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
Milestones
Received: 19 May 2014
Revised: 22 September 2016
Accepted: 26 September 2016
Published: 15 April 2017
Authors
 Daniel File Department of Mathematics Muhlenberg College Allentown, PA 18104 United States Kimball Martin Department of Mathematics University of Oklahoma Norman, OK 73019 United States Ameya Pitale Department of Mathematics University of Oklahoma Norman, OK 73019 United States