Microsolutions of differential operators and values of arithmetic Gevrey series

Abstract : We continue our investigation of E-operators, in particular their connection with G-operators; these differential operators are fundamental in understanding the dio-phantine properties of Siegel's E and G-functions. We study in detail microsolutions (in Kashiwara's sense) of Fuchsian differential operators, and apply this to the construction of basis of solutions at 0 and ∞ of any E-operator from microsolutions of a G-operator; this provides a constructive proof of a theorem of André. We also focus on the arithmetic nature of connection constants and Stokes constants between different bases of solutions of E-operators. For this, we introduce and study in details an arithmetic (inverse) Laplace transform that enables one to get rid of transcendental numbers inherent to André's original approach. As an application, we define a set of special values of arithmetic Gevrey series, and discuss its conjectural relation with the ring of exponential periods of Kontsevich-Zagier.
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Stéphane Fischler, Tanguy Rivoal. Microsolutions of differential operators and values of arithmetic Gevrey series. American Journal of Mathematics, Johns Hopkins University Press, 2018, 140 (2), pp.317-348. ⟨hal-01160765⟩

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