Difference of a Hauptmodul for ${\mathrm{\Gamma }}_0(N)$ and certain Gross-Zagier type CM value formulas

In this work, we show that the difference of a Hauptmodul for a genus zero group ${\mathrm{\Gamma }}_0(N)$ as a Hilbert modular function on $Y_{0}(N)\times Y_{0}(N)$isaBorcherdsliftoftype$(2,2)$. As applications, we derive Monster denominator formula like product expansions for these Hilbert modular functions and certain Gross-Zagier type CM value formulas.