Submission date: 19 June 2015.

Abstract:

Let K be a field with G_K(2) ≅ G_{Q_2}(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 F_2. Furthermore, v(2) is a minimal positive element in the value group Γ_v and [Γ_v:2Γ_v]=2. This forms the first positive result on a more general conjecture about the structure of pro-p Galois groups. As an application, we prove a strong version of the birational section conjecture for smooth, complete curves X over Q_2, as well as an analogue for varieties.

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Full text arXiv 1506.05956: pdf, ps.