Action of the Galois group on the periods of certain Mumford curves
In this paper, \(C\) is a cyclic Galois covering of \(\mathbb{P}^1_K\), where \(K\) is an algebraically closed field, complete with respect to a nonarchimedean valuation. It is assumed that \(C\) is a Mumford curve, i.e. admits a uniformization by a Schottky group \(\Gamma \subset \text{PGL}_2 (K)\). The Jacobian of \(C\) can be analytically constructed as \((K^\times)^g/R\); here \(g\) is the genus of \(C\), and the ``period lattice'' \(R\) is canonically isomorphic to the abelianized group \(\Gamma_{\text{ab}} = \Gamma/[\Gamma, \Gamma]\) [see e.g. \textit{L. Gerritzen} and \textit{M. van der Put}, ``Schottky groups and Mumford curves'', Lect. Notes Math. 817 (1980; Zbl 0442.14009)]. The aim of this paper is to study the structure of \(\Gamma_{\text{ab}}\) as a module under the Galois group \(\langle \sigma \rangle\) of the covering \(f : C \to \mathbb{P}^1_K\). The main result is that, as a \(\mathbb{Z} [b]/ \text{Tr} (\sigma)\)-module, \(\Gamma_{\text{ab}}\) is projective, more precisely a direct sum of \({r \over 2} - 1\) cyclic modules, where \(r\) is the number of ramification points of \(f\); it is free (of rank \({r \over 2} - 1)\) if and only if \(f\) is totally ramified. The proof consists essentially in a careful analysis of the map \(\overline f\) that is induced by \(f\) on suitable prestable reductions of \(C\) and \(\mathbb{P}^1_K\), and of properties of the associated intersection graphs. The paper is a somewhat condensed version of the author's Ph. D. thesis; for some of the proofs the reader is referred to this thesis (in particular lemma 4.18 and the relation to the de Rham cohomology of \(C\) in section 6).
scientific article; zbMATH DE number 754098