Global automorphic applications of p-adic analytic representation theory

Abstract

In this two-part thesis, we apply p-adic analytic representation theoretic techniques to study questions about automorphic forms for various groups via the completed homology, cohomology, or complex. The two parts are written independently of each other. In the first part, we construct a derived variant of Emerton's eigenvarieties using the locally analytic representation theory of p-adic groups. The main innovations include comparison and exploitation of two homotopy equivalent completed complexes associated to the locally symmetric spaces of a quasi-split reductive group G, comparison to overconvergent cohomology, proving exactness of finite slope part functor, together with some representation-theoretic statements. As a global application, we exhibit an eigenvariety coming from data of GLn over a CM field as a subeigenvariety for a quasi-split unitary group. In the second part, we prove a degree-one saving bound for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on SL2 over any number field that is not totally real.In particular, we establish a sharp bound on the growth of cuspidal Bianchi modular forms. We transfer our problem into a question over the completed universal enveloping algebras by applying an algebraic microlocalisation of Ardakov and Wadsley to the completed homology. We prove finitely generated Iwasawa modules under the microlocalisation are generic, solving the representation theoretic question by estimating growth of Poincare–Birkhoff–Witt filtrations on such modules.