Zagier duality for harmonic weak Maass forms of integral weight

TL;DR

This paper establishes a duality relationship between vector valued harmonic weak Maass forms and weakly holomorphic modular forms of integral weight, revealing new structural insights in the theory of modular forms.

Contribution

It demonstrates the existence of Zagier duality for vector valued harmonic weak Maass forms and derives scalar valued duality from the vector valued case.

Findings

Zagier duality exists between vector valued harmonic weak Maass forms and weakly holomorphic modular forms.

The duality leads to an isomorphism between scalar and vector valued harmonic weak Maass forms.

The results build on recent developments in harmonic weak Maass forms by Bruinier, Funke, Ono, and Rhoades.

Abstract

We show the existence of "Zagier duality" between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass forms as developed in recent works by Bruinier, Funke, Ono, and Rhoades. Concerning the isomorphism between the spaces of scalar and vector valued harmonic weak Maass forms of integral weight, "Zagier duality" between scalar valued ones is derived.