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

Abstract

AbstractLet 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.