Classicality of overconvergent automorphic forms on some Shimura varieties

Abstract

This thesis consists of two parts. In Part 1 we study the rigid cohomology of the ordinary locus in some compact PEL Shimura varieties of type C with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent Hecke eigenforms, generalizing work of Coleman. In part 2 we compare the conjecture of Buzzard-Gee on the association of Galois representations to C-algebraic automorphic representations with the conjectural description of the cohomology of Shimura varieties due to Kottwitz, and the reciprocity law at in nity due to Arthur. This is done by extending Langlands's representation of the L-group associated with a Shimura datum to a representation of the C-group of Buzzard-Gee. The approach o ers an explanation of the explicit Tate twist appearing in Kottwitz's description.