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On the index theorem for \(p\)-adic differential equations. IV. (Sur le théorème de l’indice des équations différentielles \(p\)-adiques. IV.) (French) Zbl 1078.12501

Let \(K\) be a complete ultrametric extension of \(\mathbb{Q}_p\) having a discrete valuation. \({\mathcal A}_{K(1)}\) is the ring of power series converging for \(|x|<1\) which the authors call analytic functions in the unit disk and \({\mathcal R}_{K(1)}\) is the ring of Laurent series converging in an annulus \(1-\varepsilon <|x|<1\), for some \(\varepsilon >0\) (the elements of \({\mathcal R}_{K(1)}\) are called analytic functions at the edge of 1). Note that the domain of convergence of series should be seen in another field whose valuation is dense.
Following three previous papers of theirs [Ann. Inst. Fourier 43, No. 5, 1545–1574 (1993; Zbl 0834.12005); Ann. Math. (2) 146, No. 2, 345–410 (1997; Zbl 0929.12003); Ann. Math. (2) 151, No. 2, 385–457 (2000; Zbl 1078.12501)], here the authors use the \(p\)-adic structure of a singular point in a differential equation in order to show the existence of lattices for differential modules on \({\mathcal R}_{K(1)}\), assuming certain arithmetic conditions involving exponents of the \(p\)-adic monodromy. Thus, the authors find a sufficient condition to ensure that given an \({\mathcal R}_{K(1)}\)-differential module \(\mathcal M\) soluble at 1, then there exists a lattice on \({\mathcal A}_{K(1)}\) which is a free \({\mathcal A}_{K(1)}\)-module of rank \(m\) with a connection, extending \(\mathcal M\) and admitting zero as the only polar singular point. Hypotheses and conclusions are based on a non-Liouville property for exponents. In particular, it applies to modules having a Frobenius structure.
Among other intermediate results, the authors show a theorem of c-compactness in a ring of analytic functions when the ground field is spherically complete.

MSC:

12H25 \(p\)-adic differential equations
14F30 \(p\)-adic cohomology, crystalline cohomology
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
14G20 Local ground fields in algebraic geometry
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