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The differential Galois theory of regular singular \(p\)-adic differential equations. (English) Zbl 0854.14009
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.
Reviewer: G.Christol (Paris)

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
12H25 \(p\)-adic differential equations
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
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