J. Ramanujan Math. Soc. 32, No.4 (2017) 397–416
Adjoint L-functions, period integrals, and a converse theorem for SL 2 Vinayak Vatsal Department of Mathematics, University of British Columbia, Vancouver, BC, Canada e-mail:
[email protected] Communicated by: Prof. Dipendra Prasad Received: August 8, 2016 Abstract. The purpose of this paper is two-fold. Our main new result is a converse theorem for automorphic forms on the group S L 2 . 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 PG L 2 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 S L 2 , which in turn leads to the converse theorem. 2000 Mathematics Subject Classification: 11F11, 11F33, 11F66.
1. Introduction This paper consists of two parts, with a certain degree 3 L-function playing a unifying role. The goal of the first part is to draw attention to a conjecture concerning indivisibility of quadratic twists of L-functions for automorphic forms on the group P G L 2 (see Conjecture 2.1 below). This conjecture seems to be part of the folklore, but seems not to be well-known beyond a brief discussion in [Pra10], where it arises in the study of the Shimura lift from a quaternion algebra to the metaplectic group. A key role in the statement of the conjecture is played by the special value of the degree three L-function associated to the adjoint action on trace zero matrices for automorphic forms on P G L 2 , and it is this degree 3 L-function which is our principal object of study. The second part of the paper moves to the group S L 2 . The connection with the first part is that the existence of nonzero quadratic twists has a simple reformulation in terms of period integrals on the group S L 2. The 397