The local pro-p anabelian geometry of curves
Let \(p\) be a prime number. We call sub-\(p\)-adic any subfield of a finitely generated extension of \(\mathbb{Q}_p\). In this paper we prove a certain version of Grothendieck's anabelian conjecture on the arithmetic fundamental groups of hyperbolic curves over sub-\(p\)-adic fields. (Note that this includes number fields, as well as finite extensions of \(\mathbb{Q}_p\).) The proof of this result builds on the previous anabelian work of H. Nakamura and A. Tamagawa, as well as on the \(p\)-adic Hodge theory of G. Faltings and Bloch-Kato. Our main result may he stated as follows: Let \(K\) be a sub-\(p\)-adic field. Let \(X_K\) be a hyperbolic curve over \(K\) -- i.e., if \(\overline K\) is an algebraic closure of \(K\), then \(X_{\overline K}{\overset {def} =}X_K \otimes_K \overline K\) is obtained by removing \(r\) points from a smooth, proper, geometrically connected curve of genus \(g\), where \(2g-2+r>0\). Let \(\Gamma_K {\overset {def}=} \text{Gal}(\overline K/K)\) be the absolute Galois group of \(K\). Note that we have a natural projection \(\pi_1(X_K)\to\Gamma_K\), whose kernel may be identified with \(\pi_1 (X_{\overline K})\) (where we omit the base-points, which are not important here). Let us write \(\Pi_{X_K}\) for the quotient of \(\pi_1(X_K)\) by the kernel of the projection \(\pi_1(X_{\overline K}) \to\pi_1^{(p)} (X_{\overline K})\) from \(\pi_1 (X_{\overline K})\) to its maximal pro-\(p\) quotient. Write \(\Delta_X\subseteq \Pi_{X_K}\) for the image of \(\pi_1 (X_{\overline K})\) in \(\Pi_{X_K}\). Let \(S_K\) be a smooth variety over \(K\); thus, we may similarly form \(\Pi_{S_K}\). Then the main result states that the natural morphism \[ \Hom_K^{dom}(S_K,X_K) \to\Hom_{\Gamma_K}^{\text{open}} (\Pi_{S_K}, \Pi_{X_K}) \] (induced by functoriality) from the set of dominant \(K\)-morphisms \(S_K\to X_K\) to the set of open continuous homomorphisms \(\Pi_{S_K} \to\Pi_{X_K}\) compatible with the projections to \(\Gamma_K\), and considered up to composition with an inner automorphism arising from \(\Delta_X\) is a bijection.
hyperbolic curves over sub-\(p\)-adic fields
scientific article; zbMATH DE number 1383529