Explicit application of Waldspurger's Theorem

TL;DR

This paper makes Waldspurger's theorem more explicit by translating its automorphic and adelic formulations into practical congruence conditions involving Dirichlet characters, enabling easier computation of critical L-values.

Contribution

It provides a simplified, explicit version of Waldspurger's recipes for modular forms of half-integral weight, making the theorem more accessible for computational purposes.

Findings

Derived explicit congruence conditions for Waldspurger's theorem

Demonstrated the practicality with several computational examples

Enhanced the usability of Waldspurger's theorem for explicit calculations

Abstract

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 $n$-th quadratic twist of $\phi$ to the $n$-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 (adelic) 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.