Recovering \(p\)-adic valuations from pro-\(p\) Galois groups
type: Item
instance of: scholarly article
title: Recovering \(p\)-adic valuations from pro-\(p\) Galois groups
author: Jochen Koenigsmann
author: Kristian Strommen
published in: Journal of the London Mathematical Society. Second Series
publication date: 2024-05-13T00:00:00Z
review text: Let \(K\) be a field and \(p\) be a prime. A \(p\)-henselian valuation on \(K\) is a valuation on \(K\) which extends uniquely to every Galois extension of degree \(p\). Now we assume that \(G_K(2)\simeq G_{\mathbb{Q}_2}\), where \(G_K(2)\) denotes the maximal pro-\(2\) quotient of the absolute Galois group of \(K\). Let \(T=\mbox{Norm}_{K(\sqrt{5})/K}\left(K(\sqrt{5})^{\times}\right)\) \(\mathcal{O}_1=\{x\in K\backslash T\; :\; 1+x\in T\}\), \(\mathcal{O}_2=\{ x\in T\; :\; x\mathcal{O}_1\subseteq \mathcal{O}_1\}\) and \(\mathcal{O}=\mathcal{O}_1\cup \mathcal{O}_2\) (the authors show that \(T\) is generated by \(-1\) and \(5\) modulo \(K^2\)). The main theorem proves that \(\mathcal{O}\) is a valuation ring of \(K\), with residue field \(\mathbb{F}_2\). The set of units is \(\mathcal{O}^{\times} =\{x\in T\; :\; 1+2x\in T\; \text{ and } \; 2+x\in T\}\). Denote by \(v\) the valuation associated to \(\mathcal{O}\), then \(v\) is \(2\)-Henselian, \([vK^{\times}:2vK^{\times}]=2\) and \(v(2)\) is the minimal positive element of \(vK\). In a corollary of this theorem they let \(X\) be a smooth complete variety over \(\mathbb{Q}_2\). They prove that every section of the canonical projection \(G_{\mathbb{Q}_2(X)}(2)\twoheadrightarrow G_{\mathbb{Q}_2}(2)\) lies above a unique \(\mathbb{Q}_2\)-valuation \(v\) (so \(X(\mathbb{Q}_2)\neq \emptyset\)); if \(X\) is a curve, then \(v\) corresponds to a \(\mathbb{Q}_2\)-rational point.
reviewed by: Gerard Leloup
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 11G25
zbMATH DE Number: 7846304
zbMATH Keywords: \(p\)-adic valuation
zbMATH Keywords: pro-\(p\) Galois group
zbMATH Keywords: \(p\)-henselian valuation
zbMATH Keywords: smooth curve
zbMATH Keywords: rational point
MaRDI profile type: Publication
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DOI: 10.1112/JLMS.12901
description: scientific article; zbMATH DE number 7846304