Recovering \(p\)-adic valuations from pro-\(p\) Galois groups
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.
scientific article; zbMATH DE number 7846304