Recovering p $p$ -adic valuations from pro- p $p$ Galois groups

Abstract

Let $K$ be a field with $G_K\left(2\right)\simeq G_{Q_2}\left(2\right)$, where $G_F\left(2\right)$ 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 $F_2$. Furthermore, $v\left(2\right)$ 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 $Q_2$, as well as an analogue for varieties.