Arithmetic differential equations on $GL_n$, III: Galois groups

TL;DR

This paper explores the solutions and Galois groups of linear arithmetic differential equations on $GL_n$, providing insights into their structure and symmetries within an arithmetic differential framework.

Contribution

It advances the understanding of arithmetic differential equations by analyzing their solutions and associated Galois groups, extending previous work on their foundational properties.

Findings

Characterization of solutions to linear arithmetic differential equations

Identification of differential Galois groups for these equations

Extension of arithmetic differential theory to $GL_n$

Abstract

Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations themselves were introduced in a previous paper. In the present paper we deal with the solutions of these equations as well as with the differential Galois groups attached to the solutions.