Reversible linear differential equations

TL;DR

This paper investigates symmetries of meromorphic connections on vector bundles over compact Riemann surfaces, introducing the Fano Group and relating it to the differential Galois group under certain conditions.

Contribution

It introduces the Fano Group to study symmetries of connections and establishes a relationship between the Galois group, Fano group, and symmetries via an exact sequence under specific hypotheses.

Findings

Defined the Fano Group for symmetries of connections

Established an exact sequence relating Galois, Fano groups, and symmetries

Connected symmetries with the structure of the Galois group

Abstract

Let $\nabla$ be a meromorphic connection on a vector bundle over a compact Riemann surface $\Gamma$. An automorphism $\sigma:\Gamma\to\Gamma$ is called a symmetry of $\nabla$ if the pull-back bundle and the pull-back connection can be identified with $\nabla$. We study the symmetries by means of what we call the Fano Group; and, under the hypothesis that $\nabla$ has a unimodular reductive Galois group, we relate the differential Galois group, the Fano group and the symmetries by means of an exact sequence.