On generalized modular forms supported on cuspidal and elliptic points

TL;DR

This paper proves that certain generalized modular forms with rational Fourier expansions and divisors supported only at cusps and specific points are actually classical modular forms, exploring limitations and open questions.

Contribution

It extends previous results to show that these generalized forms are classical, and discusses potential limitations and open problems in the field.

Findings

Generalized modular forms with specified support are classical.

Limitations to the extension are identified and discussed.

Open questions about zeros of modular forms of prime level are posed.

Abstract

In this paper, we extend previous results to prove that generalized modular forms with rational Fourier expansions whose divisors are supported only at the cusps and certain other points in the upper half plane are actually classical modular forms. We discuss possible limitations to this extension and pose questions about possible zeroes for modular forms of prime level.