Analogue \(p\)-adique du théorème de Turrittin et le théorème de la monodromie \(p\)-adique.
type: Item
instance of: scholarly article
title: Analogue \(p\)-adique du théorème de Turrittin et le théorème de la monodromie \(p\)-adique.
zbMATH Open document ID: 1071.12004
published in: Inventiones Mathematicae
publication date: 2002-08-08T00:00:00Z
review text: The aim of this paper is to prove the basic structure theorem about \(p\)-adic differential equations. Another proof of the same theorem is available in [\textit{Y. André}, ``Filtrations de type Hasse-Arf et monodromie \(p\)-adique.'' Invent. Math. 148, No. 2, 285--317 (2002; Zbl 1081.12003)]. Even if both proofs are based on the same properties (mainly on the integrality of the \(p\)-adic irregularity) they are very distinct. The one given here follows rather faithfully (with many extra difficulties) the classical proof of Turrittin theorem for differential equations over a formal power series field. It is then as effective as possible, even if it uses several tools, for instance the decomposition theorem, which are far from being effective. Let \(K\) be a discretely valued complete field of characteristic \(0\) with residue field \(k\) of characteristic \(p\) and let \(\mathcal{R}=\mathcal{R}_{K,x}\) be the so called Robba ring (namely the ring of power series \(\sum_{n\in \mathbb{Z}} a_n\,x^n\), \((a_n\in K)\) that converges in some annulus \(1-\varepsilon<| x| <1\)). Any finite separable extension of \(k((x))\) can be written \(k'((t))\), for some finite extension \(k'\) of \(k\) and with \(x\) in \(t\,k'[t]\). It can then be lifted in an extension \(\mathcal{R}_{K',t}\) of \(\mathcal{R}\). The \(p\)-adic analog of Turrittin theorem says that for any \(\mathcal{R}\)-differential module (namely \(\mathcal{R}[d/dx]\)-module, free of finite rank as \(\mathcal{R}\)-module) \(\mathcal{M}\) which is endowed with a Frobenius structure (namely such that, for \(\varphi(x)=x^p\), \(\mathcal{M}\) and \(\varphi^{*}(\mathcal{M}): = \mathcal{R}_{\;\vrule height 9pt width0pt\llap{}_\varphi}\)\!\!\!\nwarrow _\cal{R}
reviewed by: Gilles Christol
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1777247
zbMATH Keywords: \(p\)-adic monodromy
author: Zoghman Mebkhout
MaRDI profile type: Publication
full work available at URL: s002220100208
OpenAlex ID: W2013644282
Wikidata QID: Q105582704
DOI: 10.1007/S002220100208
description: scientific article; zbMATH DE number 1777247