On harmonic weak Maass forms of half integral weight

TL;DR

This paper establishes an isomorphism between vector valued and scalar valued harmonic weak Maass forms of half integral weight, extending classical results for holomorphic modular forms to a broader context.

Contribution

It introduces a new isomorphism for harmonic weak Maass forms of half integral weight, generalizing Eichler and Zagier's results to non-holomorphic cases.

Findings

Vector valued harmonic weak Maass forms are isomorphic to scalar valued forms with Fourier coefficients supported on specific progressions.

The result extends classical modular form theory to harmonic weak Maass forms of half integral weight.

Provides a framework for understanding the structure of harmonic weak Maass forms in relation to scalar forms.

Abstract

We show that certain space of vector valued harmonic weak Maass forms of half integral weight is isomorphic to a space of scalar valued ones whose Fourier coefficients are supported on suitable progressions. This kind of result for holomorphic modular forms was proved by Eichler and Zagier.