paper

Difference of modular functions and their CM value factorization

arXiv:1711.02983

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 $I^¨ (\frac{d _{1} + d _{1}}{2}) - I^¨ (\frac{d _{2} + d _{2}}{2})$ , where $I^¨$ is the $j$ -invariant or the Weber invariant $I ¨ _{2}$ . 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 $I ¨ _{2}$ . The method used here could be extended to deal with other modular functions on a genus zero modular curve.

accepted to appear in Trans. AMS