On globally nilpotent differential equations

TL;DR

This paper demonstrates that the middle convolution operation preserves global nilpotence in Fuchsian systems, leading to new examples of globally nilpotent systems with unique properties, including one with an exceptional G_2 Galois group.

Contribution

It shows that middle convolution preserves global nilpotence and constructs new globally nilpotent Fuchsian systems with novel features.

Findings

Preservation of global nilpotence under middle convolution

Construction of a rank two globally nilpotent system outside known classes

Example of a rank seven system with G_2 Galois group

Abstract

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven whose differential Galois group is isomorphic to the exceptional simple algebraic group of type $G_2.$