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

Let K$K$ be a field with GK(2)≃GQ2(2)$G_K(2) \simeq G_{\mathbb {Q}_2}(2)$, where GF(2)$G_F(2)$ denotes the maximal pro‐2 quotient of the absolute Galois group of a field F$F$. We prove that then K$K$ admits a (non‐trivial) valuation v$v$ which is 2‐henselian and has residue field F2$\mathbb {F}_2$. Furthermore, v(2)$v(2)$ is a minimal positive element in the value group Γv$\Gamma _v$ and [Γv:2Γv]=2$[\Gamma _v:2\Gamma _v]=2$. This forms the first positive result on a more general conjecture about recovering p$p$‐adic valuations from pro‐p$p$ 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$X$ over Q2$\mathbb {Q}_2$, as well as an analogue for varieties.