p-arithmetic cohomology and p-adic automorphic forms

Abstract

The cohomology of an arithmetic group with coefficients in finite-dimensional representations can be described in terms of automorphic representations of the group. In this thesis, we prove similar results for the cohomology of an *S*-arithmetic groups (where *S* is a finite set of primes) with coefficients in different types of representations. For example, we show that the cohomology of (duals of) locally algebraic representations of the local groups at places in *S* can be described in terms of automorphic representations satisfying certain conditions determined by the locally algebraic representation. We show that the cohomology with coefficients in (duals of) locally analytic representations can be used to define *p*-adic automorphic forms and families of them (eigenvarieties). In particular, we are able to give constructions of these objects in many new cases, such as when the reductive group is not quasi-split at *p*. We also prove that these constructions are equivalent, in the cases where they are defined, to those obtained using overconvergent cohomology and to the Bernstein eigenvarieties constructed by Breuil-Ding.