On the Levelt's theorem

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.