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Kloosterman sums and monodromy of a $p$-adic hypergeometric equation

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Kloosterman sums and monodromy of a p-adic hypergeometric equation COMPOSITIO MATHEMATICA RICHARDCREW Kloosterman sums andmonodromy of a p- adic hypergeometric equation Compositio Mathematica, tome 91, no 1 (1994), p. 1-36. <> © Foundation Compositio Mathematica, 1994, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 1 Kloosterman sums and monodromy of a p-adic hypergeometric equation RICHARD CREW* Department of Mathematics, University of Florida, Gainesville, FL 32611 USA Received 10 May 1990; accepted in final form 30 November 1992 Compositio Mathematica 91: 1-36, 1994. (Ç) 1994 Kluwer Academic Publishers. Printed in the Netherlands. Suppose that X is a separated, geometrically connected scheme over a finite field k of characteristic p, and that {03C1l}l is a comparable system of semisimple 1-adic representations of 03C01(X), in the sense of Serre [21, 1-10]. Let n!eom(x) denote the geometric fundamental group rcl(X Q kalg), where kalg is the algebraic closure of k, and let G, be the Zariski closure of the image of P /n1eom. Are the G, in any sense "the same" for all l, or for almost all l? This kind of question goes back to Serre [21], and does not yet have a complete answer, although significant results in this direction have been obtained by Larsen and Pink [18]. Suppose now that (M, F) is an overconvergent isocrystal on X. In [7], we defined a kind of differential Galois group DGal(M) attached to M, and showed that in many ways it behaves li

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