Le Th\'eor\`eme de Levelt

TL;DR

This paper discusses a partial generalization of Levelt's theorem, which relates to the properties of solutions of Fuchsian differential equations through the study of associated local systems.

Contribution

It extends Levelt's theorem to a broader class of differential equations, providing new insights into the structure of their local systems.

Findings

Partial generalization of Levelt's theorem achieved

Enhanced understanding of local systems for Fuchsian equations

Potential applications to hypergeometric functions and differential equations

Abstract

Let $(E)$ a homogeneous linear differential equation of order $n$ Fuchsien over $\mathbb{P}^{1}(\mathbb{C}) $. The idea of Riemann (1857) was to obtain the properties of solutions of (E) by studying the local system. Thus, he obtained some properties of Gauss hypergeometric functions by studying the assocated rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash {3 points} $. For example, he obtained the Kummer transformations of the hypergeometric functions without any calculation. The success of the Riemann's methods is due to the fact that the irreducible rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash {3 points} $ are "rigid". Levelt theorem, see \cite{B} Theorem 1.2.3 proves this result. In this work, we propose a partial generalization of this theorem.