Differential equations and finite groups

The classical solution of the Riemann-Hilbert problem attaches to a given representation of the fundamental group a regular singular linear differential equation. We present a method to compute this differential equation in the case of a representation with finite image. The approach uses Galois coverings of P-1 \ {0, 1, infinity}, differential Galois theory, and a formula for the character of the Galois action of the space of holomorphic differentials. Examples are produced for the finite primitive unimodular groups of degree two and three. (C) 2000 Academic Press.