Generalized modular forms with a cuspidal divisor
In 2005, Kohnen proved 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 $-quotient. Under the assumption of 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.