$p$ -adic $L$ -functions for $text{GL}_{2}$

Abstract

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct p-adic L-functions for non-critical slope rational modular forms, the theory has been extended to construct p-adic L-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then there is a canonical way of constructing a ray class distribution from the corresponding overconvergent symbol, that moreover interpolates critical values of the L-function of the eigenform. We thus define its p-adic L-function to be this distribution.