the companion article [18, Theorem 1.1] gives a\nfurther arithmetic test for an elliptic target over Q: vanishing of its full p-power Selmer corank\nat one prime. Section 8 states the precise hypotheses and deductions for these cases.\nThe localization description also yields constraints without assuming that the finite-cover\nlocus consists of rational points. Every ordinary adelic representative of a localization tuple\nsurvives linear descent and lies in the Brauer–Manin set, by further results of Harari–Stix [11,"
        },
        {
          "blockId": "p5-d4b9ec4b22",
          "page": 5,
          "quote": "preprint version, Corollary 3.1 and Theorem 4.1]. If F contains no CM subfield, Betts–Stix’s\ntheorem makes the projection of the localization image to C(Fv) finite at every finite place v [4,\npreprint version, Theorem A]. These statements constrain the localization image; the finite-cover\nequality remains a separate arithmetic hypothesis in the global section-conjecture criterion."
        },
        {
          "blockId": "p8-c97559e01e",
          "page": 8,
          "quote": "Proof. All valuations in this statement are the horizontal, discrete closed-point valuations,\nwhich are trivial on ¯k. Their residue characteristic is zero. The completed base field at such\na point is A = ¯k((t)). Every finite extension of A has residue field ¯k and is of the form ¯k((u))\nfor a uniformizer u: successive subtraction of residue representatives gives the power-series\nexpansion in its complete valuation ring. If its ramification index is e, then t = uea(u) for a unit\na(u) ∈¯k[[u]]×. The unit has an e-th root in ¯k[[u]], by Hensel’s lemma in residue characteristic\nzero. Changing the parameter gives t = ze. Thus, inside a fixed algebraic closure, the local field\nis A(t1/e), which is cyclic over A since ¯k contains all roots of unity.\nFor e | n, this is a subfield of A(t1/n). All local factors of a finite compositum, or of a normal\nclosure formed from conjugates, therefore remain subfields of this same cyclic extension. This\nproves the assertions about subextensions, composita, and Galois closures.\nFor pullback, write the local base map as t = sra(s), where r ≥1 and a(s) is a unit.\nExtracting its e-th root in ¯k[[s]] shows that every normalized local factor after pullback has\nramification index e/ gcd(e, r), which divides e. Finally, ramification indices multiply in a tower,\nand an étale upper cover contributes index 1. Apply the Galois-closure assertion to a cover\nwith indices dividing ℓand embed that closure into the fixed Ksep. Its field lies in the defining\ncompositum Lℓ."
        },
        {
          "blockId": "p8-62d6c4abb0",
          "page": 8,
          "quote": "Proof. On a circle |S| = ρ of radius ρ ∈|Ω×|, the Laurent Gauss norm is multiplicative.\nNormalize u and its inverse to have norm one. Their reductions are inverse units in eΩ[s, s−1], so"
        },
        {
          "blockId": "p9-7df61f8099",
          "page": 9,
          "quote": "the reduction of u is a monomial. Thus exactly one Laurent term dominates at that radius. Its\nexponent cannot change inside the radius interval. Indeed, convergence on compact subannuli\nreduces any proposed change to finitely many competing terms; a tie between two terms occurs\nat a radius in |Ω×|, since the additive value group is Q. Such a tie contradicts the preceding\nreduction argument. A single term cSd therefore dominates throughout A, giving the asserted\nfactorization. The same argument on closed subdisks uses that the only units in eΩ[s] are\nconstants and proves the disk assertion. Its constant norm belongs to |Ω×| by evaluation at a\nrational point of the disk.\nSince ℓis a unit in the valuation ring, the binomial series for (1 + h)1/ℓconverges on every\ncompact subannulus; the resulting functions agree on overlaps. A root of c lies in Ω, and a root\nof Sd follows from a root of S, also when d < 0."
        },
        {
          "blockId": "p10-162d42f985",
          "page": 10,
          "quote": "The two extra terms are precisely those recorded by the two circuit edges. Thus the augmented\ngraph ramification is zero at every vertex; the identity copies contribute none. All ramification\nindices are prime to p. The resulting tame covering is connected, because each vertex outside\nthe circuit can be joined by a lifted path to a circuit vertex.\nLift these data by the cited theorem. The inverse image of A is exactly the disjoint union of the\nnode annuli above e. For completeness, the inverse-skeleton equality excludes off-skeleton points\nmapping to the skeleton. Each off-skeleton open ball therefore maps into a single off-skeleton\nball downstairs, and continuity identifies the images of their attachment points. Retractions\ncommute on these balls as well as on the skeleton. Since a node annulus is the inverse image of\nits open edge under retraction, this proves the asserted equality for the whole annulus. On each\nannulus above A, the pullback of S has exponent ℓor −ℓin Lemma 4.1. It therefore has an\nanalytic ℓ-th root on each of these disjoint annuli.\nNow suppose that e is separating. Each side of the graph obtained by cutting its interior\ncontains a circuit or a positive-genus vertex. Otherwise that side would be a genus-zero tree\nwith, say, N vertices. Its total valence, including the bridge half-edge, would be 2N −1, contrary\nto stability, which requires at least 3N.\nConstruct a connected unramified degree-ℓcovering of each side, retaining its boundary\nmark for the eventual gluing. If the side contains a circuit, take a connected cyclic graph\ncovering of degree ℓ, with identity residue-curve maps. Such a graph covering is obtained by a\nhomomorphism of its graph fundamental group onto Z/ℓZ. If instead a positive-genus vertex is\nused, take a connected cyclic étale cover of degree ℓof its residue curve, provided by a nonzero\nelement of its Jacobian’s ℓ-torsion [16, Theorem 8.2 and Remark 8.4]. Explicitly, for a line bundle\nL of exact order ℓ, a trivialization L⊗ℓ≃O makes Lℓ−1\nj=0 L⊗j an algebra whose relative spectrum\nis a cyclic étale cover: ℓis invertible, and the exact order of L makes the cover connected. Use ℓ\nidentity copies at the other vertices and match all marked fibres. Paths to the distinguished\nvertex show connectedness of the covered side.\nEach covered side has ℓpreimages of the bridge mark. Match them in pairs to obtain ℓ\ndistinct bridge edges, all with factor one. Every such edge is now nonseparating: after removing\none of them, the two connected covered sides are still joined by the others. The resulting covering\nof the original metrized complex is tame. Indeed the component covers are unramified, all\nfactors are one, and Riemann–Hurwitz again gives zero graph ramification. Lift it to a finite\nétale curve cover. Its triangulation remains stable: at a vertex of local degree d, the quantity\n2g′ −2 + val(v′) is d(2g −2 + val(v)) > 0.\nApply the nonseparating construction to each of these ℓedges. A connected component of\nthe fibre product of the resulting finite étale covers dominates each factor, so all the constructed\nroots persist on a common cover. On each factor-one annulus the pullback of the original S is a\nunit; Lemma 4.1 supplies its root once the new annular coordinate has a root. This proves the\nassertion also for separating edges. Finally, the same finite fibre-product argument treats finitely\nmany original annuli simultaneously."
        },
        {
          "blockId": "p12-d94954e00c",
          "page": 12,
          "quote": "The first stage used roots of order ℓ, a prime different from p. To treat arbitrary ramification, we\nnow construct an auxiliary branched cover on which a coordinate has a root of any prescribed\npositive order. Its own horizontal ramification indices still divide ℓ. The construction uses a\ncurve whose stable skeleton has parallel edges and multiplication on its Jacobian. Throughout\nthis section, v = −logp | · |.\nWe first isolate the step that must work on every sheet. For an analytic map with a specified\nlift to a uniformizing torus, pullback by multiplication by m extracts m-th roots of the lift’s\ncharacter values. The period lattice controls how local roots glue."
        },
        {
          "blockId": "p14-3a4f58d757",
          "page": 14,
          "quote": "Writing s = v(T), the ultrametric inequality consequently gives"
        },
        {
          "blockId": "p14-dc7170c439",
          "page": 14,
          "quote": "|hj| ≤p−min{s,a−s} < p−D\non Aj ∩{D < s < a −D}."
        },
        {
          "blockId": "p16-60b3674046",
          "page": 16,
          "quote": "We now use Theorem 1.2 to compare decomposition groups. The comparison lifts any section\nlocalized at a rank-one valuation to a birational section. Koenigsmann’s theorem then places\nthe same section at a rational point, and the two valuations give a contradiction. The remaining\nvaluations are horizontal refinements, where the localization theorem of Pop–Stix already\nidentifies the section."
        },
        {
          "blockId": "p17-832d186942",
          "page": 17,
          "quote": "For sufficiently large r, Hensel’s lemma applied at 1 gives a square root of qr/d using v1, and a\nsquare root of qr using v2. The sufficient inequalities are"
        },
        {
          "blockId": "p17-6aa1f30eeb",
          "page": 17,
          "quote": "Proof of the surjectivity in Theorem 1.1. Let s : Gk →ΠX be a continuous group-homomorphic\nsection. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26],\nthere is a valuation w of K extending the p-adic valuation, together with a prolongation ew to\nf\nK, such that\ns(Gk) ⊆De\nw(f\nK/K)."
        },
        {
          "blockId": "p17-7b7f5a8a8a",
          "page": 17,
          "quote": "Since Γk has rational rank one and ordered rank is at most rational rank, w has rank at most\ntwo. Rank one is excluded by Lemma 7.2. If w has rank two, let u be its rank-one coarsening.\nThe nonzero proper convex subgroup of the rank-two value group has a unique extension to\nthe value group of ew: it consists of those values some positive multiple of which lies in the\noriginal subgroup. This follows because an algebraic valued extension has torsion quotient of\nvalue groups. Every automorphism stabilizing ew preserves this subgroup, so s(Gk) stabilizes the\nresulting prolongation of u. If u is nontrivial on k, it can be normalized to extend the p-adic\nvaluation and is excluded by the same lemma."
        },
        {
          "blockId": "p18-9a49bae2ba",
          "page": 18,
          "quote": "It remains that u is trivial on k. Properness gives it a center on X. This center cannot be\ngeneric because u is nontrivial, so it is a closed point a. The valuation ring of u dominates the\ndiscrete valuation ring OX,a. Every element of K× is a power of a uniformizer times a unit of\nthis local ring; hence u is equivalent to orda. Its residue field is the finite extension k(a)/k. The\nsecond valuation in the composition w is the unique prolongation of the p-adic valuation to k(a).\nThus w is of type 2h.\nThe chosen horizontal prolongation to K¯k selects a geometric point above a. Its stabilizer\nacts on constants through Gk(a) ⊆Gk, with the embedding determined by that point. Since\ns(Gk) projects onto Gk, necessarily k(a) = k. For the full étale tower the horizontal inertia\nis trivial, and the horizontal decomposition group is the point section at a. Equivalently, the\ntype-2h conclusion of [22, preprint version, Section 6.1] gives s = sa after the corresponding\ngeometric choice of base point. This proves surjectivity.\nNotice that a type-2h valuation can lie over a non-rational closed point. Rationality here\nfollows from the surjection to Gk; no unrestricted equivalence between “not type 2h” and “not\noriginating from a rational point” is used."
        }
      ],
      "occurrenceCount": 34
    },
    {
      "symbol": "t",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p8-c97559e01e",
        "page": 8,
        "quote": "Proof. All valuations in this statement are the horizontal, discrete closed-point valuations,\nwhich are trivial on ¯k. Their residue characteristic is zero. The completed base field at such\na point is A = ¯k((t)). Every finite extension of A has residue field ¯k and is of the form ¯k((u))\nfor a uniformizer u: successive subtraction of residue representatives gives the power-series\nexpansion in its complete valuation ring. If its ramification index is e, then t = uea(u) for a unit\na(u) ∈¯k[[u]]×. The unit has an e-th root in ¯k[[u]], by Hensel’s lemma in residue characteristic\nzero. Changing the parameter gives t = ze. Thus, inside a fixed algebraic closure, the local field\nis A(t1/e), which is cyclic over A since ¯k contains all roots of unity.\nFor e | n, this is a subfield of A(t1/n). All local factors of a finite compositum, or of a normal\nclosure formed from conjugates, therefore remain subfields of this same cyclic extension. This\nproves the assertions about subextensions, composita, and Galois closures.\nFor pullback, write the local base map as t = sra(s), where r ≥1 and a(s) is a unit.\nExtracting its e-th root in ¯k[[s]] shows that every normalized local factor after pullback has\nramification index e/ gcd(e, r), which divides e. Finally, ramification indices multiply in a tower,\nand an étale upper cover contributes index 1. Apply the Galois-closure assertion to a cover\nwith indices dividing ℓand embed that closure into the fixed Ksep. Its field lies in the defining\ncompositum Lℓ."
      },
      "occurrences": [
        {
          "blockId": "p3-1d11e06d43",
          "page": 3,
          "quote": "The geometric content lies in Theorem 1.2. A finite extension of ¯k(X) is the function field of a\nfinite curve cover, which may have branch points. Near such a point, with local coordinate t, a\nbranch has the form t = ze. If another cover supplies a root of t of order divisible by all these\nbranch indices, the pulled-back cover splits over an annulus around the branch point. One can\nthen replace the part over the inner disk by a trivial cover and glue to obtain an étale cover.\nThe roots must exist on the entire inverse image of the annulus: a root on only one sheet does\nnot trivialize the full cover needed for gluing. We use this operation twice, with different sources\nof collar roots.\nThis gluing and algebraization method belongs to the formal and rigid analytic patching\ntradition of Harbater and Liu [12, 15]. Poineau treats the corresponding Berkovich framework\n[20, Section 1 and Appendix A]. Section 3 proves the precise replacement and field-embedding\nstatements used here. In particular, evaluating the resulting exterior analytic map retains the\nspecified valued field embedding; the map itself need not descend to ¯k.\nFix an odd prime ℓ̸= p. In the first stage, coordinates on the node annuli of a stable model\nacquire ℓ-th roots on suitable finite étale covers. The construction uses the lifting theorem"
        },
        {
          "blockId": "p8-c97559e01e",
          "page": 8,
          "quote": "Proof. All valuations in this statement are the horizontal, discrete closed-point valuations,\nwhich are trivial on ¯k. Their residue characteristic is zero. The completed base field at such\na point is A = ¯k((t)). Every finite extension of A has residue field ¯k and is of the form ¯k((u))\nfor a uniformizer u: successive subtraction of residue representatives gives the power-series\nexpansion in its complete valuation ring. If its ramification index is e, then t = uea(u) for a unit\na(u) ∈¯k[[u]]×. The unit has an e-th root in ¯k[[u]], by Hensel’s lemma in residue characteristic\nzero. Changing the parameter gives t = ze. Thus, inside a fixed algebraic closure, the local field\nis A(t1/e), which is cyclic over A since ¯k contains all roots of unity.\nFor e | n, this is a subfield of A(t1/n). All local factors of a finite compositum, or of a normal\nclosure formed from conjugates, therefore remain subfields of this same cyclic extension. This\nproves the assertions about subextensions, composita, and Galois closures.\nFor pullback, write the local base map as t = sra(s), where r ≥1 and a(s) is a unit.\nExtracting its e-th root in ¯k[[s]] shows that every normalized local factor after pullback has\nramification index e/ gcd(e, r), which divides e. Finally, ramification indices multiply in a tower,\nand an étale upper cover contributes index 1. Apply the Galois-closure assertion to a cover\nwith indices dividing ℓand embed that closure into the fixed Ksep. Its field lies in the defining\ncompositum Lℓ."
        },
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        },
        {
          "blockId": "p17-52809f792f",
          "page": 17,
          "quote": "so these valuations are inequivalent. The resulting contradiction is an instance of the classical\nuniqueness principle for henselian valuations associated with F. K. Schmidt [24]. We give\nexplicitly the two-valuation approximation needed here.\nChoose a nonsquare d ∈k×. It remains a nonsquare in M: otherwise GM = Σ would fix\na square root of d in ¯k, contrary to the surjection Σ →Gk. Let t ∈K be a uniformizer at a.\nChoose a positive integer N with N > v1(t), and put z = pN/t. Then"
        },
        {
          "blockId": "p17-4291df9365",
          "page": 17,
          "quote": "v1(z) > 0,\nv2(z) = −v2(t) < 0."
        }
      ],
      "occurrenceCount": 5
    },
    {
      "symbol": "u",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p6-0f4a412be2",
        "page": 6,
        "quote": "Here it is enough to make the valuation of a(u)/a(0) −1 greater than vp(ec) + 1/(p −1). Thus\nzc = u a(u)1/ec is analytic with a nonzero derivative at the center. Analytic inversion gives a\nsmaller disk on which it is a coordinate and tb = zec\nc .\nDo this in disjoint neighborhoods of all the finitely many points above b. The complement of\nthese neighborhoods in the proper analytic curve Can\nΩis compact. Its image under f is compact,\nclosed, and does not contain b. Consequently a sufficiently small disk about b has its entire\ninverse image in the chosen neighborhoods. Shrink it once more so that, in every zc-chart, that\ninverse image is the disk defined by the corresponding bound on |zc|ec.\nThe point ξ does not equal any point of X(¯k), since its projection to X¯k is generic. The\nshrinking disks at b form a neighborhood basis there. Hausdorffness therefore permits all disks\nto avoid ξ and one another. We may choose their radii in |Ω×|. The argument also applies when\nξ is an Ω-rational point whose projection to X¯k is generic."
      },
      "occurrences": [
        {
          "blockId": "p6-af5d5cc85a",
          "page": 6,
          "quote": "Proof. Choose a local analytic parameter u at c. The pullback of tb is ueca(u) with a(0) ̸= 0.\nAfter choosing an ec-th root of a(0) in Ωand shrinking the disk, a(u)/a(0) is sufficiently close to\n1 for its ec-th root to be given by"
        },
        {
          "blockId": "p6-a82a52e120",
          "page": 6,
          "quote": "exp\n\ne−1\nc\nlog(a(u)/a(0))\n\n."
        },
        {
          "blockId": "p6-0f4a412be2",
          "page": 6,
          "quote": "Here it is enough to make the valuation of a(u)/a(0) −1 greater than vp(ec) + 1/(p −1). Thus\nzc = u a(u)1/ec is analytic with a nonzero derivative at the center. Analytic inversion gives a\nsmaller disk on which it is a coordinate and tb = zec\nc .\nDo this in disjoint neighborhoods of all the finitely many points above b. The complement of\nthese neighborhoods in the proper analytic curve Can\nΩis compact. Its image under f is compact,\nclosed, and does not contain b. Consequently a sufficiently small disk about b has its entire\ninverse image in the chosen neighborhoods. Shrink it once more so that, in every zc-chart, that\ninverse image is the disk defined by the corresponding bound on |zc|ec.\nThe point ξ does not equal any point of X(¯k), since its projection to X¯k is generic. The\nshrinking disks at b form a neighborhood basis there. Hausdorffness therefore permits all disks\nto avoid ξ and one another. We may choose their radii in |Ω×|. The argument also applies when\nξ is an Ω-rational point whose projection to X¯k is generic."
        },
        {
          "blockId": "p8-c97559e01e",
          "page": 8,
          "quote": "Proof. All valuations in this statement are the horizontal, discrete closed-point valuations,\nwhich are trivial on ¯k. Their residue characteristic is zero. The completed base field at such\na point is A = ¯k((t)). Every finite extension of A has residue field ¯k and is of the form ¯k((u))\nfor a uniformizer u: successive subtraction of residue representatives gives the power-series\nexpansion in its complete valuation ring. If its ramification index is e, then t = uea(u) for a unit\na(u) ∈¯k[[u]]×. The unit has an e-th root in ¯k[[u]], by Hensel’s lemma in residue characteristic\nzero. Changing the parameter gives t = ze. Thus, inside a fixed algebraic closure, the local field\nis A(t1/e), which is cyclic over A since ¯k contains all roots of unity.\nFor e | n, this is a subfield of A(t1/n). All local factors of a finite compositum, or of a normal\nclosure formed from conjugates, therefore remain subfields of this same cyclic extension. This\nproves the assertions about subextensions, composita, and Galois closures.\nFor pullback, write the local base map as t = sra(s), where r ≥1 and a(s) is a unit.\nExtracting its e-th root in ¯k[[s]] shows that every normalized local factor after pullback has\nramification index e/ gcd(e, r), which divides e. Finally, ramification indices multiply in a tower,\nand an étale upper cover contributes index 1. Apply the Galois-closure assertion to a cover\nwith indices dividing ℓand embed that closure into the fixed Ksep. Its field lies in the defining\ncompositum Lℓ."
        },
        {
          "blockId": "p8-36813d5eeb",
          "page": 8,
          "quote": "Lemma 4.1 (Units on disks and annuli). An analytic unit on an open disk over Ωhas constant\nabsolute value in |Ω×|. If A = {r < |S| < R} is an open annulus and u ∈O(A)×, then"
        },
        {
          "blockId": "p8-6b360aa50e",
          "page": 8,
          "quote": "u = cSd(1 + h),\nc ∈Ω×,\nd ∈Z,\nh ∈O(A),\n|h(x)| < 1\n(x ∈A),"
        },
        {
          "blockId": "p8-fcb18b4841",
          "page": 8,
          "quote": "where cSd is a term of the Laurent expansion of u. The function 1 + h has an analytic ℓ-th root\non A. Consequently, on any analytic cover of A on which S has an ℓ-th root, so does u."
        },
        {
          "blockId": "p8-62d6c4abb0",
          "page": 8,
          "quote": "Proof. On a circle |S| = ρ of radius ρ ∈|Ω×|, the Laurent Gauss norm is multiplicative.\nNormalize u and its inverse to have norm one. Their reductions are inverse units in eΩ[s, s−1], so"
        },
        {
          "blockId": "p9-7df61f8099",
          "page": 9,
          "quote": "the reduction of u is a monomial. Thus exactly one Laurent term dominates at that radius. Its\nexponent cannot change inside the radius interval. Indeed, convergence on compact subannuli\nreduces any proposed change to finitely many competing terms; a tie between two terms occurs\nat a radius in |Ω×|, since the additive value group is Q. Such a tie contradicts the preceding\nreduction argument. A single term cSd therefore dominates throughout A, giving the asserted\nfactorization. The same argument on closed subdisks uses that the only units in eΩ[s] are\nconstants and proves the disk assertion. Its constant norm belongs to |Ω×| by evaluation at a\nrational point of the disk.\nSince ℓis a unit in the valuation ring, the binomial series for (1 + h)1/ℓconverges on every\ncompact subannulus; the resulting functions agree on overlaps. A root of c lies in Ω, and a root\nof Sd follows from a root of S, also when d < 0."
        },
        {
          "blockId": "p17-7b7f5a8a8a",
          "page": 17,
          "quote": "Since Γk has rational rank one and ordered rank is at most rational rank, w has rank at most\ntwo. Rank one is excluded by Lemma 7.2. If w has rank two, let u be its rank-one coarsening.\nThe nonzero proper convex subgroup of the rank-two value group has a unique extension to\nthe value group of ew: it consists of those values some positive multiple of which lies in the\noriginal subgroup. This follows because an algebraic valued extension has torsion quotient of\nvalue groups. Every automorphism stabilizing ew preserves this subgroup, so s(Gk) stabilizes the\nresulting prolongation of u. If u is nontrivial on k, it can be normalized to extend the p-adic\nvaluation and is excluded by the same lemma."
        },
        {
          "blockId": "p18-9a49bae2ba",
          "page": 18,
          "quote": "It remains that u is trivial on k. Properness gives it a center on X. This center cannot be\ngeneric because u is nontrivial, so it is a closed point a. The valuation ring of u dominates the\ndiscrete valuation ring OX,a. Every element of K× is a power of a uniformizer times a unit of\nthis local ring; hence u is equivalent to orda. Its residue field is the finite extension k(a)/k. The\nsecond valuation in the composition w is the unique prolongation of the p-adic valuation to k(a).\nThus w is of type 2h.\nThe chosen horizontal prolongation to K¯k selects a geometric point above a. Its stabilizer\nacts on constants through Gk(a) ⊆Gk, with the embedding determined by that point. Since\ns(Gk) projects onto Gk, necessarily k(a) = k. For the full étale tower the horizontal inertia\nis trivial, and the horizontal decomposition group is the point section at a. Equivalently, the\ntype-2h conclusion of [22, preprint version, Section 6.1] gives s = sa after the corresponding\ngeometric choice of base point. This proves surjectivity.\nNotice that a type-2h valuation can lie over a non-rational closed point. Rationality here\nfollows from the surjection to Gk; no unrestricted equivalence between “not type 2h” and “not\noriginating from a rational point” is used."
        }
      ],
      "occurrenceCount": 11
    },
    {
      "symbol": "w",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p16-d357ebb858",
        "page": 16,
        "quote": "Corollary 7.1. Let k/Qp be finite and let X/k be a smooth proper geometrically connected\ncurve of genus at least two. Put K = k(X) and let f\nK ⊂Ksep be the extension corresponding to\nthe full arithmetic étale fundamental group of X. Let w be a real-valued rank-one valuation of\nK extending the p-adic valuation, normalized by w(p) = 1, and fix a prolongation wsep to Ksep.\nRestriction induces an isomorphism of profinite groups"
      },
      "occurrences": [
        {
          "blockId": "p16-d357ebb858",
          "page": 16,
          "quote": "Corollary 7.1. Let k/Qp be finite and let X/k be a smooth proper geometrically connected\ncurve of genus at least two. Put K = k(X) and let f\nK ⊂Ksep be the extension corresponding to\nthe full arithmetic étale fundamental group of X. Let w be a real-valued rank-one valuation of\nK extending the p-adic valuation, normalized by w(p) = 1, and fix a prolongation wsep to Ksep.\nRestriction induces an isomorphism of profinite groups"
        },
        {
          "blockId": "p16-0a9614cda9",
          "page": 16,
          "quote": "Dwsep(Ksep/K)\n∼\n−−→De\nw(f\nK/K),\new = wsep| e\nK."
        },
        {
          "blockId": "p16-f889b64629",
          "page": 16,
          "quote": "Lemma 7.2. With k, X, K, and f\nK as in Corollary 7.1, let w be any rank-one valuation of K\nextending the p-adic valuation and let ew be any prolongation to f\nK. The projection"
        },
        {
          "blockId": "p16-1e032d255e",
          "page": 16,
          "quote": "De\nw(f\nK/K) −→Gk"
        },
        {
          "blockId": "p16-4611b28177",
          "page": 16,
          "quote": "Proof. Normalize w(p) = 1 and prolong the valuation to Ksep. Suppose that a section exists.\nCorollary 7.1 lifts it to a continuous section"
        },
        {
          "blockId": "p17-6aa1f30eeb",
          "page": 17,
          "quote": "Proof of the surjectivity in Theorem 1.1. Let s : Gk →ΠX be a continuous group-homomorphic\nsection. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26],\nthere is a valuation w of K extending the p-adic valuation, together with a prolongation ew to\nf\nK, such that\ns(Gk) ⊆De\nw(f\nK/K)."
        },
        {
          "blockId": "p17-26f38acfac",
          "page": 17,
          "quote": "We classify the possibilities for w.\nThe valuation transcendence inequality gives"
        },
        {
          "blockId": "p17-9b0b095290",
          "page": 17,
          "quote": "ratrank(Γw/Γk) ≤1."
        },
        {
          "blockId": "p17-7b7f5a8a8a",
          "page": 17,
          "quote": "Since Γk has rational rank one and ordered rank is at most rational rank, w has rank at most\ntwo. Rank one is excluded by Lemma 7.2. If w has rank two, let u be its rank-one coarsening.\nThe nonzero proper convex subgroup of the rank-two value group has a unique extension to\nthe value group of ew: it consists of those values some positive multiple of which lies in the\noriginal subgroup. This follows because an algebraic valued extension has torsion quotient of\nvalue groups. Every automorphism stabilizing ew preserves this subgroup, so s(Gk) stabilizes the\nresulting prolongation of u. If u is nontrivial on k, it can be normalized to extend the p-adic\nvaluation and is excluded by the same lemma."
        },
        {
          "blockId": "p18-9a49bae2ba",
          "page": 18,
          "quote": "It remains that u is trivial on k. Properness gives it a center on X. This center cannot be\ngeneric because u is nontrivial, so it is a closed point a. The valuation ring of u dominates the\ndiscrete valuation ring OX,a. Every element of K× is a power of a uniformizer times a unit of\nthis local ring; hence u is equivalent to orda. Its residue field is the finite extension k(a)/k. The\nsecond valuation in the composition w is the unique prolongation of the p-adic valuation to k(a).\nThus w is of type 2h.\nThe chosen horizontal prolongation to K¯k selects a geometric point above a. Its stabilizer\nacts on constants through Gk(a) ⊆Gk, with the embedding determined by that point. Since\ns(Gk) projects onto Gk, necessarily k(a) = k. For the full étale tower the horizontal inertia\nis trivial, and the horizontal decomposition group is the point section at a. Equivalently, the\ntype-2h conclusion of [22, preprint version, Section 6.1] gives s = sa after the corresponding\ngeometric choice of base point. This proves surjectivity.\nNotice that a type-2h valuation can lie over a non-rational closed point. Rationality here\nfollows from the surjection to Gk; no unrestricted equivalence between “not type 2h” and “not\noriginating from a rational point” is used."
        }
      ],
      "occurrenceCount": 10
    },
    {
      "symbol": "z",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p17-52809f792f",
        "page": 17,
        "quote": "so these valuations are inequivalent. The resulting contradiction is an instance of the classical\nuniqueness principle for henselian valuations associated with F. K. Schmidt [24]. We give\nexplicitly the two-valuation approximation needed here.\nChoose a nonsquare d ∈k×. It remains a nonsquare in M: otherwise GM = Σ would fix\na square root of d in ¯k, contrary to the surjection Σ →Gk. Let t ∈K be a uniformizer at a.\nChoose a positive integer N with N > v1(t), and put z = pN/t. Then"
      },
      "occurrences": [
        {
          "blockId": "p17-52809f792f",
          "page": 17,
          "quote": "so these valuations are inequivalent. The resulting contradiction is an instance of the classical\nuniqueness principle for henselian valuations associated with F. K. Schmidt [24]. We give\nexplicitly the two-valuation approximation needed here.\nChoose a nonsquare d ∈k×. It remains a nonsquare in M: otherwise GM = Σ would fix\na square root of d in ¯k, contrary to the surjection Σ →Gk. Let t ∈K be a uniformizer at a.\nChoose a positive integer N with N > v1(t), and put z = pN/t. Then"
        },
        {
          "blockId": "p17-4291df9365",
          "page": 17,
          "quote": "v1(z) > 0,\nv2(z) = −v2(t) < 0."
        },
        {
          "blockId": "p17-7b97baa8f2",
          "page": 17,
          "quote": "v1(qr/d −1) = v1((1 −d)/d) + rv1(z) −→+∞,"
        },
        {
          "blockId": "p17-3d516e1394",
          "page": 17,
          "quote": "whereas\nv2(qr −1) = v2(d −1) −rv2(z) −→+∞."
        }
      ],
      "occurrenceCount": 4
    },
    {
      "symbol": "Γ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p13-3b2bd6a638",
        "page": 13,
        "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
      },
      "occurrences": [
        {
          "blockId": "p13-3b2bd6a638",
          "page": 13,
          "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
        },
        {
          "blockId": "p13-c6ed4239e0",
          "page": 13,
          "quote": "Its character lattice is M = H1(Γ, Z); its tropical target is Hom(M, R). The period lattice is\nfree, and q is a topological covering and local analytic isomorphism. These are the Jacobian\nuniformization conventions of [3, Sections 4.1–4.2 and 4.5.2]. The compatibility of the classical\nand tropical Abel–Jacobi maps [3, Proposition 6.1 and Corollary 6.6] identifies the latter with\nintegration on Γ using the edge-length pairing.\nThe open Berkovich annulus Aj strongly deformation retracts to its open skeletal interval\n[2, Section 2.3] and is locally path connected. It is thus simply connected, so i|Aj lifts to an"
        }
      ],
      "occurrenceCount": 2
    },
    {
      "symbol": "Λ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p12-8b3352aee6",
        "page": 12,
        "quote": "Lemma 5.1 (Roots after division on a totally degenerate abelian variety). Let J be an abelian\nvariety over Ωwith analytic uniformization q : Tan →Jan, where T is a split torus and the\nkernel Λ ⊂T(Ω) is a free lattice acting by translation. Suppose q is a topological covering and\na local analytic isomorphism. Let A be an analytic space, let a : A →Tan be analytic, and let\nm ≥1 be an integer. Set\nB = A ×q◦a,Jan,[m] Jan."
      },
      "occurrences": [
        {
          "blockId": "p12-8b3352aee6",
          "page": 12,
          "quote": "Lemma 5.1 (Roots after division on a totally degenerate abelian variety). Let J be an abelian\nvariety over Ωwith analytic uniformization q : Tan →Jan, where T is a split torus and the\nkernel Λ ⊂T(Ω) is a free lattice acting by translation. Suppose q is a topological covering and\na local analytic isomorphism. Let A be an analytic space, let a : A →Tan be analytic, and let\nm ≥1 be an integer. Set\nB = A ×q◦a,Jan,[m] Jan."
        },
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        },
        {
          "blockId": "p13-abfe83435d",
          "page": 13,
          "quote": "q : Tan −→Jan\nΩ,\nker(q) = Λ ⊂T(Ω)."
        }
      ],
      "occurrenceCount": 3
    },
    {
      "symbol": "Σ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p17-cd9725f405",
        "page": 17,
        "quote": "Write Σ = bs(Gk) and M = (Ksep)Σ. The subgroup Σ is compact and hence closed, so\nGM = Σ. Let v1 and v2 be the restrictions to M of the two selected prolongations. Each is\nhenselian: its absolute Galois group stabilizes a prolongation, and the transitive action on all\nprolongations therefore has a single point. Both valuations are nontrivial and have rank one.\nIndeed, value groups in an algebraic valued extension differ by a torsion quotient, which preserves\ntheir convex-subgroup chains. Moreover,"
      },
      "occurrences": [
        {
          "blockId": "p17-cd9725f405",
          "page": 17,
          "quote": "Write Σ = bs(Gk) and M = (Ksep)Σ. The subgroup Σ is compact and hence closed, so\nGM = Σ. Let v1 and v2 be the restrictions to M of the two selected prolongations. Each is\nhenselian: its absolute Galois group stabilizes a prolongation, and the transitive action on all\nprolongations therefore has a single point. Both valuations are nontrivial and have rank one.\nIndeed, value groups in an algebraic valued extension differ by a torsion quotient, which preserves\ntheir convex-subgroup chains. Moreover,"
        },
        {
          "blockId": "p17-52809f792f",
          "page": 17,
          "quote": "so these valuations are inequivalent. The resulting contradiction is an instance of the classical\nuniqueness principle for henselian valuations associated with F. K. Schmidt [24]. We give\nexplicitly the two-valuation approximation needed here.\nChoose a nonsquare d ∈k×. It remains a nonsquare in M: otherwise GM = Σ would fix\na square root of d in ¯k, contrary to the surjection Σ →Gk. Let t ∈K be a uniformizer at a.\nChoose a positive integer N with N > v1(t), and put z = pN/t. Then"
        }
      ],
      "occurrenceCount": 2
    },
    {
      "symbol": "α",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p13-2efce9f9ee",
        "page": 13,
        "quote": "and let α ∈k\n× satisfy a = v(α) > 2D. There is a smooth, proper, connected curve Q/k and a\nfinite map Q →P1"
      },
      "occurrences": [
        {
          "blockId": "p13-2efce9f9ee",
          "page": 13,
          "quote": "and let α ∈k\n× satisfy a = v(α) > 2D. There is a smooth, proper, connected curve Q/k and a\nfinite map Q →P1"
        },
        {
          "blockId": "p13-7b5280585c",
          "page": 13,
          "quote": "T −α ."
        },
        {
          "blockId": "p13-ef8b47c581",
          "page": 13,
          "quote": "The four distinct points 0, α, 1, ∞have orders 1, −1, 1, −1 in the divisor of the displayed rational\nfunction. The cyclic cover is connected, has degree ℓ, and is totally ramified at precisely these\npoints. Riemann–Hurwitz gives"
        },
        {
          "blockId": "p13-36e42bc923",
          "page": 13,
          "quote": "Here is a split semistable model of Z. Work first over the integers of a sufficiently large finite\nextension of k containing α. The blowup of the model of the line along (T, α) has two rational\ncomponents and one node, with local equation TU = α. The inner component has coordinate\nS = T/α; the four branch sections are disjoint and smooth. The Kummer function reduces\nto T −1 on the outer component and to −S/(S −1) on the inner component. Each of these\ndegree-ℓcovers of a rational component has just two totally ramified points and is rational. Near\nthe node the Kummer function is\nT −1\n1 −U ,"
        },
        {
          "blockId": "p13-3b2bd6a638",
          "page": 13,
          "quote": "a unit whose reduction at the node is −1. Since ℓis invertible, normalization is étale there,\nwith ℓsplit copies of the node after enlarging the finite constant field. Away from the node and\nbranch sections the same unit criterion gives étaleness. At a branch section the function or its\ninverse is a simple relative parameter, so the normalization is smooth there. Thus there are\nexactly two rational components, joined by ℓnodes, each of thickness v(α) = a. They are stable\nbecause ℓ≥3.\nConsequently the skeleton Γ consists of two vertices and ℓparallel edges e1, . . . , eℓ, each of\nlength a. The node annuli Aj are disjoint copies of {|α| < |T| < 1} with coordinate T. This last\nassertion can also be checked directly: on that annulus both T and α/T have absolute value\nless than one, and the two factors in −(1 −T)/(1 −α/T) have convergent ℓ-th roots. Hence the\ninverse image is exactly ℓcopies of the annulus.\nWe next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud\nand Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi\nmap i : Z →J using a point over k. Total degeneration gives a toric analytic uniformization"
        },
        {
          "blockId": "p14-c8ba572435",
          "page": 14,
          "quote": "Here c′\njT is the actual coefficient term of degree one. Dominance at every radius r ∈(|α|, 1)\ngives |hj,n|rn < 1. Taking limits at the endpoints yields, for n > 0,"
        },
        {
          "blockId": "p14-9b72f355e7",
          "page": 14,
          "quote": "|hj,n| ≤1,\n|hj,−n| ≤|α|n."
        }
      ],
      "occurrenceCount": 7
    },
    {
      "symbol": "ζ",
      "meaning": "No explicit definition located by the extractor.",
      "status": "not-located",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p7-15d8e7fa2f",
        "page": 7,
        "quote": "ζ∈µec\n(Z −ζac)."
      },
      "occurrences": [
        {
          "blockId": "p7-15d8e7fa2f",
          "page": 7,
          "quote": "ζ∈µec\n(Z −ζac)."
        },
        {
          "blockId": "p7-15cdd177d5",
          "page": 7,
          "quote": "Each ac is an analytic unit on the overlap. Distinct linear factors are comaximal, because\n(ζ −ζ′)ac is a unit. This is true also when p | ec: a nonzero constant in Ωis invertible, regardless\nof its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d\ncover, not just of one sheet.\nChoose these trivializations and glue the finite locally free analytic algebras. There is no\nfurther cocycle condition between different branch disks, as those disks are disjoint. The glued\nalgebra is coherent and finite étale over Y an, these being local properties. Proper analytic\nGAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and\nmultiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the\ncorresponding equivalence for finite covers of a proper curve.\nÉtaleness is detected after\nanalytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y .\nChoose any nonempty connected component and call it VΩ. A connected component of this\nfinite étale cover has positive locally constant degree over the connected curve Y , hence is still\nsurjective. It is smooth and proper. Restricting the original projection from the pullback over\nYU gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and\nthe same component argument applies without gluing."
        }
      ],
      "occurrenceCount": 2
    },
    {
      "symbol": "λ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p12-08e5a73bba",
        "page": 12,
        "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
      },
      "occurrences": [
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "ν",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p3-fe1bba6732",
        "page": 3,
        "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
      },
      "occurrences": [
        {
          "blockId": "p3-fe1bba6732",
          "page": 3,
          "quote": "The statement concerns the chosen copies of these fields inside one completion. Write\nDν(L/K) for the decomposition group of a valuation ν on a Galois extension L/K. Corollary 7.1\nuses Theorem 1.2 to identify the decomposition group in Ksep/K with the one in f\nK/K. Indeed,\nan automorphism in the kernel fixes a dense subfield of H and hence fixes Ksep.\nA section localized at a rank-one valuation therefore lifts to a birational section. Koenigs-\nmann’s theorem places this lifted section also at a rational point. Its fixed field would then\ncarry two inequivalent henselian rank-one valuations, one extending the p-adic valuation and the\nother trivial on constants. The contradiction is an explicit instance of the classical two-valuation\nprinciple associated with F. K. Schmidt [24]; Section 7 gives the needed approximation and\nsquare-root argument, including p = 2.\nRank-two valuations are handled by their rank-one coarsenings. The remaining case is a\nclosed-point valuation refined by the p-adic valuation on its residue field, called type 2h by\nPop–Stix. The surjection of the section onto Gk forces that closed point to be rational, and\ntheir type-2h analysis then identifies the section [22, preprint version, Section 6.1]. Section 7\ncompletes this argument and proves injectivity of the point-to-section map by Kummer theory\non the Jacobian."
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "ξ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-7f0c749ae4",
        "page": 5,
        "quote": "The inclusion of the completion H into E is isometric: it identifies H with the closure of f\nK in\nE. Since k is complete, the restriction of the chosen norm to ¯k is its unique prolongation. Thus\nΩ:= b¯k is a specified subfield of H. It is algebraically closed and has additive value group Q,\nwith the normalization v = −logp | · |. The equality of value groups before and after completion\nfollows because every nonzero limit has the same norm as sufficiently close approximants.\nWe use Berkovich analytifications over Ω. A field embedding K0 ,→H determines a point of\nX¯k(H) supported at the generic point of X¯k. Together with the specified embedding Ω,→H,\nit determines a point of XΩ(H), whose analytic image we denote by ξ. Its projection to X¯k is\ngeneric; its support on XΩneed not be generic. In particular, ξ can be an Ω-rational point. We\nretain the whole H-valued point, including the embedding of its completed residue field into H,\nwhenever we use ξ.\nIf F/K0 is finite, its normalization gives a smooth proper connected curve over ¯k with a finite\nmap to X¯k. We call the ramification indices of this map at closed points horizontal ramification\nindices. The corresponding discrete valuations are trivial on ¯k and have residue characteristic\nzero. These indices are unrelated to possible wild ramification on a characteristic-p reduction\nmodel.\nFix an odd prime ℓ̸= p. Inside Ksep, let Lℓbe the compositum of all finite Galois extensions\nF/K0 whose horizontal ramification indices divide ℓ. The unramified extensions are included.\nIn particular f\nK ⊂Lℓ: the universal étale field over K0 is the same field f\nK after constants are\nincluded. One may see this by descending any finite étale cover of X¯k to a finite extension of k\nand then composing with the finite étale constant-field cover of X.\nThe field-theoretic goal can be displayed as follows. The top row consists of the defining\ninclusions; the two right vertical arrows are the containments proved in the successive stages."
      },
      "occurrences": [
        {
          "blockId": "p5-7f0c749ae4",
          "page": 5,
          "quote": "The inclusion of the completion H into E is isometric: it identifies H with the closure of f\nK in\nE. Since k is complete, the restriction of the chosen norm to ¯k is its unique prolongation. Thus\nΩ:= b¯k is a specified subfield of H. It is algebraically closed and has additive value group Q,\nwith the normalization v = −logp | · |. The equality of value groups before and after completion\nfollows because every nonzero limit has the same norm as sufficiently close approximants.\nWe use Berkovich analytifications over Ω. A field embedding K0 ,→H determines a point of\nX¯k(H) supported at the generic point of X¯k. Together with the specified embedding Ω,→H,\nit determines a point of XΩ(H), whose analytic image we denote by ξ. Its projection to X¯k is\ngeneric; its support on XΩneed not be generic. In particular, ξ can be an Ω-rational point. We\nretain the whole H-valued point, including the embedding of its completed residue field into H,\nwhenever we use ξ.\nIf F/K0 is finite, its normalization gives a smooth proper connected curve over ¯k with a finite\nmap to X¯k. We call the ramification indices of this map at closed points horizontal ramification\nindices. The corresponding discrete valuations are trivial on ¯k and have residue characteristic\nzero. These indices are unrelated to possible wild ramification on a characteristic-p reduction\nmodel.\nFix an odd prime ℓ̸= p. Inside Ksep, let Lℓbe the compositum of all finite Galois extensions\nF/K0 whose horizontal ramification indices divide ℓ. The unramified extensions are included.\nIn particular f\nK ⊂Lℓ: the universal étale field over K0 is the same field f\nK after constants are\nincluded. One may see this by descending any finite étale cover of X¯k to a finite extension of k\nand then composing with the finite étale constant-field cover of X.\nThe field-theoretic goal can be displayed as follows. The top row consists of the defining\ninclusions; the two right vertical arrows are the containments proved in the successive stages."
        },
        {
          "blockId": "p6-c0041351b3",
          "page": 6,
          "quote": "where ec is the ramification index at c. No condition that ec be prime to p is required. For finitely\nmany distinct points b, the disks can be chosen disjoint and disjoint from the fixed point ξ."
        },
        {
          "blockId": "p6-0f4a412be2",
          "page": 6,
          "quote": "Here it is enough to make the valuation of a(u)/a(0) −1 greater than vp(ec) + 1/(p −1). Thus\nzc = u a(u)1/ec is analytic with a nonzero derivative at the center. Analytic inversion gives a\nsmaller disk on which it is a coordinate and tb = zec\nc .\nDo this in disjoint neighborhoods of all the finitely many points above b. The complement of\nthese neighborhoods in the proper analytic curve Can\nΩis compact. Its image under f is compact,\nclosed, and does not contain b. Consequently a sufficiently small disk about b has its entire\ninverse image in the chosen neighborhoods. Shrink it once more so that, in every zc-chart, that\ninverse image is the disk defined by the corresponding bound on |zc|ec.\nThe point ξ does not equal any point of X(¯k), since its projection to X¯k is generic. The\nshrinking disks at b form a neighborhood basis there. Hausdorffness therefore permits all disks\nto avoid ξ and one another. We may choose their radii in |Ω×|. The argument also applies when\nξ is an Ω-rational point whose projection to X¯k is generic."
        },
        {
          "blockId": "p7-7113083a50",
          "page": 7,
          "quote": "Lemma 3.3. In Proposition 3.2, suppose that the composite VΩ→XΩis isomorphic, over XΩ,\nto the base change of a finite morphism V →X¯k from a smooth proper connected curve, and\nassume that ξ ∈U. If ¯k(V ) admits a K0-embedding into H, then ¯k(C) admits a K0-embedding\ninto H."
        },
        {
          "blockId": "p7-a80d8ea63e",
          "page": 7,
          "quote": "Proof. The given embedding determines an H-valued point of V . Together with the fixed\ninclusion Ω⊂H, it determines an H-valued point of V ׯk Ω, and the chosen isomorphism\ntransports this point to VΩ. Its image in XΩ(H) is exactly the fixed point induced by K0 ,→H\nand Ω,→H. In particular its underlying analytic image is ξ ∈U.\nAn H-valued point of an algebraic variety over Ωis equivalently an H-valued point of its\nanalytification [29, Fact 5.1.3]. More explicitly, it gives an analytic point v together with an\nisometric embedding of the completed residue field H(v) into H. The analytic map over U in\nProposition 3.2 induces a compatible map on completed residue fields, and hence an H-valued\npoint of CΩover the same point of XΩ(H). Project this point to C. Its image on X¯k is the\ngeneric point with the specified map K0 →H. Since C →X¯k is finite, its support on C is also\ngeneric. The point therefore induces the required field embedding ¯k(C) ,→H over K0.\nNeither the analytic map over U nor a chosen local trivialization has been asserted to descend\nto ¯k."
        },
        {
          "blockId": "p12-a2f25852d0",
          "page": 12,
          "quote": "Proof. Let F/K0 be a finite Galois extension whose horizontal ramification indices divide ℓ,\nand let C →X¯k be its smooth proper curve cover. If it is unramified, its function field is\nalready contained in f\nK. Otherwise take disjoint branch-point disks avoiding ξ, small enough for\nLemma 3.1. Lemma 4.3, applied on X¯k, supplies a finite étale Y →XΩand collars on whose\nfull inverse images the branch coordinates have ℓ-th roots.\nApply Proposition 3.2 with these roots. It gives a connected finite étale cover VΩ→Y and\nan analytic map to CΩover the exterior of the smaller branch disks. The composite VΩ→XΩis\nfinite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism\nover XΩ, to a cover V →X¯k. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a\nK0-embedding into f\nK ⊂H. Lemma 3.3 therefore yields a K0-embedding F ,→H at the fixed\nH-valued point above ξ. Since F/K0 is normal and both fields lie in E, this says F ⊂H for the\nchosen copy of F. Taking their compositum proves the proposition."
        },
        {
          "blockId": "p15-0ad83e953c",
          "page": 15,
          "quote": "Proof of Theorem 1.2. Let F/K0 be any finite Galois extension inside Ksep, and let C →X¯k be\nits smooth proper curve cover. Its branch set B is finite. If B is empty, then F ⊂f\nK ⊂H by\ndefinition of the universal étale field. Suppose henceforth that B is nonempty.\nFor each b ∈B, choose a rational local parameter tb at b and disjoint disks Bb as in Lemma 3.1;\nthese disks avoid ξ. Let mb be the least common multiple of the ramification indices of C at the\npoints above b. Choose βb ∈¯k× whose absolute value is smaller than the radius of Bb in the\ncoordinate tb.\nApply Proposition 5.2 with m = mb.\nIn its notation choose a positive integer Db >\nvp(mb) + 1/(p −1) and αb ∈¯k× with ab = −logp |αb| > 2Db. It gives a smooth proper connected\ncurve cover of the T-line whose horizontal indices divide ℓand on whose full inverse image of"
        },
        {
          "blockId": "p19-b196bde152",
          "page": 19,
          "quote": "where π0 denotes the set of connected components. Complex places are omitted, as in the\ncorrected convention of Stoll’s erratum [27, Section 1]. For geometrically connected C the\ncomplex component factors in the earlier convention are singletons, but their omission matters\nfor geometrically disconnected torsors. A point of C(AK)• survives a torsor Y →C under a\nfinite K-group scheme G if it lifts to Y ξ(AK)• for some twist Y ξ, where ξ ∈H1(K, G). Write\nC(AK)f-cov\n•\nfor the points surviving every such torsor, and C(AK)f-ab\n•\nfor those surviving every\nsuch torsor with G abelian [26, Definitions 5.2 and 5.4]. We identify C(K) with its diagonal\nimage. The definitions give"
        },
        {
          "blockId": "p21-a353547873",
          "page": 21,
          "quote": "We return to an arbitrary smooth proper geometrically connected curve C/K of genus at least\ntwo over a number field, without assuming (8.3). The localization image in Proposition 8.1\nsatisfies additional descent conditions and, under a restriction on K, has finite image at each\nfinite place.\nLet ρ : C(AK) →C(AK)• be the natural projection. Write Dlin(C) for the ordinary adelic\ndescent set: P belongs to it if, for every C-torsor T →C under a linear algebraic K-group\nG, it lifts to T ξ(AK) for some global twist ξ ∈H1(K, G). Harari–Stix’s Theorem 4.1 sends\nthe finite-torsor condition established in Proposition 8.1 to this linear descent condition, while\ntheir Corollary 3.1, applied to the geometric abelianization of the interpolating section, gives\nBrauer–Manin membership [11, preprint version, Corollary 3.1 and Theorem 4.1]. Consequently"
        }
      ],
      "occurrenceCount": 9
    },
    {
      "symbol": "π",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p12-08e5a73bba",
        "page": 12,
        "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
      },
      "occurrences": [
        {
          "blockId": "p1-356e8bf501",
          "page": 1,
          "quote": "1 −→∆X −→ΠX −→Gk −→1,\n∆X = π1(X¯k),\nΠX = π1(X),\nGk = Gal(¯k/k).\n(1.1)"
        },
        {
          "blockId": "p2-3127881a48",
          "page": 2,
          "quote": "(\ns : Gk →π1(X) a continuous group homomorphism"
        },
        {
          "blockId": "p2-3c30ed1a23",
          "page": 2,
          "quote": ")\nπ1(X¯k)"
        },
        {
          "blockId": "p2-b83772449f",
          "page": 2,
          "quote": "such that (π1(X) →Gk) ◦s = idGk"
        },
        {
          "blockId": "p11-aac65728ac",
          "page": 11,
          "quote": "Proof. All algebraic data descend to a finite extension of k. After a further finite extension,\nchoose a split semistable model with the bi as disjoint smooth sections. Shrinking the coordinate\ndisks causes no problem, since the desired collars need only lie inside the originally specified disks.\nIn an integral local parameter ui at a section, successive blowups at its reductions introduce\nnode annuli\n|π|j+1 < |ui| < |π|j."
        },
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        },
        {
          "blockId": "p18-162c716e80",
          "page": 18,
          "quote": "1 −→π1(D ¯F ) −→π1(D) −→GF −→1,\nGF = Gal( ¯F/F),"
        },
        {
          "blockId": "p19-c6c884927f",
          "page": 19,
          "quote": "v real\nπ0\nV (Kv)\n,"
        },
        {
          "blockId": "p19-b196bde152",
          "page": 19,
          "quote": "where π0 denotes the set of connected components. Complex places are omitted, as in the\ncorrected convention of Stoll’s erratum [27, Section 1]. For geometrically connected C the\ncomplex component factors in the earlier convention are singletons, but their omission matters\nfor geometrically disconnected torsors. A point of C(AK)• survives a torsor Y →C under a\nfinite K-group scheme G if it lifts to Y ξ(AK)• for some twist Y ξ, where ξ ∈H1(K, G). Write\nC(AK)f-cov\n•\nfor the points surviving every such torsor, and C(AK)f-ab\n•\nfor those surviving every\nsuch torsor with G abelian [26, Definitions 5.2 and 5.4]. We identify C(K) with its diagonal\nimage. The definitions give"
        },
        {
          "blockId": "p19-91a0c7fc05",
          "page": 19,
          "quote": "π0\nC(Kv)\n −→Sec(C/Kv)"
        }
      ],
      "occurrenceCount": 10
    },
    {
      "symbol": "ρ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p12-08e5a73bba",
        "page": 12,
        "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
      },
      "occurrences": [
        {
          "blockId": "p8-62d6c4abb0",
          "page": 8,
          "quote": "Proof. On a circle |S| = ρ of radius ρ ∈|Ω×|, the Laurent Gauss norm is multiplicative.\nNormalize u and its inverse to have norm one. Their reductions are inverse units in eΩ[s, s−1], so"
        },
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
        },
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          "blockId": "p21-a353547873",
          "page": 21,
          "quote": "We return to an arbitrary smooth proper geometrically connected curve C/K of genus at least\ntwo over a number field, without assuming (8.3). The localization image in Proposition 8.1\nsatisfies additional descent conditions and, under a restriction on K, has finite image at each\nfinite place.\nLet ρ : C(AK) →C(AK)• be the natural projection. Write Dlin(C) for the ordinary adelic\ndescent set: P belongs to it if, for every C-torsor T →C under a linear algebraic K-group\nG, it lifts to T ξ(AK) for some global twist ξ ∈H1(K, G). Harari–Stix’s Theorem 4.1 sends\nthe finite-torsor condition established in Proposition 8.1 to this linear descent condition, while\ntheir Corollary 3.1, applied to the geometric abelianization of the interpolating section, gives\nBrauer–Manin membership [11, preprint version, Corollary 3.1 and Theorem 4.1]. Consequently"
        },
        {
          "blockId": "p21-3cdbd6f10d",
          "page": 21,
          "quote": "ρ−1loc\nSec(C/K)\n ⊆Dlin(C) ∩C(AK)Br.\n(8.4)"
        }
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      "source": {
        "blockId": "p12-08e5a73bba",
        "page": 12,
        "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
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          "quote": "For every character χ of T, the pullback of χ ◦a to B has an analytic m-th root."
        },
        {
          "blockId": "p12-08e5a73bba",
          "page": 12,
          "quote": "Proof. Let π : B →A and b : B →Jan be the projections, and write aB = a ◦π. Locally on B,\nlift b to an analytic map t to Tan. The identity [m]b = q ◦aB says that tm/aB takes values in\nΛ. This is a locally constant function because Λ is discrete. Its image in Λ/Λm is independent\nof the local lift: replacing t by tρ changes tm/aB by ρm, for ρ ∈Λ. We therefore obtain an\nanalytic-domain decomposition into finitely many clopen subsets B¯λ, indexed by Λ/Λm.\nChoose a representative λ ∈Λ for each coset. On B¯λ, adjust each local lift by a locally\nconstant period so that tm = λaB. Two adjusted lifts on an overlap differ by a period ρ with\nρm = 1. The lattice Λ is torsion free, so ρ = 1. Thus the adjusted lifts glue to a single analytic\nmap t¯λ : B¯λ →Tan. Choose dλ ∈Ω× with dm\nλ = χ(λ). The function χ(t¯λ)/dλ is an m-th root of\nχ(aB). These functions glue across the disjoint clopen subsets. No assumption that m is prime\nto p is used."
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