A problem of Petersson about weight 0 meromorphic modular forms

TL;DR

This paper constructs weight 0 meromorphic modular forms explicitly using Poincaré series, addressing divergence issues with Hecke's trick and employing polar harmonic Maass forms for linear combinations.

Contribution

It introduces a novel method for constructing weight 0 meromorphic modular forms via Poincaré series and polar harmonic Maass forms, improving upon Petersson's approach.

Findings

Explicit construction of weight 0 meromorphic modular forms.

Resolution of divergence issues using Hecke's trick.

Connection between polar harmonic Maass forms and meromorphic modular forms.

Abstract

In this paper, we provide an explicit construction of weight $0$ meromorphic modular forms. Following work of Petersson, we build these via Poincar\'e series. There are two main aspects of our investigation which differ from his approach. Firstly, the naive definition of the Poincar\'e series diverges and one must analytically continue via Hecke's trick. Hecke's trick is further complicated in our situation by the fact that the Fourier expansion does not converge everywhere due to singularities in the upper half-plane so it cannot solely be used to analytically continue the functions. To explain the second difference, we recall that Petersson constructed linear combinations from a family of meromorphic functions which are modular if a certain principal parts condition is satisfied. In contrast to this, we construct linear combinations from a family of non-meromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied.