Adjoint L-functions, period integrals, and a converse theorem for S L2

Abstract

The purpose of this paper is two-fold. Our main new result is a converse theorem for automorphic forms on the group SL2. However, we would also like to discuss an apparently unrelated question in number theory which motivated our study, namely, the question of whether or not a given automorphic form on PGL2 admits a quadratic twist whose special value is nonzero modulo a given prime p. The question seems extremely hard, but it leads naturally to the study of period integrals on the group SL2, which in turn leads to the converse theorem.