# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] P. Berthelot: G?ometrie rigide et cohomologie des vari?t?s alg?briques de caracteristiquep. Journ?es d’analysep-adique (Luminy 1982). M?moire de la S.M.F. no. 23, suppl. au Bull. S.M.F.114 (1986) fasc. 2, pp. 7-32. [2] P. Berthelot: Cohomologie rigide et cohomologie rigide ? support propre. To appear in Ast?risque [3] H. Cartan, S. Eilenberg: Homological Algebra. Princeton University Press 1956 · Zbl 0075.24305 [4] G. Christol: Un th?or?me de transfert pour les disques singuli?res reguli?res. Ast?risque119-120 (1984) 151-168 [5] R. Crew: F-isocrystals and their monodromy groups. Ann. Sc. Ec. Norm. Sup. 4 e ser.25 (1992) 420-464 · Zbl 0783.14008 [6] R. Crew: Thep-adic monodromy of a generic abelian scheme in characteristicp. In:p-adic Methods in Number Theory and Algebraic Geometry. Contemporary Mathematics133 (1992), AMS, pp. 59-74 · Zbl 0785.14009 [7] R. Crew: Kloosterman sums and the monodromy of ap-adic hypergeometric equation. Compositio Mathematica91 (1994) 1-36 · Zbl 0806.14018 [8] P. Deligne: Equations Diff?rentielles ? points singuli?res r?guliers. Lecture Notes in Math.163 (1970) Springer-Verlag [9] P. Deligne: La conjecture de Weil II. Publ. Math. IHES52 (1980) 137-252 · Zbl 0456.14014 [10] P. Deligne: Categories Tannakiennes. In: The Grothendieck Festschrift. Birkh?user 1989 [11] P. Deligne J. Milne: Tannakian Categories. Lecture Notes in Math.900 (1982) Springer-Verlag · Zbl 0477.14004 [12] B. Dwork: On the uniqueness of Frobenius operator on Differential Equations, in Algebraic Number Theory?in honor of K. Iwasawa. Adv. Studies in Pure Math.17 (1989) 89-96 [13] N. Katz: On the calculation of some differential galois groups. Inv. Math.87 (1987) 13-61 · Zbl 0609.12025 · doi:10.1007/BF01389152 [14] N. Katz: Exponential sums and differential equations. Annals of Math. Studies124. Princeton University Press 1990 · Zbl 0731.14008 [15] M. Lazard: Let z?roes des fonctions analytiques d’une variable sur un corps valu? complet. Publ. Math. IHES14 (1962) 47-75 [16] W. L?tkebohmert: Formal-algebraic and rigid-analytic geometry. Math. Ann.286 (1990) 341-371 · Zbl 0716.32022 · doi:10.1007/BF01453580 [17] M. van der Put:p-adic differential equations, in Proceedings of a conference onp-adic analyis. Hengelhoef (1986) 181-187 [18] S. Sperber:p-adic hypergeometric functions and their cohomology. Duke Math. J.44 (1977) 535-589 · Zbl 0408.12025 · doi:10.1215/S0012-7094-77-04424-6 [19] A. Grothendieck et al.: Rev?tements ?tales et groupe fondamental. Lecture Notes in Math.224 (1971) Springer-Verlag
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.