Test vectors and central L-values for GL(2)

Abstract

We determine local test vectors for Waldspurger functionals for GL 2 , in the case where both the representation of GL 2 and the character of the degree two extension are ramified, with certain restrictions. We use this to obtain an explicit version of Waldspurger's formula relating twisted central L-values of automorphic representations on GL 2 with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results. 1

Central values of twisted base change L-functions associated to Hilbert modular forms

“…where we have L(1/2, π E ⊗ Ω) = 0 for those π ∈ F new (2k, N) − F new (2k, N, Θ). We remark that for Σ 3 = ∅, a much wider result has been obtained by File, Martin and Pitale in [3], where they determined the local test vectors for Waldspurger functionals for GL 2 . Assume Σ 3 = ∅.…”

Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

“…Acknowledgments We thank Jeanine van Order for providing us with the reference [11]. While writing this note Santiago Molina informed us that he proved a similar result on the equality of automorphic and algebraic periods.…”

“…The second assertion follows from the vanishing criterion of toric periods integrals by Saito and Tunnell (see [18] and [20]). The first assertion is a direct consequence of the calculations in [11].…”

“…The crucial input is a computation of toric period integrals by File, Martin and Pitale (cf. [11]). We keep the notations from the previous section.…”

“…[11]) on toric period integrals to show that our Stickelberger elements interpolate (square roots of) special values of the L-function of the base change of π B to PGL 2 (E), i.e. for every character χ : G → C * of conductor m we have…”