Difference of modular functions and their CM value factorization

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.