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Explicit application of Waldspurger’s theorem

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 August 2013

Soma Purkait
Soma Purkait*
Affiliation:
Mathematics Institute,Zeeman Building,University of Warwick,Coventry, CV4 7AL,United Kingdom email [email protected]

Abstract

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For a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger’s theorem relates the critical value of the $\mathrm{L} $-function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger’s recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our ‘simplified Waldspurger’ by giving several examples.

MSC classification

Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 216 - 245
Copyright
© The Author(s) 2013 

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