Abstract

In the last few decades, the theory of p-adic modular forms has seen many applications to different problems in number theory. This theory is well-understood, its central objects of study being overconvergent p-adic modular forms. However, when attempting to generalize the theory to automorphic forms for more general reductive groups, the picture is less clear. For example, there are several different proposed definitions for spaces of p-adic automorphic forms, such as overconvergent and completed cohomology. In this talk I will explain more precisely what properties we want such spaces to satisfy and discuss a different proposal, based on the study of the cohomology of p-arithmetic groups with coefficients in p-adic locally analytic representations.