Arithmetic differential equations on $$GL_n$$ G L n , III Galois groups

Abstract

Differential equations have arithmetic analogues (Buium in Arithmetic differential equations, Mathematical Surveys and Monographs, vol 118. American Mathematical Society, Providence 2005) 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 (Buium and Dupuy, in Arithmetic differential equations on $$GL_{n}$$ , II: arithmetic Lie–Cartan theory, arXiv:1308.0744 ). In the present paper we deal with the solutions of these equations as well as with the Galois groups attached to the solutions.