Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Recovering p-adic valuations from pro-p Galois groups

Jochen Koenigsmann, Kristian Strommen

Abstract

Let $K$ be a field with $G_K(2) \simeq G_{\mathbb{Q}}(2)$, where $G_F(2)$ denotes the maximal pro-2 quotient of the absolute Galois group of a field $F$. We prove that then $K$ admits a (non-trivial) valuation $v$ which is 2-henselian and has residue field $\mathbb{F}_2$. Furthermore, $v(2)$ is a minimal positive element in the value group $Γ_v$ and $[Γ_v:2Γ_v]=2$. This forms the first positive result on a more general conjecture about recovering $p$-adic valuations from pro-$p$ Galois groups which we formulate precisely. As an application, we show how this result can be used to easily obtain number-theoretic information, by giving an independent proof of a strong version of the birational section conjecture for smooth, complete curves $X$ over $\mathbb{Q}_2$, as well as an analogue for varieties.

Recovering p-adic valuations from pro-p Galois groups

Abstract

Let be a field with , where denotes the maximal pro-2 quotient of the absolute Galois group of a field . We prove that then admits a (non-trivial) valuation which is 2-henselian and has residue field . Furthermore, is a minimal positive element in the value group and . This forms the first positive result on a more general conjecture about recovering -adic valuations from pro- Galois groups which we formulate precisely. As an application, we show how this result can be used to easily obtain number-theoretic information, by giving an independent proof of a strong version of the birational section conjecture for smooth, complete curves over , as well as an analogue for varieties.

Paper Structure

This paper contains 18 sections, 36 theorems, 60 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $p$-Henselian valuations
  4. Demushkin groups and their invariants
  5. Demushkin fields and field-theoretic interpretation of the Demushkin invariants
  6. The structure of $G_{\mathbb{Q}_2}(2)$.
  7. Step 1 of the main result: recovery of multiplicative structure and norm groups
  8. Step 2 of the main result: construction of the valuation via norm combinatorics
  9. Application: the Birational Section Conjecture
  10. Basic statement of the birational section conjecture
  11. The pro-$p$ version of the birational section conjecture
  12. Description of norm-combinatorics algorithm
  13. Proof of Lemma \ref{['lemma24']} and Proposition \ref{['propxy']}
  14. Ruling out the exceptional case
  15. Preliminary observations about Case B.
  16. ...and 3 more sections

Key Result

Lemma \oldthetheorem

There exists a unique $s \in \mathbb{N} \cup \{\infty\}$ such that with $n = \text{rank}(G)$ and $p^{\infty} = 0$ by convention.

Theorems & Definitions (71)

  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • ...and 61 more