The $\mathbb{Z}/p$ metabelian birational $p$-adic section conjecture for varieties

Abstract

We generalize the $\mathbb{Z}/p$metabelian birational $p$-adic section conjecture for curves, as introduced and proved in Pop [On the birational$p$-adic section conjecture, Compos. Math. 146 (2010), 621–637], to all complete smooth varieties, provided $p>2$. The condition $p>2$ seems to be of technical nature only, and might be removable.

1 Introduction

The (birational) ( $p$ -adic) section conjecture (SC) originates from Grothendieck [Gro83, Gro84] (see [SL98]), and weaker/conditional forms of the SC are a part of the local theory in anabelian geometry, see e.g. Faltings [Fal98] and Szamuely [Sza04]. In spite of serious efforts to tackle the SC, only the full Galois birational $p$ -adic SC is completely resolved, see Koenigsmann [Koe05] for the case of curves and Stix [Sti13] for higher dimensional varieties. On the other hand, a much stronger form of the birational $p$ -adic SC for curves, to be precise, the $\mathbb{Z}/p$ metabelian birational $p$ -adic SC for curves, was proved in Pop [Pop10]. The aim of this note is to prove a similarly strong result for the higher dimensional varieties, at least in the case $p>2$ .

For the reader’s sake and to make the presentation self contained (to some extent), we begin by recalling a few notations and well-known facts; see e.g. the Introduction in [Pop10]. First, for an arbitrary (perfect) base field $k$ and complete smooth geometrically integral $k$ -varieties $X$ , let $K=k(X)$ be the function field of $X$ . Let $\tilde{K}|K$ be some Galois extension, $\tilde{k}\subseteq \tilde{K}$ be the relative algebraic closure of $k$ in $\tilde{K}$ , and consider the resulting canonical exact sequence of Galois groups:

$$\begin{eqnarray}1\rightarrow \operatorname{Gal}(\tilde{K}|K\tilde{k})\longrightarrow \operatorname{Gal}(\tilde{K}|K)\xrightarrow[{}]{\tilde{p}_{K}}\operatorname{Gal}(\tilde{k}|k)\rightarrow 1.\end{eqnarray}$$

Let $\tilde{X}\rightarrow X$ be the normalization of $X$ in the field extension $K{\hookrightarrow}\tilde{K}$ . For $x\in X$ and $\tilde{x}\in \tilde{X}$ above  $x$ , let $T_{x}\subseteq Z_{x}$ be the inertia/decomposition groups of $\tilde{x}|x$ and $G_{x}:=\operatorname{Aut}(\unicode[STIX]{x1D705}(\tilde{x})|\unicode[STIX]{x1D705}(x))$ be the residual automorphism group. By decomposition theory, one has a canonical exact sequence

Next suppose that $x$ is $k$ -rational, i.e., $\unicode[STIX]{x1D705}(x)=k$ . Since $\tilde{k}\subset \unicode[STIX]{x1D705}(\tilde{x})$ , the projection $Z_{x}\xrightarrow[{}]{\tilde{p}_{K}}\operatorname{Gal}(\tilde{k}|k)$ gives rise to a canonical surjective homomorphism $G_{x}\rightarrow \operatorname{Gal}(\tilde{k}|k)$ , which in general is not injective. On the other hand, if $\tilde{k}{\hookrightarrow}\unicode[STIX]{x1D705}(\tilde{x})$ is purely inseparable, then $G_{x}\rightarrow \operatorname{Gal}(\tilde{k}|k)$ is an isomorphism. Hence, if the sequence (∗) splits, then $\tilde{p}_{K}$ has sections $\tilde{s}_{x}:\operatorname{Gal}(\tilde{k}|k)\rightarrow Z_{x}\subset \operatorname{Gal}(\tilde{K}|K)$ , which we call sections above  $x$ . And, notice that the conjugacy classes of sections $\tilde{s}_{x}$ above $x$ build a ‘bouquet’, which is in a canonical bijection with the (non-commutative) continuous cohomology pointed set $\text{H}_{\text{cont}}^{1}(\operatorname{Gal}(\tilde{k}|k),T_{x})$ defined via the split exact sequence (∗).

Parallel to the case of points $x\in X$ , one has a similar situation for $k$ -valuations $v$ of $K$ as follows. For any prolongation $\tilde{v}$ of $v$ to $\tilde{K}$ , we denote by $T_{v}\subseteq Z_{v}$ the inertia/decomposition groups of $\tilde{v}|v$ and by $G_{v}=Z_{v}/T_{v}$ the residual automorphism group. As above, if $\unicode[STIX]{x1D705}(v)=k$ and $\tilde{k}{\hookrightarrow}\unicode[STIX]{x1D705}(\tilde{v})$ is purely inseparable, one has: first, the canonical homomorphism $G_{v}\rightarrow \operatorname{Gal}(\tilde{k}|k)$ is an isomorphism. Second, if the exact sequence $1\rightarrow T_{v}\rightarrow Z_{v}\rightarrow G_{v}\rightarrow 1$ splits, then the projection $\tilde{p}_{K}:\operatorname{Gal}(\tilde{K}|K)\rightarrow \operatorname{Gal}(\tilde{k}|k)$ has a section $\tilde{s}_{v}:\operatorname{Gal}(\tilde{k}|k)\rightarrow Z_{v}\subseteq \operatorname{Gal}(\tilde{K}|K)$ , which we call a section above  $v$ . And, the conjugacy classes of sections $\tilde{s}_{v}$ above $v$ build a ‘bouquet’, which is in a canonical bijection with the (non-commutative) continuous cohomology pointed set $\text{H}_{\text{cont}}^{1}(\operatorname{Gal}(\tilde{k}|k),T_{v})$ defined via the canonical split exact sequence above.

Finally, if $\tilde{K}|K$ contains a separable closure $K^{\text{s}}|K$ of $K$ and hence $\tilde{k}$ contains a corresponding separable closure $k^{\text{s}}$ of $k$ , then $\unicode[STIX]{x1D705}(s)^{\text{s}}\subseteq \unicode[STIX]{x1D705}(\tilde{x}),\unicode[STIX]{x1D705}(\tilde{v})$ and $G_{x}$ and $G_{v}$ are the absolute Galois groups of $\unicode[STIX]{x1D705}(x)$ and $\unicode[STIX]{x1D705}(v)$ , respectively. Further, in this situation, $1\rightarrow T_{v}\rightarrow Z_{v}\rightarrow G_{v}\rightarrow 1$ is split; see e.g. [KPR86]. Thus, if $\unicode[STIX]{x1D705}(v)=k$ , sections above $v$ exist. In particular, if $x\in X(k)$ is a $k$ -rational point, then choosing $v$ such that $\unicode[STIX]{x1D705}(x)=\unicode[STIX]{x1D705}(v)$ , it follows that sections above $x$ exist as well, because every section above $v$ is a section above $x$ as well. We mention though that in general the bouquet of sections above $x$ is much richer than the one of sections above $v$ . Namely, by general decomposition theory, one has $T_{v}\subset T_{x}$ , and $\text{H}_{\text{cont}}^{1}(\operatorname{Gal}(\tilde{k}|k),T_{v})\rightarrow \text{H}_{\text{cont}}^{1}(\operatorname{Gal}(\tilde{k}|k),T_{x})$ is a strict inclusion in general.

Next let $p$ be a fixed prime number. We denote by $K^{\prime }|K$ the (maximal) $\mathbb{Z}/p$ elementary abelian extension of $K$ , and by $K^{\prime \prime }$ the maximal $\mathbb{Z}/p$ elementary abelian extension of $K^{\prime }$ (in some fixed algebraic closure of $K$ ). Then $K^{\prime \prime }|K$ is a Galois extension, which we call the $\mathbb{Z}/p$ metabelian extension of $K$ , and its Galois group $\operatorname{Gal}(K^{\prime \prime }|K)$ is called the metabelian Galois group of $K$ . Note that $k^{\prime }:=\overline{k}\cap K^{\prime }$ and $k^{\prime \prime }:=\overline{k}\cap K^{\prime \prime }$ are the $\mathbb{Z}/p$ elementary abelian extension and the $\mathbb{Z}/p$ metabelian extension, respectively, of $k$ . Finally, consider the canonical surjective projections:

$$\begin{eqnarray}\operatorname{pr}_{K}^{\prime }:\operatorname{Gal}(K^{\prime }|K)\rightarrow \operatorname{Gal}(k^{\prime }|k),\quad \operatorname{pr}_{K}^{\prime \prime }:\operatorname{Gal}(K^{\prime \prime }|K)\rightarrow \operatorname{Gal}(k^{\prime \prime }|k).\end{eqnarray}$$

We will say that a group theoretical (continuous) section $s^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ of $\operatorname{pr}_{K}^{\prime }$ is liftable if there exists a section $s^{\prime \prime }:\operatorname{Gal}(k^{\prime \prime }|k)\rightarrow \operatorname{Gal}(K^{\prime \prime }|K)$ of $\operatorname{pr}_{K}^{\prime \prime }$ which lifts $s^{\prime }$ to $\operatorname{Gal}(k^{\prime \prime }|k)$ .

Note that if $p\neq \text{char}$ , and the $p$ th roots of unity $\unicode[STIX]{x1D707}_{p}$ are contained in $k$ and hence in $K$ , then by Kummer theory we have $K^{\prime }=K[\sqrt[p]{K}]$ and $K^{\prime \prime }=K^{\prime }[\sqrt[p]{K^{\prime }}]$ and similarly for $k$ .

Theorem A. In the above notation, let $k|\mathbb{Q}_{p}$ be finite with $\unicode[STIX]{x1D707}_{p}\subset k$ . Then the following hold.

1. (1) Every $k$ -rational point $x\in X$ gives rise to a bouquet of conjugacy classes of liftable sections $s_{x}^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ above  $x$ , which is in bijection with $\text{H}^{1}(\operatorname{Gal}(k^{\prime }|k),T_{x})$ .

2. (2) Let $p>2$ and $s^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ be a liftable section. Then there exists a unique $k$ -rational point $x\in X$ such that $s^{\prime }$ equals one of the sections $s_{x}^{\prime }$ as defined above.

Actually one can reformulate the question addressed by Theorem A in terms of $p$ -adic valuations, and get the following stronger result; see § 2(C) for notation, definitions, and a few facts on (formally) $p$ -adic valuations  $v$ , e.g., the $p$ -adic rank $d_{v}$ of  $v$ and $p$ -adically closed fields, and Ax and Kochen [AK66] and Prestel and Roquette [PR85], respectively, for proofs.

Theorem B. Let $k$ be a $p$ -adically closed field with $p$ -adic valuation $v$ of $p$ -adic rank $d_{v}$ , and suppose that $\unicode[STIX]{x1D707}_{p}\subset k$ . Let $K|k$ be an arbitrary field extension. Then the following hold.

1. (1) Let $w$ be a $p$ -adic valuation of $K$ of $p$ -adic rank $d_{w}=d_{v}$ . Then $w$ prolongs $v$ to $K$ , and gives rise to a bouquet of conjugacy classes of liftable sections $s_{w}^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ above  $w$ .

2. (2) Let $p>2$ and $s^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ be a liftable section. Then there exists a unique $p$ -adic valuation $w$ of $K$ of $p$ -adic rank $d_{w}=d_{v}$ such that $s^{\prime }=s_{w}^{\prime }$ for some $s_{w}^{\prime }$ as above.

Remark/Definition. As mentioned in Pop [Pop10], the condition $\unicode[STIX]{x1D707}_{p}\subset k$ is a necessary condition in the above theorems. Nevertheless, as mentioned in [Pop10], if $\unicode[STIX]{x1D707}_{p}$ is not contained in the base field, assertions similar to Theorems A and B above hold in the following form: let $l|\mathbb{Q}_{p}$ be some finite extension and $Y\rightarrow l$ a complete geometrically integral smooth variety with function field $L=\unicode[STIX]{x1D705}(Y)$ . Let $k|l$ be a finite Galois extension with $\unicode[STIX]{x1D707}_{p}\subset k$ . Setting $K:=Lk$ , consider the field extensions $K^{\prime }|K{\hookrightarrow}K^{\prime \prime }|K$ and $k^{\prime }|k{\hookrightarrow}k^{\prime \prime }|k$ as above. Then $k^{\prime }=K^{\prime }\cap \overline{l}$ and $k^{\prime \prime }=K^{\prime \prime }\cap \overline{l}$ , and $K^{\prime }|L$ and $K^{\prime \prime }|L$ , as well as $k^{\prime }|l$ and $k^{\prime \prime }|l$ , are Galois extensions too, and one gets surjective canonical projections

$$\begin{eqnarray}\operatorname{pr}_{L}^{\prime }:\operatorname{Gal}(K^{\prime }|L)\rightarrow \operatorname{Gal}(k^{\prime }|l),\quad \operatorname{pr}_{L}^{\prime \prime }:\operatorname{Gal}(K^{\prime \prime }|L)\rightarrow \operatorname{Gal}(k^{\prime \prime }|l).\end{eqnarray}$$

In these notations and context we will say that a section $s_{L}^{\prime }:\operatorname{Gal}(k^{\prime }|l)\rightarrow \operatorname{Gal}(K^{\prime }|L)$ of $\operatorname{pr}_{L}^{\prime }$ is liftable if there exists a section $s_{L}^{\prime \prime }:\operatorname{Gal}(k^{\prime \prime }|l)\rightarrow \operatorname{Gal}(K^{\prime \prime }|L)$ of $\operatorname{pr}_{L}^{\prime \prime }$ which lifts $s_{L}^{\prime }$ .

This being said, one has the following extensions of Theorems A and B.

Theorem A⁰. In the above notation and hypothesis, the following hold.

1. (1) Every $l$ -rational point $y\in Y$ gives rise to a bouquet of conjugacy classes of liftable sections $s_{y}^{\prime }:\operatorname{Gal}(k^{\prime }|l)\rightarrow \operatorname{Gal}(K^{\prime }|L)$ above  $y$ , which is in bijection with $\text{H}^{1}(\operatorname{Gal}(k^{\prime }|l),T_{y})$ .

2. (2) Let $p>2$ and $s_{L}^{\prime }:\operatorname{Gal}(k^{\prime }|l)\rightarrow \operatorname{Gal}(K^{\prime }|L)$ be a liftable section. Then there exists a unique $l$ -rational point $y\in Y$ such that $s_{L}^{\prime }$ equals one of the sections $s_{y}^{\prime }$ as defined above.

Theorem B⁰. Let $l$ be a $p$ -adically closed field with $p$ -adic valuation $v$ and let $L|l$ be an arbitrary field extension. Then in the above notation the following hold.

1. (1) Let $w$ be a $p$ -adic valuation of $L$ with $d_{w}=d_{v}$ . Then $w$ prolongs $v$ to $L$ , and gives rise to a bouquet of conjugacy classes of liftable sections $s_{w}^{\prime }:\operatorname{Gal}(k^{\prime }|l)\rightarrow \operatorname{Gal}(K^{\prime }|L)$ above $w$ .

2. (2) Let $p>2$ and $s_{L}^{\prime }:\operatorname{Gal}(k^{\prime }|l)\rightarrow \operatorname{Gal}(K^{\prime }|L)$ be a liftable section. Then there exists a unique $p$ -adic valuation $w$ of $L$ such that $d_{w}=d_{v}$ , and $s_{L}^{\prime }$ equals one of the sections $s_{w}^{\prime }$ as above.

Remark.

As mentioned in Pop [Pop10], the $\mathbb{Z}/p$ metabelian form of the birational $p$ -adic SC for curves implies the corresponding full Galois SC, which was proved in Koenigsmann [Koe05]. The same holds correspondingly for higher dimensional varieties, provided $p>2$ , thus implying Stix’s [Sti13] result in this case. Since the proof of the implication under discussion in the case of general varieties is word-by-word the same as that from [Pop10], we will not reproduce it here.

An interesting application of the results and techniques developed here is the following fact concerning the $p$ -adic section conjecture for varieties: let $k|\mathbb{Q}_{p}$ be a finite extension and $X$ a complete smooth $k$ -variety. Then there exists a finite effectively computable family of finite geometrically $\mathbb{Z}/p$ elementary abelian (ramified) covers $\unicode[STIX]{x1D711}_{i}:X_{i}\rightarrow X$ , $i\in I$ , satisfying:

1. (i) $\bigcup _{i}\unicode[STIX]{x1D711}_{i}(X_{i}(k))=X(k)$ , i.e., every $x\in X(k)$ ‘survives’ in at least one of the covers $X_{i}\rightarrow X$ ;

2. (ii) a section $s:G_{k}\rightarrow \unicode[STIX]{x1D70B}_{1}(X,\ast )$ can be lifted to a section $s_{i}:G_{k}\rightarrow \unicode[STIX]{x1D70B}_{1}(X_{i},\ast )$ for some $i\in I$ if and only if $s$ arises from a $k$ -rational point $x\in X(k)$ in the way described above.

The main technical tools for the proof of the above theorems are:

• ∙ the techniques developed in Pop [Pop10] (which refine facts/methods initiated in [Pop88]);

• ∙ the theory of rigid elements, as developed by several people: Ware [War81], Arason et al. [AEJ87], Koenigsmann [Koe95], Efrat [Efr99], etc. See Topaz [Top15] as the definitive reference.

As a final remark, we notice that the condition $p>2$ in the results above originates from the weaker results about recovering valuations from rigid elements in the case $p=2$ . This technical condition might be removable, but some new ideas/techniques might be necessary to do so; see the comment at the beginning of the proof of assertion (2) of Theorem B in § 3.

2 Reviewing a few known facts

For the reader’s sake, in this section we review a few known facts about valuation theory, decomposition theory, and (formally) $p$ -adic fields, but do not reproduce proofs.

(A) Generalities about valuations and their Hilbert decomposition theory

For an arbitrary field $K$ and an arbitrary valuation $v$ of $K$ , we denote usually by ${\mathcal{O}}_{v},\mathfrak{m}_{v}$ the valuation ring/ideal of $v$ , by $vK=K^{\times }/{\mathcal{O}}^{\times }$ the value group of $v$ , and by $Kv=:{\mathcal{O}}_{v}/\mathfrak{m}_{v}:=\unicode[STIX]{x1D705}(v)$ the residue field of $v$ . Further, $U_{v}^{1}:=1+\mathfrak{m}_{v}\subset U_{v}$ denote the groups of principal $v$ -units and $v$ -units, respectively. One has the following canonical exact sequences:

$$\begin{eqnarray}1\rightarrow \mathfrak{m}_{v}\rightarrow {\mathcal{O}}_{v}\rightarrow Kv\rightarrow 0\quad \text{and}\quad 1\rightarrow U_{v}^{1}\rightarrow {\mathcal{O}}_{v}^{\times }\rightarrow (Kv)^{\times }\rightarrow 1.\end{eqnarray}$$

The set of ideals of ${\mathcal{O}}_{v}$ is totally ordered with respect to inclusion. The subrings ${\mathcal{O}}_{1}\subseteq K$ with ${\mathcal{O}}_{v}\subseteq {\mathcal{O}}_{1}$ are precisely the localizations ${\mathcal{O}}_{1}:=({\mathcal{O}}_{v})_{\mathfrak{m}_{1}}$ with $\mathfrak{m}_{1}\in \operatorname{Spec}({\mathcal{O}}_{v})$ and, moreover, $\mathfrak{m}_{1}\subset {\mathcal{O}}_{v}$ , and $({\mathcal{O}}_{v})_{\mathfrak{m}_{1}}$ is a valuation ring with valuation ideal $\mathfrak{m}_{1}$ . Further, if $v_{1}$ is the corresponding valuation of $K$ , then ${\mathcal{O}}_{0}:={\mathcal{O}}_{v}/\mathfrak{m}_{1}$ is a valuation ring of $Kv_{1}$ with valuation ideal $\mathfrak{m}_{0}:=\mathfrak{m}_{v}/\mathfrak{m}_{1}$ , say of a valuation $v_{0}$ of $Kv_{1}$ . We say that $v_{1}$ is a coarsening of $v$ , and denote $v_{1}\leqslant v$ and $v_{0}:=v/v_{1}$ .

Conversely, if $v_{1}$ is a valuation of $K$ and $v_{0}$ is a valuation on the residue field $Kv_{1}$ , then the preimage of the valuation ring ${\mathcal{O}}_{v_{0}}\subseteq Kv_{1}$ under ${\mathcal{O}}_{v_{1}}\rightarrow Kv_{1}$ is a valuation ring ${\mathcal{O}}\subseteq {\mathcal{O}}_{v_{1}}$ having as valuation ideal the preimage $\mathfrak{m}\subset {\mathcal{O}}$ of $\mathfrak{m}_{v_{0}}$ . Hence, if $v$ is the valuation defined by ${\mathcal{O}}$ on $K$ , then $Kv=(Kv_{1})v_{0}$ and one has a canonical exact sequence of totally ordered groups:

$$\begin{eqnarray}0\rightarrow v_{0}(Kv_{1})\rightarrow vK\rightarrow v_{1}K\rightarrow 0.\end{eqnarray}$$

The relation between coarsening and decomposition theory is as follows. Let $\tilde{K}|K$ be a Galois extension and $\tilde{v}|v$ be a prolongation of $v$ to $\tilde{K}$ . Then the coarsenings $\tilde{v}_{1}$ of $\tilde{v}$ are in a canonical bijection with the coarsenings $v_{1}$ of $v$ via ${\mathcal{O}}_{\tilde{v}_{1}}\mapsto {\mathcal{O}}_{v_{1}}:={\mathcal{O}}_{\tilde{v}_{1}}\cap K$ ; thus, ${\mathcal{O}}_{\tilde{v}_{1}}={\mathcal{O}}_{\tilde{v}}\cdot {\mathcal{O}}_{v_{1}}$ . Let $\tilde{v}_{1}|v_{1}$ be given coarsenings of $\tilde{v}|v$ and $\tilde{K}\tilde{v}_{1}|Kv_{1}$ be the corresponding residue field extension. Then $\tilde{v}_{0}:=\tilde{v}/\tilde{v}_{1}$ is canonically a prolongation of $v_{0}:=v/v_{1}$ .

Fact 1. Let $T_{\tilde{v}}\subseteq Z_{\tilde{v}}$ and $T_{\tilde{v}_{1}}\subseteq Z_{\tilde{v}_{1}}$ be the corresponding inertia/decomposition groups, and set $G_{\tilde{v}_{1}}=\operatorname{Aut}(\tilde{K}\tilde{v}_{1}|Kv_{1})$ . Then one has a canonical exact sequence $1\rightarrow T_{\tilde{v}_{1}}\rightarrow Z_{\tilde{v}_{1}}\rightarrow G_{\tilde{v}_{1}}\rightarrow 1$ , and the inertia/decomposition groups satisfy:

1. (a) $Z_{\tilde{v}}\subseteq Z_{\tilde{v}_{1}}$ and $T_{\tilde{v}}\supseteq T_{\tilde{v}_{1}}$ . Further, $T_{\tilde{v}_{1}}$ is a normal subgroup of $Z_{\tilde{v}}$ ;

2. (b) via $1\rightarrow T_{\tilde{v}_{1}}\rightarrow Z_{\tilde{v}_{1}}\rightarrow G_{\tilde{v}_{1}}=Z_{\tilde{v}_{1}}/T_{\tilde{v}_{1}}\rightarrow 1$ , one has that $T_{\tilde{v}_{0}}=T_{\tilde{v}}/T_{\tilde{v}_{1}}$ and $Z_{\tilde{v}_{0}}=Z_{\tilde{v}}/T_{\tilde{v}_{1}}$ .

(B) Hilbert decomposition in elementary abelian extensions

Let $K$ be a field of characteristic prime to $p$ containing $\unicode[STIX]{x1D707}_{n}$ , where $n=p^{e}$ is a power of the prime number $p$ , and let $\tilde{K}=K[\sqrt[n]{K}]$ be the maximal $\mathbb{Z}/n$ elementary abelian extension of $K$ . Let $v$ be a valuation of $K$ , $\tilde{v}$ be some prolongation of $v$ to $\tilde{K}$ , and $V_{\tilde{v}}\subseteq T_{\tilde{v}}\subseteq Z_{\tilde{v}}$ be the ramification, the inertia, and the decomposition, groups of $\tilde{v}|v$ , respectively. We remark that because $\operatorname{Gal}(\tilde{K}|K)$ is commutative, the groups $V_{\tilde{v}}$ , $T_{\tilde{v}}$ , and $Z_{\tilde{v}}$ depend on $v$ only. Therefore, we will simply denote them by $V_{v}$ , $T_{v}$ , and $Z_{v}$ . Finally, we denote by $K^{Z}\subseteq K^{T}\subseteq K^{V}$ the corresponding fixed fields in $\tilde{K}$ . One has the following, see e.g. Pop [Pop10, §2] (where the case $n=p$ is dealt with; but the proof is similar for general $n=p^{e}$ and we will not reproduce the details here).

Fact 2. In the above notation, the following hold.

1. (1) Let $U^{v}:=1+p^{2e}\mathfrak{m}_{v}$ . Then $\sqrt[n]{U^{v}}\subset K^{Z}$ , and $K^{Z}=K[\sqrt[n]{1+\mathfrak{m}_{v}}]$ , provided $\text{char}(Kv)\neq p$ . In particular, if $w_{1}$ and $w_{2}$ are independent valuations of $K$ , then $Z_{w_{1}}\cap Z_{w_{2}}=1$ .

2. (2) If $p\neq \operatorname{char}(Kv)$ , then $V_{v}=1$ and $\tilde{K}\tilde{v}=\widetilde{Kv}$ and hence $G_{v}:=Z_{v}/T_{v}=\operatorname{Gal}(\widetilde{Kv}|Kv)$ in this case. And, if $p=\operatorname{char}(Kv)$ , then $V_{v}=T_{v}$ , and the residue field $\tilde{K}\tilde{v}$ contains $(Kv)^{1/n}$ and a maximal $\mathbb{Z}/n$ elementary abelian extension of $Kv$ .

3. (3) Let $L:=K_{v}^{\text{h}}$ be the Henselization of $K$ with respect to $v$ . Then $\tilde{L}=L\tilde{K}$ is a maximal $\mathbb{Z}/n$ elementary extension of $L$ . Therefore, we have $\operatorname{Gal}(\tilde{L}|L)\cong Z_{\tilde{v}}$ canonically.

(C) Formally $p$ -adic fields and $p$ -adic valuations

We recall a few basic facts about $p$ -adic valuations and (formally) $p$ -adically closed fields; see Ax and Kochen [AK66] and Prestel and Roquette [PR85] for more details.

1. (1) A valuation $v$ of a field $k$ is called (formally) $p$ -adic if its residue field $kv$ is a finite field, say $\mathbb{F}_{q}$ with $q=p^{f_{v}}$ elements, and the value group $vk$ has a minimal positive element $1_{v}$ such that $v(p)=e_{v}\cdot 1_{v}$ for some natural number $e_{v}>0$ . The number $d_{v}:=e_{v}f_{v}$ is called the $p$ -adic rank (or degree) of the $p$ -adic valuation $v$ . Note that a field $k$ carrying a $p$ -adic valuation $v$ must necessarily have $\operatorname{char}(k)=0$ , as $v(p)\neq \infty$ , and $\operatorname{char}(kv)=p$ .

2. (2) Let $v$ be a $p$ -adic valuation of $k$ with valuation ring ${\mathcal{O}}_{v}$ . Then ${\mathcal{O}}_{1}:={\mathcal{O}}_{v}[1/p]$ is the valuation ring of the unique maximal proper coarsening $v_{1}$ of $v$ , which is called the canonical coarsening of  $v$ . Note that setting $k_{0}:=kv_{1}$ , and $v_{0}:=v/v_{1}$ the corresponding valuation on $k_{0}$ , we have: $v_{0}$ is a $p$ -adic valuation of $k_{0}$ with $e_{v_{0}}=e_{v}$ and $f_{v_{0}}=f_{v}$ ; hence, $d_{v_{0}}=d_{v}$ and, moreover, $v_{0}$ is a discrete valuation of $k_{0}$ . In particular, the following hold.

   1. (a) $v$ has rank one if and only if $v_{1}$ is the trivial valuation if and only if $v=v_{0}$ .

   2. (b) Giving a $p$ -adic valuation $v$ of a field $k$ of $p$ -adic rank $d_{v}=e_{v}f_{v}$ is equivalent to giving a place $\mathfrak{p}$ of $k$ with values in a finite extension $\boldsymbol{k}_{0}$ of $\mathbb{Q}_{p}$ such that the residue field $k_{0}:=k\mathfrak{p}$ of $\mathfrak{p}$ is dense in $\boldsymbol{k}_{0}$ , and $\boldsymbol{k}_{0}|\mathbb{Q}_{p}$ has ramification index $e_{v}$ and residual degree $f_{v}$ .

   3. (c) If $v_{i}<v$ is a strict coarsening of $v$ , then $v_{i}\leqslant v_{1}$ , and the quotient valuation $v/v_{i}$ on the residue field $kv_{i}$ is a $p$ -adic valuation with $e_{v/v_{i}}=e_{v}$ and $f_{v/v_{i}}=f_{v}$ ; thus, $d_{v/v_{i}}=d_{v}$ . (Actually, $\unicode[STIX]{x1D705}(v_{i}/v_{1})\cong kv_{1}$ and $\unicode[STIX]{x1D705}(v_{i}/v)\cong kv$ canonically.)

3. (3) Let $v$ be a $p$ -adic valuation of $k$ , $l|k$ a finite field extension, and denote by $w|v$ the prolongations of $v$ to $l$ . Then the following hold.

   1. (a) All prolongations $w|v$ are $p$ -adic valuations. Further, the fundamental equality holds for the finite extension $l|k$ , i.e., $[l:k]=\sum _{w|v}e(w|v)f(w|v)$ , where $e(w|v)$ and $f(w|v)$ are the ramification index and the residual degree, respectively, of $w|v$ .

   2. (b) For each $w|v$ , let $w_{1}$ be the canonical coarsening of $w$ , and $w_{0}=w/w_{1}$ be the canonical quotient on the residue field $lw_{1}$ . Then by general decomposition theory of valuations one has $e(w|v)=e(w_{1}|v_{1})e(w_{0}|v_{0})$ and $f(w|v)=f(w_{0}|v_{0})$ . Further, $e_{w}=e(w_{0}|v_{0})e_{v}$ and $f_{w}=f(w|v)f_{v}$ ; thus, $d_{w}=e(w_{0}|v_{0})f(w|v)d_{v}$ .

   3. (c) In particular, if $l|k$ is Galois, and $w^{Z}$ is the restriction of $w$ to the decomposition field $l^{Z}$ of $w$ , then $e(w|w^{Z})=e(w|v)$ and $f(w|w^{Z})=f(w|v)$ ; thus, $w^{Z}$ is a $p$ -adic valuation having $p$ -adic rank equal to the one of $v$ . Further, the same is true for infinite Galois extensions  $l|k$ .

4. (4) A field $k$ is called (formally) $p$ -adically closed if $k$ carries a $p$ -adic valuation $v$ such that for every finite extension $\tilde{k}|k$ , one has: if $v$ has a prolongation $\tilde{v}$ to $\tilde{k}$ with $d_{\tilde{v}}=d_{v}$ , then $\tilde{k}=k$ . One has the following characterization of the $p$ -adically closed fields: for a field $k$ endowed with a $p$ -adic valuation $v$ , and its canonical coarsening $v_{1}$ , the following are equivalent.

   1. (i) $k$ is $p$ -adically closed with respect to $v$ .

   2. (ii) $v$ is Henselian and $v_{1}k$ is divisible (maybe trivial).

   3. (iii) $v_{1}$ is Henselian, $v_{1}k$ is divisible (maybe trivial), and the residue field $k_{0}:=kv_{1}$ is relatively algebraically closed in its $v_{0}=v/v_{1}$ completion $\boldsymbol{k}_{0}$ (itself a finite extension of  $\mathbb{Q}_{p}$ ).

   Further, the $p$ -adic valuation of a $p$ -adically closed field is definable and unique.

5. (5) Finally, for every field $k$ endowed with a $p$ -adic valuation $v$ , there exist $p$ -adic closures $\widehat{k},\widehat{v}$ such that $d_{\widehat{v}}=d_{v}$ . Moreover, the space of the $k$ -isomorphy classes of $p$ -adic closures of $k,v$ has a concrete description as follows: let $v_{1}$ be the canonical coarsening of $v$ , and $\boldsymbol{k}_{0}|\mathbb{Q}_{p}$ the completion of the residue field of $k_{0}=kv_{1}$ with respect to the discrete valuation $v_{0}=v/v_{1}$ . Recalling the canonical exact sequence $1\rightarrow I_{v_{1}}\xrightarrow[{}]{}D_{v}\xrightarrow[{}]{\text{pr}}G_{\boldsymbol{k}_{0}}\rightarrow 1$ , one has that the space of the isomorphy classes of $p$ -adic closures of $k,v$ is in bijection with the space of conjugacy classes of sections of $\text{pr}$ and thus with $\text{H}_{\text{cont}}^{1}(G_{\boldsymbol{k}_{0}},I_{v_{1}})$ .

6. (6) In the above notation, the following hold.

   1. (a) Let $k,v$ be a $p$ -adically closed field. Then $k_{0}=kv_{1}$ is $p$ -adically closed (with respect to $v_{0}$ ), and $k^{\text{abs}}$ is actually the relative algebraic closure of $\mathbb{Q}$ in $k_{0}$ . Further, $\overline{k}=k\overline{\mathbb{Q}}$ .

   2. (b) The elementary equivalence class of a $p$ -adically closed field $k$ is determined by both the absolute subfield $k^{\text{abs}}:=k\cap \overline{\mathbb{Q}}=k_{0}\cap \overline{\mathbb{Q}}$ of $k$ and the completion $\boldsymbol{k}_{0}$ of $k_{0}=kv_{1}$ with respect to $v_{0}$ (which equals the completion of $k^{\text{abs}}$ with respect to $v_{0}$ as well).

   3. (c) If $N$ is $p$ -adically closed with respect to the $p$ -adic valuation $w$ , and $k\subseteq N$ is a subfield which is relatively closed in $N$ , then $k$ is $p$ -adically closed with respect to $v:=w|_{k}$ , $v$ and $w$ have equal $p$ -adic ranks, and $N$ and $k$ are elementary equivalent.

   4. (d) If $N|k$ is an extension of $p$ -adically closed fields of the same rank, the following hold.

      • ∙ $\tilde{k}|k\mapsto N\tilde{k}$ defines a bijection from the set of algebraic extensions $\tilde{k}|k$ of $k$ onto the set of algebraic extensions of $N$ .

      • ∙ The canonical projection $G_{N}\rightarrow G_{k}$ is an isomorphism.

   5. (e) In particular, if $L|l$ is an extension of $p$ -adically closed fields of the same rank, in the notation from the Introduction, the following canonical projections are isomorphisms:

      

(D) Valuations and rigid elements

We recall the result of Arason et al. [AEJ87, Theorem 2.16]; see also Koenigsmann [Koe95], Ware [War81], Efrat [Efr99], and especially Topaz [Top15] for much more about this. The point is that one can recover valuations of a field $K$ from particular subgroups $T\subset K^{\times }$ as follows: let $T\subset K^{\times }$ be a subgroup with $-1\in T$ . We say that $x\in K^{\times }\backslash T$ is $T$ -rigid if $1+x\in T\cup xT$ ; and, by abuse of language, we say that $K$ is $T$ -rigid if all $x\in K^{\times }\backslash T$ are $T$ -rigid.

Theorem 3 (Arason et al.).

In the above notation, let $T\subset K^{\times }$ be a subgroup with $-1\in T$ such that $K$ is $T$ -rigid. Then there exists a valuation $v$ of $K$ whose valuation ideal $\mathfrak{m}_{v}$ satisfies $1+\mathfrak{m}_{v}\subseteq T$ , and whose valuation ring ${\mathcal{O}}_{v}$ has the property that $|{\mathcal{O}}_{v}^{\times }/(T\cap {\mathcal{O}}_{v}^{\times })|\leqslant 2$ .

3 Proof of Theorem B

To (1): Let $\widehat{K},\widehat{w}$ be a $p$ -adic closure of $K,w$ . Then $\widehat{w}$ prolongs $w$ and has $p$ -adic rank $d_{\widehat{w}}=d_{w}$ and thus equal to $d_{v}$ by the fact that $d_{w}=d_{v}$ . Therefore, since $k$ is $p$ -adically closed, $k$ must be relatively algebraically closed in $\widehat{K}$ . We conclude by using (†) from § 2(C)(6)(e), with $l:=k$ and $L:=\widehat{K}$ , and taking into account that the isomorphism $\operatorname{Gal}(\widehat{K}^{\prime \prime }|\widehat{K})\rightarrow \operatorname{Gal}(k^{\prime \prime }|k)$ factors through $\operatorname{Gal}(K^{\prime \prime }|K)\rightarrow \operatorname{Gal}(k^{\prime \prime }|k)$ and thus gives rise to a liftable section of $\operatorname{Gal}(K^{\prime }|K)\rightarrow \operatorname{Gal}(k^{\prime }|k)$ .

To (2): The proof of assertion (2) is divided into three main steps, whereas the hypothesis $p>2$ is used only in Step 2. This might be relevant when trying to address the case  $p=2$ .

Step 1. By Kummer theory, $\operatorname{pr}_{K}^{\prime }:\operatorname{Gal}(K^{\prime }|K)\rightarrow \operatorname{Gal}(k^{\prime }|k)$ is Pontrjagin dual to the canonical embedding $k^{\times }/p\rightarrow K^{\times }/p$ . Second, given a liftable section $s^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ of $\operatorname{pr}_{K}^{\prime }$ , it follows by Kummer theory that the Pontrjagin dual of $s^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ is a surjective projection $K^{\times }/p\rightarrow k^{\times }/p$ , whose kernel $\unicode[STIX]{x1D6F4}/p\subset K^{\times }/p$ is a complement of $k^{\times }/p\subset K^{\times }/p$ . That means that $s^{\prime }$ gives rise canonically to a presentation of $K^{\times }/p$ as a direct sum

For every $k$ -subfield $K_{\unicode[STIX]{x1D6FC}}\subset K$ which is relatively algebraically closed in  $K$ , one has a commutative diagram of surjective projections

and $s^{\prime }$ gives rise canonically to a liftable section $s_{\unicode[STIX]{x1D6FC}}^{\prime }$ of $\operatorname{pr}_{\unicode[STIX]{x1D6FC}}^{\prime }:\operatorname{Gal}(K_{\unicode[STIX]{x1D6FC}}^{\prime }|K_{\unicode[STIX]{x1D6FC}})\rightarrow \operatorname{Gal}(k^{\prime }|k)$ , etc. In particular, one has corresponding canonical presentations as direct sums

defined by the sections $s_{\unicode[STIX]{x1D6FC}}^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K_{\unicode[STIX]{x1D6FC}}^{\prime }|K_{\unicode[STIX]{x1D6FC}})$ induced by $s^{\prime }:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K)$ .

Claim.

In the above notions, one has $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}/p=\unicode[STIX]{x1D6F4}/p\cap K_{\unicode[STIX]{x1D6FC}}^{\times }/p$ and thus $\unicode[STIX]{x1D6F4}/p$ determines $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}/p$ .

Indeed, let $D_{\unicode[STIX]{x1D6FC}}:=\operatorname{im}(s_{\unicode[STIX]{x1D6FC}}^{\prime })\subset \operatorname{Gal}(K_{\unicode[STIX]{x1D6FC}}^{\prime }|K_{\unicode[STIX]{x1D6FC}})$ . Then, by the definition of $s_{\unicode[STIX]{x1D6FC}}^{\prime }$ , it follows that $D_{\unicode[STIX]{x1D6FC}}$ is the image of $D=\operatorname{im}(s^{\prime })$ under the canonical projection $\operatorname{Gal}(K^{\prime }|K)\rightarrow \operatorname{Gal}(K_{\unicode[STIX]{x1D6FC}}^{\prime }|K_{\unicode[STIX]{x1D6FC}})$ . In other words, by Pontrjagin duality, the projection $K_{\unicode[STIX]{x1D6FC}}^{\times }/p\rightarrow k^{\times }/p$ factors through the inclusion $K_{\unicode[STIX]{x1D6FC}}^{\times }/p{\hookrightarrow}K^{\times }/p$ . Hence, $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}/p$ is mapped into $\unicode[STIX]{x1D6F4}/p$ under $K_{\unicode[STIX]{x1D6FC}}^{\times }/p{\hookrightarrow}K^{\times }/p$ , which proves the claim.

Now let $T/p:=\unicode[STIX]{x1D6F4}/p\cdot {\mathcal{O}}_{v}^{\times }/p$ and $T\subset K^{\times }$ be the corresponding subgroup (thus containing the $p$ th powers in $K^{\times }$ ). Then, for every $k$ -subfield $K_{\unicode[STIX]{x1D6FC}}\subset K$ which is relatively algebraically closed in $K$ , by the remarks above one has that $T_{\unicode[STIX]{x1D6FC}}:=T\cap K_{\unicode[STIX]{x1D6FC}}^{\times }\subset K_{\unicode[STIX]{x1D6FC}}^{\times }$ satisfies $T_{\unicode[STIX]{x1D6FC}}/p=\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}/p\cdot {\mathcal{O}}_{v}^{\times }/p$ .

Finally, let $(K_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}}$ be the family of all the $k$ -subfields $K_{\unicode[STIX]{x1D6FC}}\subset K$ which are relatively algebraically closed in $K$ and satisfy $\text{tr}.\text{deg}(K_{\unicode[STIX]{x1D6FC}}|k)=1$ . Then, by Pop [Pop10, Theorem B], for every subfield $K_{\unicode[STIX]{x1D6FC}}$ , there exists a unique $p$ -adic valuation $w_{\unicode[STIX]{x1D6FC}}$ of $K_{\unicode[STIX]{x1D6FC}}$ prolonging the $p$ -adic valuation $v$ of $k$ to $K_{\unicode[STIX]{x1D6FC}}$ and having the same $p$ -adic rank as  $v$ . Our final aim is to show that there exists a (unique) $p$ -adic valuation $w$ of $K$ such that $w_{\unicode[STIX]{x1D6FC}}$ is the restriction of $w$ to $K_{\unicode[STIX]{x1D6FC}}$ for each $K_{\unicode[STIX]{x1D6FC}}$ .

Lemma 4. In the above notation, $K_{\unicode[STIX]{x1D6FC}}$ is $T_{\unicode[STIX]{x1D6FC}}$ -rigid. Further, $T=\bigcup _{\unicode[STIX]{x1D6FC}}T_{\unicode[STIX]{x1D6FC}}$ , and $K$ is $T$ -rigid.

Proof. We first show that ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}\subset T_{\unicode[STIX]{x1D6FC}}$ . Indeed, let $v$ be the $p$ -adic valuation of $k$ , and further consider: first, the canonical coarsening $v_{1}$ of $v$ and the canonical $p$ -adic valuation $v_{0}:=v/v_{1}$ on the residue field $k_{0}:=kv_{1}$ of $v_{1}$ . Second, consider the $p$ -adic valuation $w_{\unicode[STIX]{x1D6FC}}$ of $K_{\unicode[STIX]{x1D6FC}}$ , and let $w_{\unicode[STIX]{x1D6FC}1}$ and $w_{\unicode[STIX]{x1D6FC}0}:=w_{\unicode[STIX]{x1D6FC}}/w_{\unicode[STIX]{x1D6FC}1}$ and $K_{\unicode[STIX]{x1D6FC}0}$ be correspondingly defined. Notice that $w_{\unicode[STIX]{x1D6FC}}|_{k}=v$ implies that $w_{\unicode[STIX]{x1D6FC}1}|_{k}=v_{1}$ and $w_{\unicode[STIX]{x1D6FC}0}|_{k_{0}}=v_{0}$ . The following hold.

1. (a) First, by Fact 2, it follows that $\sqrt[p]{1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}}$ is contained in the decomposition field of $w_{\unicode[STIX]{x1D6FC}}$ over $K$ , which is actually the fixed field of $Z_{w_{\unicode[STIX]{x1D6FC}}}$ in $K_{\unicode[STIX]{x1D6FC}}^{\prime }$ . Second, the fixed field of $\operatorname{im}(s^{\prime })$ in $K_{\unicode[STIX]{x1D6FC}}^{\prime }$ is, by the mere definitions, generated as a field extension of $K$ by $\sqrt[p]{\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}^{^{\prime }}}$ . Thus, since $\operatorname{im}(s^{\prime })\subset Z_{w_{\unicode[STIX]{x1D6FC}}}$ , it follows by Kummer theory that $1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}\subset \unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}$ .

2. (b) Since, by the mere definition, one has $\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}1}}\subset \mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}$ and $p$ is invertible in ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}1}}$ , it follows that $1+\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}1}}\subset 1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}$ . Thus, one has finally $1+\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}1}}\subset \unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}$ as well.

3. (c) Since $w_{\unicode[STIX]{x1D6FC}}$ and $v$ have the same $p$ -adic rank, it follows by the discussion in § 2(C)(5) that $w_{\unicode[STIX]{x1D6FC}0}$ and $v_{0}$ are discrete $p$ -adic valuations of the same $p$ -adic rank and hence $k_{0}$ is dense in $K_{\unicode[STIX]{x1D6FC}0}$ . Therefore, since $w_{\unicode[STIX]{x1D6FC}0}|v_{0}$ are discrete valuations, and $k_{0}$ is dense in $K_{\unicode[STIX]{x1D6FC}0}$ under $k_{0}{\hookrightarrow}K_{\unicode[STIX]{x1D6FC}0}$ , one has that ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}0}}^{\times }={\mathcal{O}}_{v_{0}}^{\times }\cdot (1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}0}})$ and $K_{\unicode[STIX]{x1D6FC}0}^{\times }=k_{0}^{\times }\cdot (1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}0}})$ as well.

4. (d) Since $K_{\unicode[STIX]{x1D6FC}0}^{\times }={\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}1}}^{\times }/(1+\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}1}})$ , $k_{0}^{\times }={\mathcal{O}}_{v_{1}}^{\times }/(1+\mathfrak{m}_{v_{1}})$ , and $1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}0}}=(1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}})/(1+\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}1}})$ , from the equality $K_{\unicode[STIX]{x1D6FC}0}^{\times }=k_{0}^{\times }\cdot (1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}0}})$ above, it follows that ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}1}}^{\times }={\mathcal{O}}_{v_{1}}^{\times }\cdot (1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}})$ .

5. (e) Similarly, the equalities ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}0}}^{\times }={\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }/(1+\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}1}})$ and ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}0}}^{\times }={\mathcal{O}}_{v_{0}}^{\times }\cdot (1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}0}})$ imply that ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }={\mathcal{O}}_{v}^{\times }\cdot (1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}})$ .

Hence, since ${\mathcal{O}}_{v}^{\times },1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}\subset T_{\unicode[STIX]{x1D6FC}}$ , one finally has ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }={\mathcal{O}}_{v}^{\times }\cdot (1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}})\subset T_{\unicode[STIX]{x1D6FC}}$ , as claimed.

We next show that $K_{\unicode[STIX]{x1D6FC}}$ is $T_{\unicode[STIX]{x1D6FC}}$ -rigid. To do so, we first notice that by the discussion above, for any fixed element $\unicode[STIX]{x1D70B}\in {\mathcal{O}}_{v}$ of minimal positive value $1_{v}\in vk$ , the following holds: let $x\in {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}1}}^{\times }$ be an arbitrary $w_{\unicode[STIX]{x1D6FC}1}$ -unit. Then there exist $m\in \mathbb{Z}$ , $\unicode[STIX]{x1D716}\in {\mathcal{O}}_{v}^{\times }$ , and $x_{1}\in 1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}$ such that

Now let $x\in K_{\unicode[STIX]{x1D6FC}}^{\times }\backslash T_{\unicode[STIX]{x1D6FC}}$ be given. Then one has the following possibilities.

1. (1) $w_{\unicode[STIX]{x1D6FC}1}(x)>0$ . Then $1+x$ is a principal $w_{\unicode[STIX]{x1D6FC}1}$ -unit and, therefore, $1+x\in \unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}$ by assertion (b) above. Since $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}\subset T_{\unicode[STIX]{x1D6FC}}$ , we conclude that $1+x\in T_{\unicode[STIX]{x1D6FC}}$ .

2. (2) $w_{\unicode[STIX]{x1D6FC}1}(x)<0$ . Then $1+x=x(1+x^{-1})$ . Since $w_{\unicode[STIX]{x1D6FC}1}(x^{-1})>0$ , by the discussion above, it follows that $1+x^{-1}\in T_{\unicode[STIX]{x1D6FC}}$ . Therefore, one finally has that $1+x\in xT_{\unicode[STIX]{x1D6FC}}$ .

3. (3) $w_{\unicode[STIX]{x1D6FC}1}(x)=0$ or, equivalently, $x\in {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}1}}^{\times }$ . Let $x=\unicode[STIX]{x1D70B}^{m}\unicode[STIX]{x1D716}x_{1}$ be as given in (♯) above. One has:

$(\unicode[STIX]{x1D6FC})$

if $m>0$ , then $x\in \unicode[STIX]{x1D70B}^{m}\cdot {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }$ and thus $1+x\in {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }$ as well. Hence, by the relation (♯) above, $1+x=\unicode[STIX]{x1D702}_{1}\cdot \unicode[STIX]{x1D702}_{0}$ for some $\unicode[STIX]{x1D702}_{1}\in 1+p^{2}\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}\subset \unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D702}_{0}\in {\mathcal{O}}_{v}^{\times }$ . Thus, finally, $1+x\in T_{\unicode[STIX]{x1D6FC}}$ ;

$(\unicode[STIX]{x1D6FD})$

if $m<0$ , then $1+x=x(1+x^{-1})$ , and $x^{-1}$ has value $-m>0$ . But then, by the first case above, $1+x^{-1}\in T_{\unicode[STIX]{x1D6FC}}$ . Hence, $1+x=x(1+x^{-1})\in xT_{\unicode[STIX]{x1D6FC}}$ and thus $1+x\in xT_{\unicode[STIX]{x1D6FC}}$ ;

$(\unicode[STIX]{x1D6FE})$

if $m=0$ , then $x\in {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }\subset T_{\unicode[STIX]{x1D6FC}}$ and thus $x\not \in K_{\unicode[STIX]{x1D6FC}}^{\times }\backslash T_{\unicode[STIX]{x1D6FC}}$ .

For the $T$ -rigidity of $K$ , let $x\in K\backslash T$ be given. If $x\in k$ , then $x\in k\backslash {\mathcal{O}}_{v}^{\times }$ (by the definition of  $T$ ). An easy case by case analysis, namely $v(x)>0$ or $v(x)<0$ , shows that $1+x\in {\mathcal{O}}_{v}^{\times }\cup x{\mathcal{O}}_{v}^{\times }$ , etc. Finally, if $x\not \in k$ , then letting $K_{\unicode[STIX]{x1D6FC}}\subset K$ be the relative algebraic closure of $k(x)$ in $K$ , one has: since $x\in K\backslash T$ , one must have $x\in K_{\unicode[STIX]{x1D6FC}}\backslash T_{\unicode[STIX]{x1D6FC}}$ . Thus, by the discussion above, it follows that $1+x\in T_{\unicode[STIX]{x1D6FC}}\cup xT_{\unicode[STIX]{x1D6FC}}$ and, therefore, $1+x\in T\cup xT$ , etc.

This concludes the proof of Lemma 4. ◻

Step 2. Using Lemma 4 above and applying the Arason–Elman–Jacob theorem 3, we get: there exists a valuation $w$ on $K$ such that $|{\mathcal{O}}_{w}^{\times }/({\mathcal{O}}_{w}\cap T)|\leqslant 2$ and $1+\mathfrak{m}_{w}\subset T$ . Hence, letting ${\mathcal{O}}_{w}^{\times }T\subset K^{\times }$ be the subgroup generated by $T$ and ${\mathcal{O}}_{w}$ , one has ${\mathcal{O}}_{w}/({\mathcal{O}}_{w}\cap T)=({\mathcal{O}}_{w}^{\times }T)/T$ and thus $|({\mathcal{O}}_{w}^{\times }T)/T|\leqslant 2$ . We claim that ${\mathcal{O}}_{w}^{\times }\subset T$ . Indeed, first, one has $k^{\times }={\mathcal{O}}_{v}^{\times }\cdot \unicode[STIX]{x1D70B}^{\mathbb{Z}}$ as direct sum and hence $(k^{\times }/p)/({\mathcal{O}}_{v}^{\times }/p)=\unicode[STIX]{x1D70B}^{\mathbb{Z}/p}$ . Second, by definitions, one has that $K^{\times }/p=\unicode[STIX]{x1D6F4}/p\cdot k^{\times }/p$ and $T/p=\unicode[STIX]{x1D6F4}/p\cdot {\mathcal{O}}_{v}^{\times }/p$ , both of which being direct sums. Thus, finally one gets that

$$\begin{eqnarray}K^{\times }/p=\unicode[STIX]{x1D6F4}/p\cdot k^{\times }/p=\unicode[STIX]{x1D6F4}/p\cdot {\mathcal{O}}_{v}^{\times }/p\cdot \unicode[STIX]{x1D70B}^{\mathbb{Z}/p}=T/p\cdot \unicode[STIX]{x1D70B}^{\mathbb{Z}/p},\end{eqnarray}$$

where the dot denotes direct sums; in particular, one has $|K^{\times }/T|=|(K^{\times }/p)/(T/p)|=p$ . Hence, considering the canonical inclusions of groups $T\subseteq {\mathcal{O}}_{w}^{\times }T\subseteq K^{\times }$ , we get

$$\begin{eqnarray}p=|K^{\times }/T|=|K^{\times }/({\mathcal{O}}_{w}^{\times }T)|\cdot |({\mathcal{O}}_{w}^{\times }T)/T|.\end{eqnarray}$$

Since $|({\mathcal{O}}_{w}^{\times }T)/T|\leqslant 2$ and $2<p$ , it follows that $|({\mathcal{O}}_{w}^{\times }T)/T|=1$ is the only possibility; hence, $T={\mathcal{O}}_{w}^{\times }T$ , and, finally, ${\mathcal{O}}_{w}^{\times }\subseteq T$ . Hence, we conclude that $|K^{\times }/{\mathcal{O}}_{w}^{\times }|\geqslant p$ and therefore we have the following result.

$\bullet$ The valuation $w$ is a non-trivial valuation of $K$ .

Step 3. Recalling that ${\mathcal{O}}_{w}^{\times }\subset T$ , one has that the canonical projection $K^{\times }/{\mathcal{O}}_{w}^{\times }\rightarrow K^{\times }/T$ is surjective. Therefore, if $b\in K$ is a generator of $K^{\times }/T$ , e.g., $b=\unicode[STIX]{x1D70B}\in k_{0}$ has $v_{0}(\unicode[STIX]{x1D70B})=1$ , then $b$ is not a $w$ -unit and $w(b)$ is not divisible by  $p$ in $wK=K^{\times }/{\mathcal{O}}_{w}^{\times }$ and hence $wK$ is not divisible by  $p$ .

For every subfield $K_{\unicode[STIX]{x1D6FC}}\subset K$ as in the proof of Lemma 4, let $v_{\unicode[STIX]{x1D6FC}}:=w|_{K_{\unicode[STIX]{x1D6FC}}}$ be the restriction of $w$ to  $K_{\unicode[STIX]{x1D6FC}}$ . Then ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}={\mathcal{O}}_{w}\cap K_{\unicode[STIX]{x1D6FC}}$ and, therefore, ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }$ is contained in $T_{\unicode[STIX]{x1D6FC}}=T\cap K_{\unicode[STIX]{x1D6FC}}$ .

Lemma 5. The restriction $v_{\unicode[STIX]{x1D6FC}}:=w|_{K_{\unicode[STIX]{x1D6FC}}}$ of $w$ to $K_{\unicode[STIX]{x1D6FC}}$ equals the $p$ -adic valuation $w_{\unicode[STIX]{x1D6FC}}$ .

Proof. By the first part of the proof of Lemma 4, we have that ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }\subset T_{\unicode[STIX]{x1D6FC}}$ . Since ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }\subseteq T_{\unicode[STIX]{x1D6FC}}$ as well, it follows that the element-wise product ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }{\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }$ is contained in $T_{\unicode[STIX]{x1D6FC}}$ . Since $T_{\unicode[STIX]{x1D6FC}}$ is a proper subgroup of $K_{\unicode[STIX]{x1D6FC}}^{\times }$ , it follows that ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }{\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }\neq K^{\times }$ as well. The following is well-known valuation theoretical nonsense: let $\mathfrak{n}$ be the largest common ideal of ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}$ and ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ . Then ${\mathcal{O}}:={\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}{\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ equals both the localization of ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}$ at $\mathfrak{n}$ and the localization of ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ at $\mathfrak{n}$ . Further, ${\mathcal{O}}$ is the smallest valuation ring of $K$ which contains both ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}$ and ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ ; or, equivalently, ${\mathcal{O}}$ is the valuation ring of the finest common coarsening of $v_{\unicode[STIX]{x1D6FC}}$ and $w_{\unicode[STIX]{x1D6FC}}$ . We now claim that one has

$$\begin{eqnarray}{\mathcal{O}}^{\times }={\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }{\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }.\end{eqnarray}$$

Indeed, let $v_{\unicode[STIX]{x1D6FC}}^{1}$ and $w_{\unicode[STIX]{x1D6FC}}^{1}$ be the valuations of $\unicode[STIX]{x1D705}(\mathfrak{n}):={\mathcal{O}}/\mathfrak{n}$ defined by ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}/\mathfrak{n}$ and ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}/\mathfrak{n}$ , respectively. Then $v_{\unicode[STIX]{x1D6FC}}^{1}$ and $w_{\unicode[STIX]{x1D6FC}}^{1}$ are independent, and one has exact sequences

$$\begin{eqnarray}1\rightarrow (1+\mathfrak{n})\rightarrow {\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }\rightarrow {\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}^{1}}^{\times }\rightarrow 1\quad and\quad 1\rightarrow (1+\mathfrak{n})\rightarrow {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }\rightarrow {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}^{1}}^{\times }\rightarrow 1.\end{eqnarray}$$

Since $v_{\unicode[STIX]{x1D6FC}}^{1}$ and $w_{\unicode[STIX]{x1D6FC}}^{1}$ are independent valuations of $\unicode[STIX]{x1D705}(\mathfrak{n})$ , one has that ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}^{1}}^{\times }{\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}^{1}}^{\times }=\unicode[STIX]{x1D705}(\mathfrak{n})^{\times }$ and therefore

$$\begin{eqnarray}({\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }{\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times })/(1+\mathfrak{n})=\unicode[STIX]{x1D705}(\mathfrak{n})^{\times }.\end{eqnarray}$$

On the other hand, one also has ${\mathcal{O}}^{\times }/(1+\mathfrak{n})=\unicode[STIX]{x1D705}(\mathfrak{n})^{\times }$ . Further, $1+\mathfrak{n}$ is contained in both ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }$ and ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }$ and hence we conclude that ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }{\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }={\mathcal{O}}^{\times }$ , as claimed.

By contradiction, suppose that ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}\neq {\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ . Recall that the valuation ring ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ has finite residue field and hence ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ is minimal among the valuation rings of $K_{\unicode[STIX]{x1D6FC}}$ and, in particular, ${\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}$ cannot be contained in ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ . Therefore, in the above notation, one has that ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}\subset {\mathcal{O}}$ strictly or, equivalently, $\mathfrak{n}\subset \mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}$ is a strict inclusion. On the other hand, if $b\in k$ is any element of minimal positive value  $1_{v}$ , then $\mathfrak{m}_{w_{\unicode[STIX]{x1D6FC}}}=b{\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}$ and, therefore, $b\not \in \mathfrak{n}$ . Thus, we have

$$\begin{eqnarray}b\in {\mathcal{O}}^{\times }={\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}^{\times }{\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}^{\times }\subseteq T_{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

contradicting the fact that $w(b)$ generates $wK/w(T)\cong \mathbb{Z}/p$ . Thus, we conclude that one must have ${\mathcal{O}}_{w_{\unicode[STIX]{x1D6FC}}}={\mathcal{O}}_{v_{\unicode[STIX]{x1D6FC}}}$ , and Lemma 5 is proved.◻

We next claim that $w$ is a $p$ -adic valuation of $K$ having $p$ -adic rank $d_{w}=d_{v}$ . Indeed, for $t\in {\mathcal{O}}_{w}$ , let $K_{\unicode[STIX]{x1D6FC}}\subset K$ be the relative algebraic closure of $k(t)$ in $K$ . Then $K_{\unicode[STIX]{x1D6FC}}|k$ has transcendence degree ${\leqslant}1$ and, therefore, $w|_{K_{\unicode[STIX]{x1D6FC}}}=w_{\unicode[STIX]{x1D6FC}}$ is the $p$ -adic valuation $w_{\unicode[STIX]{x1D6FC}}$ by Lemma 5. In particular, if $b\in k$ is such that $v(b)=1_{v}$ is the minimal positive element of $v(k^{\times })$ , it follows that $w_{\unicode[STIX]{x1D6FC}}(b)$ is the minimal positive element of $w_{\unicode[STIX]{x1D6FC}}K_{\unicode[STIX]{x1D6FC}}$ under $vk{\hookrightarrow}w_{\unicode[STIX]{x1D6FC}}K_{\unicode[STIX]{x1D6FC}}$ and, further, $kv=K_{\unicode[STIX]{x1D6FC}}w_{\unicode[STIX]{x1D6FC}}$ is the finite field of cardinality $f_{v}=f_{w_{\unicode[STIX]{x1D6FC}}}$ . One has the following.

1. (a) $w(b)$ is the minimal positive element of $w(K^{\times })$ . Indeed, for $t\in \mathfrak{m}_{w}$ , in the above notation one has $w(t)=w_{\unicode[STIX]{x1D6FC}}(t)\geqslant w_{\unicode[STIX]{x1D6FC}}(b)=w(b)$ .

2. (b) $kv=Kw$ and thus $f_{v}=f_{w_{\unicode[STIX]{x1D6FC}}}$ . Indeed, if $t\in {\mathcal{O}}_{w}$ , then in the above notation the residue $\overline{t}\in Kw$ satisfies $\overline{t}\in K_{\unicode[STIX]{x1D6FC}}w_{\unicode[STIX]{x1D6FC}}=kv$ .

Therefore, $w$ is a $p$ -adic valuation of rank $d_{w}=d_{v}$ , which is unique, by the uniqueness of $w_{\unicode[STIX]{x1D6FC}}=w|_{K_{\unicode[STIX]{x1D6FC}}}$ for every subfield $K_{\unicode[STIX]{x1D6FC}}$ . This concludes the proof of Theorem B.

4 Proof of the other announced results

(A) Proof of Theorem A

The following stronger assertion holds (from which Theorem A immediately follows).

Theorem 6. Let $k|\mathbb{Q}_{p}$ be a finite extension containing the $p$ th roots of unity, and let $k_{0}\subseteq k$ be a subfield which is relatively algebraically closed in $k$ . Let $X_{0}$ be a complete smooth $k_{0}$ -variety, and $K_{0}=k_{0}(X)$ be the function field of $X_{0}$ . The following hold.

1. (1) Every $k$ -rational point $x\in X_{0}$ gives rise to a bouquet of conjugacy classes of liftable sections  $s_{x}^{\prime }$ of  $\operatorname{Gal}(K_{0}^{\prime }|K_{0})\rightarrow \operatorname{Gal}(k_{0}^{\prime }|k_{0})$ above  $x$ .

2. (2) Suppose that $p>2$ and let $s^{\prime }$ be a liftable section of $\operatorname{Gal}(K_{0}^{\prime }|K_{0})\rightarrow \operatorname{Gal}(k_{0}^{\prime }|k_{0})$ . Then there exists a unique $k$ -rational point $x\in X_{0}$ such that $s^{\prime }$ equals one of the sections $s_{x}^{\prime }$ above.

Proof. The proof is very similar to the proof of Pop [Pop10, Theorem A]. We repeat here the arguments briefly for the reader’s sake.

To (1): Let $v$ be the valuation of $k$ . We notice that by § 2(C)(b), there exists a bijection from the set of (equivalence classes of) $p$ -adic valuations $w$ of $K_{0}=\unicode[STIX]{x1D705}(X_{0})$ with $d_{w}=d_{v}$ onto the set of bouquets of liftable sections above $k$ -rational points $x$ of $X_{0}$ , which sends each $w$ to the corresponding bouquet of liftable sections above the center $x$ of the canonical coarsening $w_{1}$ on $X=X_{0}\times _{k_{0}}k$ . We conclude by applying assertion (1) of Theorem B.

To (2): Since $k_{0}\subseteq k$ is relatively algebraically closed, it follows that $k_{0}$ is $p$ -adically closed. Let $v$ be the valuation of $k$ and of all subfields of $k$ . Since $k_{0}$ is $p$ -adically closed, we can apply Theorem B and get: for every liftable section $s^{\prime }$ of $\operatorname{Gal}(K_{0}^{\prime }|K_{0})\rightarrow \operatorname{Gal}(k_{0}^{\prime }|k_{0})$ , there exists a unique $p$ -adic valuation $w$ of $K_{0}$ which prolongs $v$ to $K_{0}$ and has $p$ -adic rank equal to the $p$ -adic rank of $v$ , such that $s^{\prime }$ is a section above $w$ . Let $w_{1}$ be the canonical coarsening of $w$ . Then we have the following cases.

Case 1. The valuation $w_{1}$ is trivial.

Then $w$ is a discrete $p$ -adic valuation of $K$ prolonging $v$ to $K$ , having the same residue field and the same value group as $v$ . Equivalently, the completions of $k_{0}$ and $K_{0}$ are equal and hence equal to $k$ . Therefore, $w$ is uniquely determined by the embedding $\imath _{w}:(K_{0},w){\hookrightarrow}(k,v)$ . In geometric terms, $\imath _{w}$ defines a $k$ -rational point $x$ of $X_{0}$ , etc.

Case 2. The valuation $w_{1}$ is not trivial.

Then $w_{1}$ is a $k_{0}$ -rational place of $K_{0}$ and hence defines a $k_{0}$ -rational point $x_{0}$ of $X_{0}$ ; hence, by base change, a $k$ -rational point $x$ of $X_{0}$ as well, etc.◻

(B) Proof of Theorem B⁰

The proof is almost identical with the one of Theorem B⁰ from Pop [Pop10]. The proof of assertion (1) is identical with the proof of assertion (1) of Theorem B; thus, we omit it. Concerning the proof of assertion (2), let $s_{L}^{\prime }:\operatorname{Gal}(k^{\prime }|l)\rightarrow \operatorname{Gal}(K^{\prime }|L)$ be a given liftable section of $\operatorname{pr}_{L}^{\prime }:\operatorname{Gal}(K^{\prime }|L)\rightarrow \operatorname{Gal}(k^{\prime }|l)$ . Then considering the restriction

$$\begin{eqnarray}s^{\prime }:=\operatorname{pr}_{L}^{\prime }|_{\operatorname{ Gal}(k^{\prime }|k)}:\operatorname{Gal}(k^{\prime }|k)\rightarrow \operatorname{Gal}(K^{\prime }|K),\end{eqnarray}$$

it follows by mere definitions that $s^{\prime }$ is a liftable section of $\operatorname{pr}_{K}^{\prime }:\operatorname{Gal}(K^{\prime }|K)\rightarrow \operatorname{Gal}(k^{\prime }|k)$ . Hence, by Theorem B, there exists a unique $p$ -adic valuation $w^{1}$ of $K$ which prolongs the $p$ -adic valuation $v_{k}$ of $k$ to $K$ and has $d_{w^{1}}=d_{v_{k}}$ , and $s^{\prime }=s_{w^{1}}$ in the usual way.

Let $w=w^{1}|_{L}$ be the restriction of $w^{1}$ to  $L$ . Then $w$ prolongs the valuation $v$ of $l$ to $L$ . We claim that $w^{1}$ is the unique prolongation of $w$ to  $K$ . Indeed, let $w^{2}:=w^{1}\circ \unicode[STIX]{x1D70E}_{0}$ , with $\unicode[STIX]{x1D70E}_{0}\in \operatorname{Gal}(k|l)$ , be a further prolongation of $w$ to $K$ . Then, if $(w^{i})^{\prime }$ is a prolongation of $w^{i}$ to $K^{\prime }$ , $i=1,2$ , and $\unicode[STIX]{x1D70E}\in \operatorname{im}(s_{L}^{\prime })$ is a preimage of $\unicode[STIX]{x1D70E}_{0}$ , then $(w^{2})^{\prime }:=(w^{1})^{\prime }\circ \unicode[STIX]{x1D70E}$ is a prolongation of $w^{2}$ to $K^{\prime }$ . Therefore, if $Z_{w^{1}}\subset \operatorname{Gal}(K^{\prime }|K)$ is the decomposition group above $w^{1}$ , then $Z_{w^{2}}:=\unicode[STIX]{x1D70E}Z_{w^{1}}\unicode[STIX]{x1D70E}^{-1}$ is the decomposition group above $w^{2}$ . On the other hand, $\operatorname{im}(s^{\prime })\subseteq Z_{w^{1}}$ by Theorem B. Since $\operatorname{Gal}(k^{\prime }|k)$ is a normal subgroup of $\operatorname{Gal}(k^{\prime }|l)$ , it follows that $\operatorname{im}(s^{\prime })$ is normal in $\operatorname{im}(s_{L}^{\prime })$ . Hence, if $\unicode[STIX]{x1D70E}\in \operatorname{im}(s_{L}^{\prime })$ , it follows that $\unicode[STIX]{x1D70E}(\operatorname{im}(s^{\prime }))\unicode[STIX]{x1D70E}^{-1}=\operatorname{im}(s^{\prime })$ and, therefore, one has

$$\begin{eqnarray}Z_{w^{1}}\supseteq \operatorname{im}(s^{\prime })=\unicode[STIX]{x1D70E}(\operatorname{im}(s^{\prime }))\unicode[STIX]{x1D70E}^{-1}\subseteq \unicode[STIX]{x1D70E}Z_{w^{1}}\unicode[STIX]{x1D70E}^{-1}=Z_{w^{2}}.\end{eqnarray}$$

Hence, $\operatorname{im}(s^{\prime })\subset Z_{w^{1}},Z_{w^{2}}$ ; thus, by the uniqueness assertion of Theorem B, we must have $w^{1}=w^{2}$ . Equivalently, if $\unicode[STIX]{x1D70E}\in \operatorname{im}(s_{L}^{\prime })$ , then $\unicode[STIX]{x1D70E}Z_{w^{1}}\unicode[STIX]{x1D70E}^{-1}=Z_{w^{1}}$ and therefore $\unicode[STIX]{x1D70E}\in Z_{w^{1}}$ . Finally, we conclude that $d_{w}=d_{v}$ , as claimed, and this concludes the proof of Theorem B⁰ .

(C) Proof of Theorem A⁰

The following stronger assertion holds (from which Theorem A⁰ follows immediately).

Theorem 7. Let $l|\mathbb{Q}_{p}$ be a finite extension. Let $l_{0}\subset l$ be a relatively algebraically closed subfield, and $k_{0}|l_{0}$ a finite Galois extension with $\unicode[STIX]{x1D707}_{p}\subset k_{0}$ . Let $Y_{0}$ be a complete smooth geometrically integral variety over $l_{0}$ . Let $L_{0}=\unicode[STIX]{x1D705}(Y_{0})$ the function field of $Y_{0}$ , and $K_{0}=L_{0}k_{0}$ .

1. (1) Every $l$ -rational point $y\in Y_{0}$ gives rise to a bouquet of conjugacy classes of liftable sections  $s_{y}^{\prime }$ of  $\operatorname{Gal}(K_{0}^{\prime }|L_{0})\rightarrow \operatorname{Gal}(k_{0}^{\prime }|l_{0})$ above $y$ .

2. (2) Let $p>2$ and $s^{\prime }:\operatorname{Gal}(k_{0}^{\prime }|l_{0})\rightarrow \operatorname{Gal}(K_{0}^{\prime }|L_{0})$ be a liftable section of  $\operatorname{Gal}(K_{0}^{\prime }|L_{0})\rightarrow \operatorname{Gal}(k_{0}^{\prime }|l_{0})$ . Then there exists a unique $l$ -rational point $y\in Y_{0}(l)$ such that $s^{\prime }$ equals one of the sections $s_{y}^{\prime }$ introduced in point (1) above.

Proof. The proof is identical to the proof of Theorem A above, with the only difference that one uses Theorem B⁰ instead of Theorem B. ◻