On a completed generating function of locally harmonic Maass forms

TL;DR

This paper explores the modularity of generating functions related to harmonic Maass forms, showing how to complete non-modular functions into modular objects, advancing understanding of their structure and properties.

Contribution

It introduces a method to complete non-modular generating functions of harmonic Maass forms into genuine modular objects, extending the theory of these forms.

Findings

The generating function is not modular on its own.

A natural completion yields a half-integral weight modular form.

The approach enhances understanding of harmonic Maass forms' modularity.

Abstract

While investigating the Doi-Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itelf modular, it can be naturally completed to obtain a half-integral weight modular object.