On Emerton’s p-adic Banach Spaces

Abstract

Given a linear algebraic group G defined over a number field, Emerton [10] has defined a sequence of p-adic Banach Spaces . These spaces are representations of the group of adele points of G, and may be regarded as spaces of “p-adic automorphic forms”. The purpose of the current paper is to introduce some new methods for studying the spaces . We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emerton’s spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems.