Difference of a Hauptmodul for $\Gamma_{0}(N)$

Abstract

In this short note, by modifying the results and proofs given in the author's recent work, we simply show that the difference of a Hauptmodul for any genus zero group $\Gamma_{0}(N)$ is a Borcherds lift. This work extends Scheithauer's results in a different direction. As a consequence, we derive Monster denominator formula like product expansion for these Hilbert modular forms.