The differential Galois theory of regular singular \(p\)-adic differential equations
type: Item
instance of: scholarly article
title: The differential Galois theory of regular singular \(p\)-adic differential equations
zbMATH Open document ID: 0854.14009
DOI: 10.1007/BF01444211
author: Richard Crew
published in: Mathematische Annalen
publication date: 1996-07-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165433
review text: Let \(k\) be a perfect field of characteristic \(p>0\), \(X\) a smooth curve over \(k\) and suppose that there is a lifting \(X_K\) of \(X\) to an algebraic curve over a field \(K\) of characteristic zero. Let \(M\) be a locally free sheaf with connection on \(X_K\) satisfying some convergence condition (namely it is ``soluble'' in generic disks). Then, by restriction to strict neighborhoods of a formally smooth lifting of \(X\), \(M\) defines an ``overconvergent isocrystal'' \(M^\dagger\) on \(X\). If \(X_K\) has a \(K\)-rational point \(x\), the category of overconvergent isocrystals and the category of locally free sheaves on \(X_K\) with connection are neutral Tannakian categories. In both situations, the fiber functor associated to \(x\) restricted to the smallest tensor subcategory containing \(M\) (resp. \(M^\dagger\)) enables to define the ``monodromy group'' \(\text{DGal} (M)\) (resp. \(\text{DGal} (M^\dagger)\)). The main result of the paper is that if \(M^\dagger\) is regular, with \(p\)-adic integers exponents two of which do not differ by a \(p\)-adic Liouville number, then \(\text{DGal} (M)\) and \(\text{DGAL} (M^\dagger)\) are isomorphic. As applications, first the unicity, up to homothety, of the Frobenius structure for irreducible \(F\)-isocrystals is shown, secondly a comparison result between monodromy groups of some isocrystals arising from geometry (Gauss-Manin connections) and corresponding \(\ell\)-adic monodromy groups is established. Proofs are based on the so-called ``transfer theorem for regular singular \(p\)-adic differential equations''. The author gives two conditions for a locally free sheaf to be overconvergent. Let us point out that, at least over a curve, the second one is implied by the first one.
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 892578
zbMATH Keywords: transfer theorem for regular singular \(p\)-adic different equations
zbMATH Keywords: overconvergent isocrystals
zbMATH Keywords: Tannakian categories
reviewed by: Gilles Christol
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 892578