Difference of modular functions and their CM value factorization

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 $\mathrm{\Psi }(\frac{d_1+\sqrt{d_1}}{2})-\mathrm{\Psi }(\frac{d_2+\sqrt{d_2}}{2})$, where $\mathrm{\Psi }$ 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.