Difference of a Hauptmodul for $Γ_{0}(N)$ and certain 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.