Harmonic weak Maass forms and periods II

TL;DR

This paper explores the algebraic properties and explicit formulas for Fourier coefficients of harmonic Maass forms of negative half-integral weight, linking them to meromorphic modular forms and their periods.

Contribution

It provides new explicit formulas for harmonic Maass form coefficients and relates their algebraicity to that of meromorphic modular forms with algebraic coefficients.

Findings

Fourier coefficients of harmonic Maass forms are related to periods of meromorphic modular forms.

Explicit formulas for coefficients are derived in terms of periods.

Algebraicity of coefficients is connected to algebraic properties of associated modular forms.

Abstract

In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients.