Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case

Let X be a smooth projective curve over an algebraically closed field k of characteristic p>0. In this paper we explore the relation between algebraic D-modules on the moduli space $Bu{n}_n$ of vector bundles of rank n on X and coherent sheaves on the moduli space $Lo{c}_n$ of vector bundles endowed with a connection (in the way predicted by Beilinson and Drinfeld for k of characteristic 0). The main technical tools used in the paper are the geometry of the Hitchin system and the Azumaya property of the algebra of differential operators in characteristic p.