Weakly holomorphic modular forms for some moonshine groups

TL;DR

This paper generalizes the study of weakly holomorphic modular forms from the full modular group to specific moonshine groups, including cases with real characters, extending previous foundational work.

Contribution

It extends Duke and Jenkins' results on modular forms to moonshine groups with real characters, broadening the scope of modular form theory.

Findings

Generalization of modular form bases to moonshine groups

Inclusion of real characters in the analysis

Extension of previous results to new groups

Abstract

In an article in the Pure and Applied Mathematics Quarterly in 2008, Duke and Jenkins investigated a certain natural basis of the space of weakly holomorphic modular forms for the full modular group $SL_2({\bf Z})$. We show here that their results can be generalized to certain moonshine groups, also allowing characters that are real on the underlying subgroup $\Gamma_0(N)$.