Goursat rigid local systems of rank four

Abstract

We study the general properties of certain rank 4 rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form H when it exists. By a computer search, we find what we expect are all irreducible such systems all whose solutions are algebraic functions and give several explicit examples defined over {mathbb {Q}}. We also exhibit one example with infinite monodromy as arising from a family of genus two curves.