Test vectors and central $L$-values for ${\rm GL}(2)$

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.