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 GL2, in the case where both the representation of GL2 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 GL2 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