Fuchsian structure for differential modules on an ultrametric polyannulus
type: Item
instance of: scholarly article
title: Fuchsian structure for differential modules on an ultrametric polyannulus
zbMATH Open document ID: 0946.12001
author: F. Gachet
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2000-03-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108500
review text: This paper sets out the theory of regular singularities for \( p\)-adic differential modules in the several variables case. Even if regular singularities seem less basic in the \( p\)-adic case than in the complex case, this work is the first (and until now the only one) significant step toward a \( p\)-adic \( D\)-module theory. Let \( K \) be a complete extension of \( \mathbb{Q}_{p}\), let \( (I_{i})_{1\leq i\leq n} \) be intervals of \( \mathbb{R}^{+}\), let \( I=\prod I_{i} \) and let \( {\mathcal A}_{K}(I) \) be the ring of functions analytic in the (poly)annulus \( {\mathcal C}(I)\)=\(\{(x_{1},\cdots,x_{n});x_{i}\in I_{i}\} \) with coefficients in \( K\). A differential module, namely a free \( {\mathcal A}_{K}(I)\)-module of finite type endowed with an integrable connection, is said to have the Robba property if its local solutions near any point \( t \) in \( {\mathcal C}(I) \) do converge in the maximal polydisk \( (|x_{i}-t_{i}|<|t_{i}|)_{1\leq i<n} \) centered in \( t \) and contained in \( {\mathcal C}(I)\). The basic step in this paper is to give the definition of exponents for differential modules with the Robba property. One of the striking properties of these exponents is their invariance when specializing variables. The main result of the paper is that any differential module with the Robba property whose exponents have non-Liouville differences is Fuchsian, namely has a basis in which the matrices of the \( x_{i} {d \over dx_{i}} \) are constant. Both results and proofs are generalizations of the one variable theory as it is explained in Dwork's paper [\textit{B. Dwork}, J. Reine Angew. Math. 484, 85-126 (1997; Zbl 0870.12008)]. However this generalization is not at all straightforward. It needed, among other, to extend the explicit majoration of Dwork-Robba in the several variables setting and, above all, to prove a \( p\)-adic Hartogs theorem (analyticity with respect to each variable implies global analyticity).
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 14F10
zbMATH DE Number: 1395896
zbMATH Keywords: regular singularity
zbMATH Keywords: Fuchsian differential module
zbMATH Keywords: Hartogs theorem
reviewed by: Gilles Christol
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1395896