Christol, G.; Mebkhout, Z. On the index theorem for \(p\)-adic differential equations. I. (Sur le théorème de l’indice des équations différentielles \(p\)- adiques. I.) (French) Zbl 0834.12005 Ann. Inst. Fourier 43, No. 5, 1545-1574 (1993). This important article obtains two goals. Firstly, the authors provide a definition of the set of exponents on a disk \(D(0,r^-)\) of a differential operator \(L\) in \(\mathbb{C}_p [x,d/dx]\). Their method involves a step by step construction: the factorization theorem of B. Dwork and Ph. Robba [Trans. Am. Math. Soc. 231, 1-46 (1977; Zbl 0375.34010)] enables them to reduce to the case of “fuchsian” operators on an annulus; they then sketch a proof of the crucial fact that such operators admit a Jordan-Hölder filtration with rank one quotients, on which the “exponents” can be read off. Each of these reductions is analytic (on suitable annuli), rather than formal, and this raises the interesting question whether the exponents are algebraic when \(L\) is defined over \(\mathbb{Q}\).However, exponents are just a tool for the second (and main) goal of the paper, viz. an extension to operators of arbitrary rank of the index theorems of Robba [cf. the book by Ph. Robba and G. Christol, Equations différentielles \(p\)-adiques (Hermann, Paris, 1994)]. The authors achieve this programme precisely under the hypothesis that the exponents of \(L\) are non-Liouville numbers. The article ends with applications to \(p\)-adic analogues of the irregularity formulae of Malgrange and of Deligne [cf. Z. Mebkhout, Astérisque 130, 365-417 (1985; Zbl 0592.32009)]. Reviewer: D.Bertrand (Paris) Cited in 6 ReviewsCited in 11 Documents MSC: 12H25 \(p\)-adic differential equations 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 47E05 General theory of ordinary differential operators Keywords:Fuchsian differential operations; \(p\)-adic analogues; exponents; index theorems; irregularity formulae of Malgrange and of Deligne Citations:Zbl 0375.34010; Zbl 0592.32009 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] [BC1] , , Algebraic versus rigid cohomologie with logarithmic coefficients, In The proceeding of the Barsotti Memorial Symposium, Abano Terme Padova, june 24-27, 1991, Perspectives in Math., Academic Press (à paraître). · Zbl 0833.14010 [2] [BC2] , , Formal and p-adique theory of differential systems with logarithmic coefficients singularities depending upon parameters, Duke Math. 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