The p-adic section conjecture

Proof. Choose a local analytic parameter u at c. The pullback of t_b is u^{e_c}a(u) with a(0)\ne0. After choosing an e_c-th root of a(0) in \Omega and shrinking the disk, a(u)/a(0) is sufficiently close to 1 for its e_c-th root to be given by

Here it is enough to make the valuation of a(u)/a(0)-1 greater than v_p(e_c)+1/(p-1). Thus z_c=u\,a(u)^{1/e_c} is analytic with a nonzero derivative at the center. Analytic inversion gives a smaller disk on which it is a coordinate and t_b=z_c^{e_c}.

Do this in disjoint neighborhoods of all the finitely many points above b. The complement of these neighborhoods in the proper analytic curve C_\Omega^{\mathrm{an}} is compact. Its image under f is compact, closed, and does not contain b. Consequently a sufficiently small disk about b has its entire inverse image in the chosen neighborhoods. Shrink it once more so that, in every z_c-chart, that inverse image is the disk defined by the corresponding bound on |z_c|^{e_c}.

The point \xi does not equal any point of X(\bar k), since its projection to X_{\bar k} is generic. The shrinking disks at b form a neighborhood basis there. Hausdorffness therefore permits all disks to avoid \xi and one another. We may choose their radii in |\Omega^\times|. The argument also applies when \xi is an \Omega-rational point whose projection to X_{\bar k} is generic.□

Proof. The inverse image Y_U of U and the inverse images Y_b of the disks \{|t_b|<R_b\} form an open cover of Y^{\mathrm{an}}. Distinct Y_b are disjoint, and Y_U\cap Y_b=g^{-1}(A_b).

On Y_U, take the pullback of C_\Omega^{\mathrm{an}}\to X_\Omega^{\mathrm{an}}. It is finite étale of degree d, since U contains no branch point. On each Y_b, take the trivial degree-d cover. These two covers are isomorphic over their overlap. Indeed, write q_b^{m_b}=g^*t_b for the assumed analytic root. In the local factor corresponding to c\in f^{-1}(b), put a_c=q_b^{m_b/e_c}. The polynomial defining that factor splits as

Each a_c is an analytic unit on the overlap. Distinct linear factors are comaximal, because (\zeta-\zeta')a_c is a unit. This is true also when p\mid e_c: a nonzero constant in \Omega is invertible, regardless of its absolute value. Taking all factors in Lemma 3.1 gives a trivialization of the full degree-d cover, not just of one sheet.

Choose these trivializations and glue the finite locally free analytic algebras. There is no further cocycle condition between different branch disks, as those disks are disjoint. The glued algebra is coherent and finite étale over Y^{\mathrm{an}}, these being local properties. Proper analytic GAGA [20, Appendix A, Theorem A.1(ii)] algebraizes the coherent algebra and its unit and multiplication; full faithfulness preserves the algebra identities. Equivalently, one uses the corresponding equivalence for finite covers of a proper curve. Étaleness is detected after analytification [29, Fact 5.1.5], so its relative spectrum is a finite étale cover of Y.

Choose any nonempty connected component and call it V_\Omega. A connected component of this finite étale cover has positive locally constant degree over the connected curve Y, hence is still surjective. It is smooth and proper. Restricting the original projection from the pullback over Y_U gives the asserted analytic map. If B is empty, the whole pullback is already finite étale and the same component argument applies without gluing.□

Proof. The given embedding determines an H-valued point of V. Together with the fixed inclusion \Omega\subset H, it determines an H-valued point of V\times_{\bar k}\Omega, and the chosen isomorphism transports this point to V_\Omega. Its image in X_\Omega(H) is exactly the fixed point induced by K_0\hookrightarrow H and \Omega\hookrightarrow H. In particular its underlying analytic image is \xi\in U.

An H-valued point of an algebraic variety over \Omega is equivalently an H-valued point of its analytification [29, Fact 5.1.3]. More explicitly, it gives an analytic point v together with an isometric embedding of the completed residue field \mathcal H(v) into H. The analytic map over U in Proposition 3.2 induces a compatible map on completed residue fields, and hence an H-valued point of C_\Omega over the same point of X_\Omega(H). Project this point to C. Its image on X_{\bar k} is the generic point with the specified map K_0\to H. Since C\to X_{\bar k} is finite, its support on C is also generic. The point therefore induces the required field embedding \bar k(C)\hookrightarrow H over K_0.

Neither the analytic map over U nor a chosen local trivialization has been asserted to descend to \bar k.□

Proof. All valuations in this statement are the horizontal, discrete closed-point valuations, which are trivial on \bar k. Their residue characteristic is zero. The completed base field at such a point is A=\bar k((t)). Every finite extension of A has residue field \bar k and is of the form \bar k((u)) for a uniformizer u: successive subtraction of residue representatives gives the power-series expansion in its complete valuation ring. If its ramification index is e, then t=u^e a(u) for a unit a(u)\in\bar k[[u]]^\times. The unit has an e-th root in \bar k[[u]], by Hensel’s lemma in residue characteristic zero. Changing the parameter gives t=z^e. Thus, inside a fixed algebraic closure, the local field is A(t^{1/e}), which is cyclic over A since \bar k contains all roots of unity.

For e\mid n, this is a subfield of A(t^{1/n}). All local factors of a finite compositum, or of a normal closure formed from conjugates, therefore remain subfields of this same cyclic extension. This proves the assertions about subextensions, composita, and Galois closures.

For pullback, write the local base map as t=s^r a(s), where r\ge1 and a(s) is a unit. Extracting its e-th root in \bar k[[s]] shows that every normalized local factor after pullback has ramification index e/\gcd(e,r), which divides e. Finally, ramification indices multiply in a tower, and an étale upper cover contributes index 1. Apply the Galois-closure assertion to a cover with indices dividing \ell and embed that closure into the fixed K^{\mathrm{sep}}. Its field lies in the defining compositum L_\ell.□

Proof. On a circle |S|=\rho of radius \rho\in|\Omega^\times|, the Laurent Gauss norm is multiplicative. Normalize u and its inverse to have norm one. Their reductions are inverse units in \widetilde\Omega[s,s^{-1}], so the reduction of u is a monomial. Thus exactly one Laurent term dominates at that radius. Its exponent cannot change inside the radius interval. Indeed, convergence on compact subannuli reduces any proposed change to finitely many competing terms; a tie between two terms occurs at a radius in |\Omega^\times|, since the additive value group is \mathbb Q. Such a tie contradicts the preceding reduction argument. A single term cS^d therefore dominates throughout A, giving the asserted factorization. The same argument on closed subdisks uses that the only units in \widetilde\Omega[s] are constants and proves the disk assertion. Its constant norm belongs to |\Omega^\times| by evaluation at a rational point of the disk.

Since \ell is a unit in the valuation ring, the binomial series for (1+h)^{1/\ell} converges on every compact subannulus; the resulting functions agree on overlaps. A root of c lies in \Omega, and a root of S^d follows from a root of S, also when d<0.□

Proof. Our aim is to make the expansion factors, the absolute values of the integral slopes along edges, divisible by \ell above the chosen edge. On an inverse annulus this makes the Laurent exponent of the pulled-back coordinate divisible by \ell; its constant factor and the factor 1+h also have \ell-th roots by Lemma 4.1. We construct such a cover directly when the edge lies on a circuit. For a separating edge, we first pass to an unramified cover in which every edge above it lies on a circuit.

We use the lifting theorem for tame coverings of metrized complexes [1, Theorem 7.7]. A covering to which it applies consists of finite residue-curve maps and a finite harmonic graph map: every fibre of every marked point is represented by edge ends, their ramification indices equal the edge expansion factors, and the component degree is the sum of these factors over each base direction. The residue maps must be tamely ramified, and the augmented graph ramification must vanish away from punctures [1, Definitions 2.19 and 2.21]. We shall use no punctures. The lift is then finite étale, and has exactly the prescribed inverse images of the vertex set and skeleton [1, Definitions 4.25 and 4.31].

First suppose that the edge e corresponding to A is nonseparating. Choose a simple circuit containing e. At each vertex on this circuit, let P,Q be the two circuit marks on its residue curve D. Divisibility of \operatorname{Pic}^0(D) [16, Theorem 8.2] supplies a degree-zero divisor E such that P-Q+\ell E is principal. Extract an \ell-th root of a function with this divisor and take the smooth normalization. The resulting cyclic cover of D has degree \ell, is connected because the order at P is prime to \ell, and is totally ramified at P,Q and nowhere else. Each other marked point has \ell distinct preimages. At every vertex outside the circuit take \ell identity copies of its residue curve.

Above each circuit edge use one edge of expansion factor \ell and of length equal to the original length divided by \ell. Above each remaining edge use \ell factor-one edges, matching all the unramified marked fibres at its ends. These rules also handle loops and chords between circuit vertices. All lengths belong to \mathbb Q. At a lifted circuit vertex each base direction has total degree \ell; at an identity-copy vertex it has total degree one. The local Riemann–Hurwitz formula is

The two extra terms are precisely those recorded by the two circuit edges. Thus the augmented graph ramification is zero at every vertex; the identity copies contribute none. All ramification indices are prime to p. The resulting tame covering is connected, because each vertex outside the circuit can be joined by a lifted path to a circuit vertex.

Lift these data by the cited theorem. The inverse image of A is exactly the disjoint union of the node annuli above e. For completeness, the inverse-skeleton equality excludes off-skeleton points mapping to the skeleton. Each off-skeleton open ball therefore maps into a single off-skeleton ball downstairs, and continuity identifies the images of their attachment points. Retractions commute on these balls as well as on the skeleton. Since a node annulus is the inverse image of its open edge under retraction, this proves the asserted equality for the whole annulus. On each annulus above A, the pullback of S has exponent \ell or -\ell in Lemma 4.1. It therefore has an analytic \ell-th root on each of these disjoint annuli.

Now suppose that e is separating. Each side of the graph obtained by cutting its interior contains a circuit or a positive-genus vertex. Otherwise that side would be a genus-zero tree with, say, N vertices. Its total valence, including the bridge half-edge, would be 2N-1, contrary to stability, which requires at least 3N.

Construct a connected unramified degree-\ell covering of each side, retaining its boundary mark for the eventual gluing. If the side contains a circuit, take a connected cyclic graph covering of degree \ell, with identity residue-curve maps. Such a graph covering is obtained by a homomorphism of its graph fundamental group onto \mathbb Z/\ell\mathbb Z. If instead a positive-genus vertex is used, take a connected cyclic étale cover of degree \ell of its residue curve, provided by a nonzero element of its Jacobian’s \ell-torsion [16, Theorem 8.2 and Remark 8.4]. Explicitly, for a line bundle L of exact order \ell, a trivialization L^{\otimes\ell}\simeq\mathcal O makes \bigoplus_{j=0}^{\ell-1}L^{\otimes j} an algebra whose relative spectrum is a cyclic étale cover: \ell is invertible, and the exact order of L makes the cover connected. Use \ell identity copies at the other vertices and match all marked fibres. Paths to the distinguished vertex show connectedness of the covered side.

Each covered side has \ell preimages of the bridge mark. Match them in pairs to obtain \ell distinct bridge edges, all with factor one. Every such edge is now nonseparating: after removing one of them, the two connected covered sides are still joined by the others. The resulting covering of the original metrized complex is tame. Indeed the component covers are unramified, all factors are one, and Riemann–Hurwitz again gives zero graph ramification. Lift it to a finite étale curve cover. Its triangulation remains stable: at a vertex of local degree d, the quantity 2g'-2+\operatorname{val}(v') is d(2g-2+\operatorname{val}(v))>0.

Apply the nonseparating construction to each of these \ell edges. A connected component of the fibre product of the resulting finite étale covers dominates each factor, so all the constructed roots persist on a common cover. On each factor-one annulus the pullback of the original S is a unit; Lemma 4.1 supplies its root once the new annular coordinate has a root. This proves the assertion also for separating edges. Finally, the same finite fibre-product argument treats finitely many original annuli simultaneously.□

Proof. All algebraic data descend to a finite extension of k. After a further finite extension, choose a split semistable model with the b_i as disjoint smooth sections. Shrinking the coordinate disks causes no problem, since the desired collars need only lie inside the originally specified disks. In an integral local parameter u_i at a section, successive blowups at its reductions introduce node annuli

On a sufficiently small disk the quotient t_i/u_i is an analytic unit, since both coordinates have a simple zero at b_i. Its absolute value is constant by Lemma 4.1. Thus these annuli are also concentric annuli for t_i, and choosing j large puts them inside B_i. Perform these modifications for all the sections and denote the resulting proper semistable model by \mathcal C. It need not be stable; the markings can now be forgotten.

The RNS theorem, applied with the set of all primes, and its model-domination consequence [17, Theorem A and Proposition 2.4(iv)] give, after finite extension of the ground field, a finite étale cover whose stable model maps to \mathcal C. Pass to split reduction and a geometric connected component, and write

for the analytic-base-changed curve cover and the model morphism. No finiteness assertion about the model morphism is needed.

Fix one of the target node annuli A_i\subset B_i just constructed, and choose an irrational real number a strictly between its two additive endpoint valuations. Consider the entire inverse image

Reduction is compatible with the model morphism. A point reducing to a component generic point of \mathcal Y_0 is a model vertex, hence is type 2 and gives t_i a value in |\Omega^\times|; its additive valuation is rational. It cannot lie above this level. A point reducing to a smooth closed point lies in an open disk tube. If it lies above the chosen level, that smooth point maps to the target node, so the entire disk tube maps into A_i. The pullback of t_i is a unit there. By Lemma 4.1 its norm is constant and belongs to |\Omega^\times|, which again excludes the irrational level. All remaining points above the level lie in source node annuli whose nodes map to the target node.

Let N_i be the union of all source node annuli whose nodes map to the target node defining A_i; this is a finite union. The complement of N_i in the proper analytic curve Y_0^{\mathrm{an}} is compact. Intersect it with the inverse image of a closed concentric subannulus of A_i containing the chosen level in its interior. On this compact intersection the continuous real function -\log_p|t_i| has compact image avoiding a. Consequently there is an open interval about a with rational endpoints whose entire inverse image lies in N_i. It determines a collar A_i'\subset A_i.

Every source node annulus in N_i maps into A_i by reduction compatibility, so the pullback of t_i is an analytic unit on it. Apply Lemma 4.2 simultaneously to all these source annuli, for all i. Their coordinates acquire \ell-th roots on a common connected finite étale cover of Y_0. Lemma 4.1 then gives roots of the pullbacks of the t_i on the full inverse images of the A_i'. The resulting composite cover of C_\Omega is finite étale.□

Proof. Let F/K_0 be a finite Galois extension whose horizontal ramification indices divide \ell, and let C\to X_{\bar k} be its smooth proper curve cover. If it is unramified, its function field is already contained in \widetilde K. Otherwise take disjoint branch-point disks avoiding \xi, small enough for Lemma 3.1. Lemma 4.3, applied on X_{\bar k}, supplies a finite étale Y\to X_\Omega and collars on whose full inverse images the branch coordinates have \ell-th roots.

Apply Proposition 3.2 with these roots. It gives a connected finite étale cover V_\Omega\to Y and an analytic map to C_\Omega over the exterior of the smaller branch disks. The composite V_\Omega\to X_\Omega is finite étale. Proper base-extension invariance of finite étale covers descends it, up to isomorphism over X_\Omega, to a cover V\to X_{\bar k}. Here we use [10, Exposé X, Corollary 1.8]. Its function field has a K_0-embedding into \widetilde K\subset H. Lemma 3.3 therefore yields a K_0-embedding F\hookrightarrow H at the fixed H-valued point above \xi. Since F/K_0 is normal and both fields lie in E, this says F\subset H for the chosen copy of F. Taking their compositum proves the proposition.□

Proof. Let \pi:B\to A and b:B\to J^{\rm an} be the projections, and write a_B=a\circ\pi. Locally on B, lift b to an analytic map t to \mathbb T^{\rm an}. The identity [m]b=q\circ a_B says that t^m/a_B takes values in \Lambda. This is a locally constant function because \Lambda is discrete. Its image in \Lambda/\Lambda^m is independent of the local lift: replacing t by t\rho changes t^m/a_B by \rho^m, for \rho\in\Lambda. We therefore obtain an analytic-domain decomposition into finitely many clopen subsets B_{\bar\lambda}, indexed by \Lambda/\Lambda^m.

Choose a representative \lambda\in\Lambda for each coset. On B_{\bar\lambda}, adjust each local lift by a locally constant period so that t^m=\lambda a_B. Two adjusted lifts on an overlap differ by a period \rho with \rho^m=1. The lattice \Lambda is torsion free, so \rho=1. Thus the adjusted lifts glue to a single analytic map t_{\bar\lambda}:B_{\bar\lambda}\to\mathbb T^{\rm an}. Choose d_\lambda\in\Omega^\times with d_\lambda^m=\chi(\lambda). The function \chi(t_{\bar\lambda})/d_\lambda is an m-th root of \chi(a_B). These functions glue across the disjoint clopen subsets. No assumption that m is prime to p is used.□

Proof. Let Z be the smooth projective normalization of

The four distinct points 0,\alpha,1,\infty have orders 1,-1,1,-1 in the divisor of the displayed rational function. The cyclic cover is connected, has degree \ell, and is totally ramified at precisely these points. Riemann–Hurwitz gives

Here is a split semistable model of Z. Work first over the integers of a sufficiently large finite extension of k containing \alpha. The blowup of the model of the line along (T,\alpha) has two rational components and one node, with local equation TU=\alpha. The inner component has coordinate S=T/\alpha; the four branch sections are disjoint and smooth. The Kummer function reduces to T-1 on the outer component and to -S/(S-1) on the inner component. Each of these degree-\ell covers of a rational component has just two totally ramified points and is rational. Near the node the Kummer function is

a unit whose reduction at the node is -1. Since \ell is invertible, normalization is étale there, with \ell split copies of the node after enlarging the finite constant field. Away from the node and branch sections the same unit criterion gives étaleness. At a branch section the function or its inverse is a simple relative parameter, so the normalization is smooth there. Thus there are exactly two rational components, joined by \ell nodes, each of thickness v(\alpha)=a. They are stable because \ell\ge3.

Consequently the skeleton \Gamma consists of two vertices and \ell parallel edges e_1,\ldots,e_\ell, each of length a. The node annuli A_j are disjoint copies of \{|\alpha|<|T|<1\} with coordinate T. This last assertion can also be checked directly: on that annulus both T and \alpha/T have absolute value less than one, and the two factors in -(1-T)/(1-\alpha/T) have convergent \ell-th roots. Hence the inverse image is exactly \ell copies of the annulus.

We next apply nonarchimedean uniformization of abelian varieties, developed by Raynaud and Bosch–Lütkebohmert [23, 6, 5]. Let J be the Jacobian of Z, and choose an Abel–Jacobi map i:Z\to J using a point over \overline{k}. Total degeneration gives a toric analytic uniformization

Its character lattice is M=H_1(\Gamma,\mathbb Z); its tropical target is \operatorname{Hom}(M,\mathbb R). The period lattice is free, and q is a topological covering and local analytic isomorphism. These are the Jacobian uniformization conventions of [3, Sections 4.1–4.2 and 4.5.2]. The compatibility of the classical and tropical Abel–Jacobi maps [3, Proposition 6.1 and Corollary 6.6] identifies the latter with integration on \Gamma using the edge-length pairing.

The open Berkovich annulus A_j strongly deformation retracts to its open skeletal interval [2, Section 2.3] and is locally path connected. It is thus simply connected, so i|_{A_j} lifts to an analytic map \widehat i_j:A_j\to\mathbb T^{\rm an}: first lift continuously along the covering q, then use its analytic local inverses. Orient all edges by increasing v(T). For j'\ne j, the cycle c_j=e_j-e_{j'} is an element of M. If \chi_j is its character, then u_j=\chi_j\circ\widehat i_j is an analytic unit. The derivative of v(u_j) along e_j, with respect to v(T), is exactly one: the integral of the harmonic form represented by c_j over a segment of e_j is the length of that segment. Lifts of the tropical map differ only by a constant period, which does not change this derivative.

By Lemma 4.1, an analytic unit on an open annulus has a unique dominant Laurent monomial at every interior Gauss radius; its exponent is constant and is the slope of its log-norm. It follows that

Here c_j'T is the actual coefficient term of degree one. Dominance at every radius r\in(|\alpha|,1) gives |h_{j,n}|r^n<1. Taking limits at the endpoints yields, for n>0,

Writing s=v(T), the ultrametric inequality consequently gives

Figure 1 displays the cycle that supplies the slope-one character and the trimming that gives this bound.

On every closed subannulus this estimate has a strict uniform margin. The logarithm and exponential therefore converge locally uniformly and

is analytic there, with b_j^m=1+h_j. Indeed, since D>1/(p-1), the logarithm satisfies |\log(1+h_j)|\le |h_j|, and the valuation of the exponent is greater than D-v_p(m)>1/(p-1).

For all \ell annuli at once, form the same algebraic finite étale cover

It is finite étale for every positive integer m, since the ground field has characteristic zero. Lemma 5.1 applied to a=\widehat i_j gives an analytic m-th root of u_j on the whole inverse image of A_j. Dividing it by b_j and an m-th root of c_j' gives an analytic m-th root of T above the middle collar. The annuli A_j are disjoint, so this gives a root on its entire inverse image in Z'. Choose any connected component Q of Z'. Its map to Z is finite étale and surjective, so it is a smooth proper connected curve with the required property. Its composite with Z\to\mathbb P^1 has only ramification indices 1 or \ell. All algebraic maps used to define Q are over \overline{k}; the analytic roots and torus lifts are only required over \Omega.□

Proof of Theorem 1.2. Let F/K_0 be any finite Galois extension inside K^{\mathrm{sep}}, and let C\to X_{\bar k} be its smooth proper curve cover. Its branch set B is finite. If B is empty, then F\subset\widetilde K\subset H by definition of the universal étale field. Suppose henceforth that B is nonempty.

For each b\in B, choose a rational local parameter t_b at b and disjoint disks B_b as in Lemma 3.1; these disks avoid \xi. Let m_b be the least common multiple of the ramification indices of C at the points above b. Choose \beta_b\in\bar k^\times whose absolute value is smaller than the radius of B_b in the coordinate t_b.

Apply Proposition 5.2 with m=m_b. In its notation choose a positive integer D_b>v_p(m_b)+1/(p-1) and \alpha_b\in\bar k^\times with a_b=-\log_p|\alpha_b|>2D_b. It gives a smooth proper connected curve cover of the T-line whose horizontal indices divide \ell and on whose full inverse image of

the coordinate T has an analytic m_b-th root. Pull this cover back by the nonconstant morphism X_{\bar k}\to\mathbb P^1_{\bar k} given by T=t_b/\beta_b, and take a normalized connected component. Denote the resulting cover by W_b\to X_{\bar k}.

Lemma 3.4 shows that its horizontal indices still divide \ell. Roots pull back to every point of the normalized component, so t_b has an analytic m_b-th root on the entire inverse image in W_{b,\Omega}^{\mathrm{an}} of the collar

inside B_b. Here we multiplied the pulled-back root of T by an m_b-th root of \beta_b in \Omega.

Embed the finitely many function fields \bar k(W_b) into the fixed K^{\mathrm{sep}} over K_0 and form their compositum. Let W\to X_{\bar k} be the associated smooth proper connected curve. It dominates every W_b; its horizontal indices divide \ell by Lemma 3.4. All collar roots therefore pull back to the full inverse images in W_\Omega^{\mathrm{an}}.

Proposition 3.2, with Y=W_\Omega, produces a connected finite étale cover V_\Omega\to W_\Omega and an analytic map from its exterior region to C_\Omega^{\mathrm{an}} over X_\Omega^{\mathrm{an}}. Since W is proper over the algebraically closed field \bar k, invariance of finite étale covers under the algebraically closed extension \bar k\subset\Omega [10] gives a connected finite étale cover V\to W and an isomorphism

No descent assertion is needed for the exterior analytic map.

Although no such bound was imposed on C\to X_{\bar k}, the composite V\to X_{\bar k} still has horizontal indices dividing \ell. By Lemma 3.4, its function field embeds over K_0 in L_\ell. This embedding can, if desired, be chosen to extend the already chosen embedding of \bar k(W): extend that embedding to \bar k(V) inside K^{\mathrm{sep}} and then take its Galois closure, which still has indices dividing \ell. Proposition 4.4 places its image in H.

Lemma 3.3 now gives a K_0-embedding F=\bar k(C)\hookrightarrow H\subset E. Because F/K_0 is normal, this embedding has image the chosen subfield F of E. Indeed a primitive element of F is sent to a root of its minimal polynomial, all of whose roots already belong to F; the same holds for every element. The image, having degree [F:K_0], is therefore F itself. Consequently F\subset H.

Every element of K^{\mathrm{sep}} lies in a finite Galois extension of K_0 inside K^{\mathrm{sep}}: this is an algebraic closure of the characteristic-zero field K_0. Taking the union over those extensions proves K^{\mathrm{sep}}\subset H inside E.□

Proof. Restriction is surjective. Indeed, lift an automorphism stabilizing \widetilde w to an automorphism of K^{\mathrm{sep}}/K. Its translate of w^{\mathrm{sep}} is another prolongation of \widetilde w. Transitivity of \operatorname{Gal}(K^{\mathrm{sep}}/\widetilde K) on these prolongations allows us to adjust the lift so that it stabilizes w^{\mathrm{sep}}. The transitivity statement for an infinite normal extension follows from its finite-level version by compactness.

An automorphism stabilizing a rank-one valuation preserves its normalized absolute value. To see why normalization matters, any two order-preserving embeddings of the value group into \mathbb R differ by a positive scale; the value on p fixes that scale. Such an automorphism therefore extends as an isometry to E=\widehat{K^{\mathrm{sep}}}. If its restriction to \widetilde K is the identity, it fixes H=\widehat{\widetilde K}\subset E pointwise. Theorem 1.2 gives K^{\mathrm{sep}}\subset H, so the automorphism is the identity.

Both decomposition groups are closed subgroups of their Galois groups. For example, stabilization of a valuation ring is the intersection, over its algebraic elements, of conditions on finite Galois orbits, each of which is clopen. Restriction is consequently a continuous bijection between compact Hausdorff groups, and its inverse is continuous.□

Proof. Normalize w(p)=1 and prolong the valuation to K^{\mathrm{sep}}. Suppose that a section exists. Corollary 7.1 lifts it to a continuous section

whose image stabilizes this prolongation. The birational section theorem of Koenigsmann places the entire image of \widehat s also in the decomposition group of a prolongation of \operatorname{ord}_a for some a\in X(k) [13, preprint version, Proposition 2.4(2), Definition 1.4, and Observation 1.3]. Here \operatorname{ord}_a is the closed-point valuation of K, trivial on k.

Write \Sigma=\widehat s(G_k) and M=(K^{\mathrm{sep}})^\Sigma. The subgroup \Sigma is compact and hence closed, so G_M=\Sigma. Let v_1 and v_2 be the restrictions to M of the two selected prolongations. Each is henselian: its absolute Galois group stabilizes a prolongation, and the transitive action on all prolongations therefore has a single point. Both valuations are nontrivial and have rank one. Indeed, value groups in an algebraic valued extension differ by a torsion quotient, which preserves their convex-subgroup chains. Moreover,

so these valuations are inequivalent. The resulting contradiction is an instance of the classical uniqueness principle for henselian valuations associated with F. K. Schmidt [24]. We give explicitly the two-valuation approximation needed here.

Choose a nonsquare d\in k^\times. It remains a nonsquare in M: otherwise G_M=\Sigma would fix a square root of d in \bar k, contrary to the surjection \Sigma\to G_k. Let t\in K be a uniformizer at a. Choose a positive integer N with N>v_1(t), and put z=p^N/t. Then

For positive integers r, set

The denominator is nonzero because v_1(z^r)>0. At the first valuation q_r tends to d, and at the second it tends to 1. More precisely,

whereas

For sufficiently large r, Hensel’s lemma applied at 1 gives a square root of q_r/d using v_1, and a square root of q_r using v_2. The sufficient inequalities are

Their quotient makes d a square in M, a contradiction. This argument includes p=2 and arbitrary nondiscrete rank-one value groups.□

Proof of the surjectivity in Theorem 1.1. Let s:G_k\to\Pi_X be a continuous group-homomorphic section. By the valuative localization theorem of Pop–Stix [22, preprint version, Theorem 26], there is a valuation w of K extending the p-adic valuation, together with a prolongation \widetilde w to \widetilde K, such that

We classify the possibilities for w.

The valuation transcendence inequality gives

Since \Gamma_k has rational rank one and ordered rank is at most rational rank, w has rank at most two. Rank one is excluded by Lemma 7.2. If w has rank two, let u be its rank-one coarsening. The nonzero proper convex subgroup of the rank-two value group has a unique extension to the value group of \widetilde w: it consists of those values some positive multiple of which lies in the original subgroup. This follows because an algebraic valued extension has torsion quotient of value groups. Every automorphism stabilizing \widetilde w preserves this subgroup, so s(G_k) stabilizes the resulting prolongation of u. If u is nontrivial on k, it can be normalized to extend the p-adic valuation and is excluded by the same lemma.

It remains that u is trivial on k. Properness gives it a center on X. This center cannot be generic because u is nontrivial, so it is a closed point a. The valuation ring of u dominates the discrete valuation ring \mathcal O_{X,a}. Every element of K^\times is a power of a uniformizer times a unit of this local ring; hence u is equivalent to \operatorname{ord}_a. Its residue field is the finite extension k(a)/k. The second valuation in the composition w is the unique prolongation of the p-adic valuation to k(a). Thus w is of type 2h.

The chosen horizontal prolongation to K\bar k selects a geometric point above a. Its stabilizer acts on constants through G_{k(a)}\subseteq G_k, with the embedding determined by that point. Since s(G_k) projects onto G_k, necessarily k(a)=k. For the full étale tower the horizontal inertia is trivial, and the horizontal decomposition group is the point section at a. Equivalently, the type-2h conclusion of [22, preprint version, Section 6.1] gives s=s_a after the corresponding geometric choice of base point. This proves surjectivity.

Notice that a type-2h valuation can lie over a non-rational closed point. Rationality here follows from the surjection to G_k; no unrestricted equivalence between “not type 2h” and “not originating from a rational point” is used.□

Proof of the injectivity in Theorem 1.1. Suppose a,b\in X(k) have the same section class modulo the full geometric fundamental group. Let J be the Jacobian of X, and use a as base point for the Abel–Jacobi map

Their induced sections on J are again geometrically conjugate. For every positive integer n, multiplication [n]:J\to J is a finite étale torsor under J[n], also when p\mid n since k has characteristic zero. Conjugate sections give isomorphic G_k-actions on its geometric fiber, so the existence of a fixed point is unchanged by conjugation. The section at i_a(a)=0 has the fixed lift 0. Therefore the fiber over i_a(b) has a G_k-fixed point, which is a k-rational point of that fiber. Thus

This is also the usual Kummer argument: the class of the multiplication torsor pulled back along a section is unchanged by geometric conjugation.

The compact p-adic analytic group J(k) has an open subgroup U\cong\mathbb Z_p^{g(X)[k:\mathbb Q_p]} by the local logarithm. If q=[J(k):U], then qJ(k)\subseteq U. For each r\geq0, take n=qp^r in the displayed intersection. Any element of that intersection can then be written as p^r(qy) with y\in J(k), and hence belongs to p^rU. As U is a finite free \mathbb Z_p-module,

and hence i_a(b)=0. If a\ne b, the linear equivalence of the degree-one divisors a and b would give a degree-one morphism X\to\mathbb P^1, forcing genus zero. Hence a=b, as required.□

Proof. Let s\in\operatorname{Sec}(C/K). Choose a representative point of the determined component at each real place, and any point at each complex place. Harari–Stix’s Theorem 2.1, with U=1 and the omitted set of places S=\varnothing, shows that these localizations survive every torsor under a finite K-group scheme; their Remark 2.2(1) identifies this as finite-cover descent [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. Consequently

Conversely, let x belong to the finite-cover locus and, by properness, choose an ordinary adelic representative P=(P_v)\in C(\mathbb A_K). Stoll’s cohomological description after Definition 5.2, together with Definition 5.4, identifies survival with diagonal global H^1(K,G)-membership of the local evaluations for every torsor under a finite K-group scheme G [26]. These evaluations factor through the real components, and complex places impose no lifting condition, in the corrected convention already used above. Thus every such P satisfies the ordinary finite-torsor condition. Harari–Stix’s Theorem 2.1 with U=1 and S=\varnothing, using implication (i)\Rightarrow(iii), gives a global section whose restrictions are the point sections of P_v, up to conjugation by the full geometric fundamental group [11, preprint version, Theorem 2.1 and Remark 2.2(1)]. The finite-point and real-component uniqueness used above identify its localization with x, proving (8.2).□

Proof. Let s\in\operatorname{Sec}(C/K). By Proposition 8.1 and (8.3), there is a\in C(K) such that x(s) is the diagonal localization of a, hence equals the localization of the point section s_a.

At every finite place the point determined by s is therefore a. At each real place choose a itself in the determined component, and choose a also at each complex place. The resulting ordinary adelic representative of the localization tuple is supported on \{a\}. Thus the support Z(s) in Stix’s terminology, the Zariski-closed support of this tuple, is finite over K. Since C is proper of genus at least two, it is hyperbolic and has no cuspidal sections. Stix’s finite-support result therefore makes s a point section [25, arXiv version, Section 3 and Corollary 5]. Finally, if two points give the same global section, their sections agree after restriction to any one finite completion. Injectivity in Theorem 1.1 then identifies the two points. This proves injectivity and also identifies the point giving s with a.□

Proof. Stoll’s Theorem 8.6 gives C(\mathbb A_K)_\bullet^{\mathrm{f\text{-}ab}}=C(K) under the first condition, and his Corollary 8.1 gives the same equality under the second, without requiring a K-rational base point on C [26]. The corrected covering arguments and the repair of Proposition 8.5 used by Theorem 8.6 are in [27, Sections 2–3]; the stated equalities are retained. The inclusions (8.1) now imply (8.3), so Theorem 8.2 applies.□

Proof. The corank-zero case of the companion Selmer converse [18, Theorem 1.1] gives \operatorname{rank}_{\mathbb Z}E(\mathbb Q)=0 and finiteness of the entire group \operatorname{Sha}(E/\mathbb Q). The Mordell–Weil theorem therefore makes E(\mathbb Q) finite, while finiteness of \operatorname{Sha}(E/\mathbb Q) makes its maximal divisible subgroup zero. Corollary 8.3(i) applies.□

Proof. Stoll’s Corollary 8.8 gives C(\mathbb A_{\mathbb Q})_\bullet^{\mathrm{f\text{-}ab}}=C(\mathbb Q) for each of these projective modular curves of positive genus [26], with the corrected conventions of [27]. Apply (8.1) and Theorem 8.2.□