Difference of a Hauptmodul for ${\Gamma }_0(N)$ 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.

Accepted for publication in SCIENCE CHINA Mathematics. arXiv admin note: text overlap with arXiv:1701.08433