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

File, Daniel; Martin, Kimball; Pitale, Ameya

doi:10.2140/ant.2017.11.253

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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

DOI: 10.2140/ant.2017.11.253

Abstract

We determine local test vectors for Waldspurger functionals for ${G L}_{2}$ , in the case where both the representation of ${G L}_{2}$ 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 ${G L}_{2}$ 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

Vol. 11, No. 2, 2017