Arithmetic differential equations on GLn, I: Differential cocycles

The theory of differential equations has an arithmetic analogue [8] in which derivatives are replaced by Fermat quotients. One can then ask what is the arithmetic analogue of a linear differential equation. The study of usual linear differential equations is the same as the study of the differential cocycle from GLnGLn into its Lie algebra given by the logarithmic derivative [14] (equivalently by the Maurer–Cartan connection [17]). However we prove here that there are no such cocycles in the context of arithmetic differential equations. In sequels of this paper [10] and [11] we will remedy the situation by introducing arithmetic analogues of Lie algebras and a skew version of differential cocycles; this will lead to a theory of linear arithmetic differential equations.