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Journal of the Ramanujan Mathematical Society

Volume 32, Issue 4, December 2017  pp. 397–416.

Adjoint L-functions, period integrals, and a converse theorem for SL2

Authors:  Vinayak Vatsal
Author institution:Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Summary:  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.

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