Recovering p-adic valuations from pro-p Galois groups - Koenigsmann, J; Strommen, K (Wiley, 2024)
Let (Formula presented.) be a field with (Formula presented.), where (Formula presented.) denotes the maximal pro-2 quotient of the absolute Galois group of a field (Formula presented.). We prove that then (Formula presented.) admits a (non-trivial) valuation (Formula presented.) which is 2-henselian and has residue field (Formula presented.). Furthermore, (Formula presented.) is a minimal positive element in the value group (Formula presented.) and (Formula presented.). This forms the first positive result on a more general conjecture about recovering (Formula presented.) -adic valuations from pro- (Formula presented.) 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 (Formula presented.) over (Formula presented.), as well as an analogue for varieties.