Action of the Galois group on the periods of certain Mumford curves
type: Item
instance of: scholarly article
title: Action of the Galois group on the periods of certain Mumford curves
zbMATH Open document ID: 0836.14016
author: Christophe Brouillard
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-05-02T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93609
full work available at URL: jtnb6-2.html
review text: 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).
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 754098
zbMATH Keywords: cyclic covering
zbMATH Keywords: Galois action on periods
zbMATH Keywords: period lattice
zbMATH Keywords: Mumford curve
zbMATH Keywords: Schottky group
zbMATH Keywords: Jacobian
reviewed by: Frank Herrlich
MaRDI profile type: Publication
OpenAlex ID: W2326574049
cites work: Q1071829
cites work: Q5516858
cites work: Q3325851
cites work: Q4033874
cites work: Q3769025
cites work: Géométrie analytique rigide et applications. (Ridgid analytic geometry and applications)
cites work: Q4156864
cites work: Periods and Gauss-Manin connection for families of p-adic Schottky groups
cites work: Integrale zweiter Gattung auf Mumfordkurven
cites work: Schottky groups and Mumford curves
cites work: Q5622916
cites work: The order of the automorphism group of a \(p\)-adic Schottky curve
cites work: Q5183633
cites work: Genre et genre residuel des corps de fonctions valués. (Genus and residual genus of valued function fields)
cites work: Q5635548
cites work: Q4148144
cites work: p-adic periods of genus two Mumford-Schottky curves.
cites work: Sur les Périodes des Intégrales Abéliennes
DOI: 10.5802/JTNB.120
description: scientific article; zbMATH DE number 754098