Recovering p-adic valuations from pro-p Galois groups

TL;DR

This paper demonstrates that certain pro-2 Galois groups uniquely determine a valuation on the field, revealing deep connections between Galois theory and valuation theory, and applies this to number theory and algebraic geometry.

Contribution

It proves that fields with a specific pro-2 Galois group admit a unique 2-henselian valuation with particular properties, advancing the understanding of recovering valuations from Galois groups.

Findings

Existence of a 2-henselian valuation with residue field F_2

Valuation's value group has minimal positive element at 2

Application to birational section conjecture over Q_2

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 $\Gamma_v$ and $[\Gamma_v:2\Gamma_v]=2$. 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.