A local-global question in automorphic forms

Abstract

In this paper, we consider the SL(2) analogue of two well-known theorems about period integrals of automorphic forms on GL(2): one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on GL₂(A_F) of cuspidal automorphic representations on GL₂(A_E) where E is a quadratic extension of a number field , and the other due to Waldspurger involving toric periods of automorphic forms on GL₂(A_F). In both these cases, now involving SL(2), we analyze period integrals on globalL -packets; we prove that under certain conditions, a global automorphic L-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.