On the Levelt's theorem

TL;DR

This paper discusses a partial generalization of Levelt's theorem, which characterizes properties of solutions to certain linear differential equations using local systems and rigidity concepts.

Contribution

It introduces a partial extension of Levelt's theorem, expanding the understanding of rigid local systems associated with Fuchsian differential equations.

Findings

Partial generalization of Levelt's theorem proposed

Enhanced understanding of rigid local systems

Connections to classical hypergeometric functions

Abstract

Let $(E)$ be a homogeneous linear differential equation Fuchsian of order $n$ 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 associated 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\} $ is linearly "rigid" in the sense of Katz \cite{Katz}. This result constitute one of the best studied example of linear rigid system, it was proved by the Levelt's theorem \cite{B} Theorem 1.2.3. In this work we propose a partial generalization of the Levelt's theorem.