Solutions of algebraic linear ordinary differential equations

Abstract

A classical result of F.Klein states that, given a finite primitive group $G\subseteq SL_2(\mathbb{C})$, there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to $G$ is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group $G\subseteq SL_n(\mathbb{C})$, there exist a positive integer $d=d(G)$ and a standard equation such that any LODE whose differential Galois group is isomorphic to $G$ is gauge equivalent, over a field extension $F$ of degree $d$, to an equation projectively equivalent to the pullback by a map in $F$ of this standard equation. For $n=3$, these standard equations can be chosen to be hypergeometric.