Difference of a Hauptmodul for $\Gamma_{0}(N)$ and certain Gross-Zagier type CM value formulas

TL;DR

This paper demonstrates that the difference of a Hauptmodul for $ ext{Γ}_0(N)$ can be expressed as a Borcherds lift, leading to new product expansions and CM value formulas related to Gross-Zagier theory.

Contribution

It introduces a novel connection between Hauptmodul differences and Borcherds lifts, providing explicit formulas and applications in modular forms and CM values.

Findings

Difference of Hauptmodul is a Borcherds lift of type (2,2).

Derived Monster denominator-like product expansions.

Established Gross-Zagier type CM value formulas.

Abstract

In this work, we show that the difference of a Hauptmodul for a genus zero group $\Gamma_{0}(N)$ as a Hilbert modular function on $Y_{0}(N)\times Y_{0}(N)$ is a Borcherds lift of type $(2,2)$. As applications, we derive Monster denominator formula like product expansions for these Hilbert modular functions and certain Gross-Zagier type CM value formulas.