Explicit application of Waldspurger's theorem

Soma Purkait
2013 LMS Journal of Computation and Mathematics  
AbstractFor 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
more » ... nd 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.
doi:10.1112/s1461157013000144 fatcat:4v3ca44t3zhkpmudhjss7okz7e