Frobenius modules and Galois representations

Abstract

Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for p-adic differential equations with (strong) Frobenius structure over p-adic differential fields with algebraically closed residue field.