Harmonic Maass forms and periods

TL;DR

This paper establishes a new connection between the coefficients of weight 1/2 harmonic Maass forms and periods of algebraic differentials, extending Waldspurger's theorem to a broader class of forms.

Contribution

It proves that coefficients of weight 1/2 harmonic Maass forms are determined by periods of algebraic differentials, generalizing known results for eigenforms.

Findings

Coefficients are linked to periods of algebraic differentials of the third kind.

Extends Waldspurger's theorem to harmonic Maass forms.

Provides a new geometric interpretation of Maass form coefficients.

Abstract

According to Waldspurger's theorem, the coefficients of half-integral weight eigenforms are given by central critical values of twisted Hecke L-functions, and therefore by periods. Here we prove that the coefficients of weight 1/2 harmonic Maass forms are determined by periods of algebraic differentials of the third kind on modular and elliptic curves.