#### Vol. 11, No. 2, 2017

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

### Daniel File, Kimball Martin and Ameya Pitale

Vol. 11 (2017), No. 2, 253–318
##### 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.

##### Keywords
modular forms, test vectors, periods, $L$-values
##### Mathematical Subject Classification 2010
Primary: 11F67
Secondary: 11F41, 11F70, 11F66