Solutions of algebraic linear ordinary differential equations

TL;DR

This paper generalizes Klein's classical result by showing that for any finite primitive group in SL_n(C), there exists a standard equation to which all related linear ordinary differential equations can be reduced, extending the hypergeometric case.

Contribution

It extends Klein's result from SL_2(C) to SL_n(C), establishing the existence of standard equations for LODEs with a given differential Galois group, including hypergeometric equations for n=3.

Findings

Existence of a standard equation for any finite primitive group in SL_n(C).

Reduction of LODEs to pullbacks of standard equations over a field extension.

Hypergeometric equations serve as standard equations for n=3 cases.

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.