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

## TL;DR

This paper demonstrates that the difference of a Hauptmodul for any genus zero group _{0}(N) is a Borcherds lift, extending previous results and deriving new product expansions for related Hilbert modular forms.

## Contribution

It shows that the difference of Hauptmoduls for genus zero groups _{0}(N) can be expressed as Borcherds lifts, extending Scheithauer's work in a new direction.

## Key findings

- Difference of Hauptmodul is a Borcherds lift
- Derived Monster denominator formula-like product expansions
- Extended previous results to all genus zero groups _{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.

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Source: https://tomesphere.com/paper/1704.05990