On generalized modular forms with a cuspidal divisor

TL;DR

This paper extends Kohnen's result on modular functions with cuspidal divisors from square-free levels to prime square levels, identifying conditions for eta-product representations and providing counterexamples when rationality fails.

Contribution

It generalizes Kohnen's theorem to levels that are squares of primes and explores the necessity of rational Fourier coefficients for eta-product representations.

Findings

Extension of Kohnen's result to prime square levels.

Counterexamples showing the necessity of rationality condition.

Identification of generalized modular forms beyond eta-products.

Abstract

In [6], Kohnen proves that if $\Gamma=\Gamma_0(N)$ where $N$ is a square-free integer, then any modular function of weight $0$ for $\Gamma$ having a divisor supported at the cusps is an $\eta$-product. Under the condition of having rational Fourier coefficients, we are able to extend Kohnen's result to the case where $N$ is the square of a prime. If the rationality condition does not hold, we show that the statement is no longer true by providing a family of counter-examples that generalizes naturally the Dedekind $\eta$-function. This paper fits within the framework of generalized modular forms in the sense of Knopp and Mason.