Microsolutions of differential operators and values of arithmetic Gevrey series

TL;DR

This paper investigates the microsolutions of differential operators related to $E$- and $G$-functions, providing new insights into their solutions, connection constants, and special values of arithmetic Gevrey series, with implications for exponential periods.

Contribution

It offers a constructive method to analyze $E$-operators via microsolutions of $G$-operators and introduces an arithmetic Laplace transform to study connection constants.

Findings

Constructed bases of solutions at 0 and infinity for $E$-operators.

Introduced an arithmetic Laplace transform to analyze connection constants.

Defined special values of arithmetic Gevrey series related to exponential periods.

Abstract

We continue our investigation of $E$-operators, in particular their connection with $G$-operators; these differential operators are fundamental in understanding the diophantine 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 $\infty$ of any $E$-operator from microsolutions of a $G$-operator; this provides a constructive proof of a theorem of Andr\'e. 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 Andr\'e'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.