Adjoint L-Functions, Period Integrals, and A Converse theorem for SL2

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.