Abstract
Waldspurger’s formula gives an identity between the norm of a torus period and an$L$-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding$L$-functions agree, (the norms of) these periods—which occur on different quaternion algebras—are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.
Highlights
The key to our approach is the construction of a seesaw of dual reductive pairs that precisely realizes the quaternion algebras B1 and B2
We prove an explicit Rallis inner product formula relating θφ(χ ) to L(1, χ ), which in particular shows that
We identify ResE/F (W0 ) = V and let ι : UE (W0 ) → Sp(ResE/F (W0 )) = Sp(V ) be the natural embedding
Summary
Waldspurger’s work in 1985 sparked the beginnings of a rich theory studying the relationship between special values of L-functions and automorphic periods. We will employ the theta correspondence to construct automorphic forms and compare the resulting torus periods. To this end, the key to our approach is the construction of a seesaw of dual reductive pairs that precisely realizes the quaternion algebras B1 and B2. We prove (Theorem 6.1) that the global theta lifts can be described in terms of automorphic induction and Jacquet– Langlands and that the global theta lift vanishes if and only if the Jacquet– Langlands transfer does not exist
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
