Abstract

The (de Rham) geometric Langlands correspondence for GL(n) asserts that to an irreducible rank n integrable connection on a complex smooth projective curve X, we may naturally associate a D-module on Bun_n(X), the moduli stack of rank n vector bundles on X. Making appropriate changes to the formulation there are also Betti and l-adic versions of the above correspondence. In this talk we consider the rigid (and ultimately motivic) side of the story. In particular we conjecture the existence of a p-adic geometric Langlands correspondence relating rank n F-isocrystals on X (now a curve over F_p) to arithmetic D-modules on Bun_n(X). We will also explore the potential of a motivic version of the GLC which should specialize to each of the above correspondences under appropriate realisations. We will assume no specific background in the geometric Langlands correspondence.