Solutions of algebraic linear ordinary differential equations

Abstract

A classical result of Klein states that, given a finite primitive group G⊆SL2(C), there exists a hypergeometric equation such that every 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⊆SLn(C), there exist a positive integer d=d(G) and a standard equation such that every 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.