Difference of modular functions and their CM value factorization

TL;DR

This paper derives explicit factorization formulas for differences of modular functions' CM values, providing new proofs of classical results and conjectures using Borcherds lifting and CM value formulas.

Contribution

It introduces a method to explicitly factorize differences of modular functions' CM values, extending to other functions on genus zero modular curves.

Findings

Provides a new proof of Gross-Zagier factorization for singular moduli.

Proves the Yui-Zagier conjecture for Weber invariant $oldsymbol{ extomega_2}$.

Method can be extended to other modular functions on genus zero curves.

Abstract

In this paper, we use Borcherds lifting and the big CM value formula of Bruinier, Kudla, and Yang to give an explicit factorization formula for the norm of $\Psi(\frac{d_1+\sqrt{d_1}}2) -\Psi(\frac{d_2+\sqrt{d_2}}2)$, where $\Psi$ is the $j$-invariant or the Weber invariant $\omega_2$. The $j$-invariant case gives another proof of the well-known Gross-Zagier factorization formula of singular moduli, while the Weber invariant case gives a proof of the Yui-Zagier conjecture for $\omega_2$. The method used here could be extended to deal with other modular functions on a genus zero modular curve.