On the modular completion of certain generating functions

TL;DR

This paper constructs non-holomorphic modular completions of certain generating functions related to meromorphic and weakly holomorphic modular forms, revealing new modular properties and connections to Zagier's work.

Contribution

It introduces a method to complete generating functions to non-holomorphic modular forms of specific weights, linking meromorphic and weakly holomorphic forms in a novel way.

Findings

Generated non-holomorphic modular forms of weights 3/2 and 2

Connected generating functions to Zagier's traces of singular moduli

Provided explicit completions for families of meromorphic modular forms

Abstract

We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights 3/2 and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight 3/2, which prominently appear in work of Zagier on traces of singular moduli.