# 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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