Goursat rigid local systems of rank four

TL;DR

This paper investigates rank four Goursat rigid local systems, providing criteria for irreducibility, explicit descriptions, and examples of algebraic solutions, including one with infinite monodromy from genus two curves.

Contribution

It offers a comprehensive analysis of Goursat's rank four rigid local systems, including irreducibility conditions, explicit integral and Hermitian forms, and classification of algebraic solutions.

Findings

Identified all irreducible systems with algebraic solutions.

Provided explicit examples over the rationals.

Constructed a system with infinite monodromy from genus two curves.

Abstract

We study the general properties of certain rank four rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form 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 the rationals. We also exhibit one example with infinite monodromy as arising from a family of genus two curves.