Central value of automorphic $L-$functions

TL;DR

This paper generalizes Waldspurger's formula to totally real fields, linking Fourier coefficients of half-integral weight forms with central L-values, and explores implications for conjectures and spaces in automorphic forms.

Contribution

It extends Waldspurger's formula to totally real fields and provides new interpretations and applications in automorphic form theory.

Findings

Generalized Waldspurger's formula to totally real fields

Connected Fourier coefficients with central L-values in new settings

Established equivalence between Ramanujan conjecture and Lindelöf hypothesis cases

Abstract

We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula in terms of equality between global distributions. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.