In the 1980s, Gross and Zagier discovered a beautiful factorization formula for norm of difference of singular moduli $j(\tau_1)-j(\tau_2)$, where $j$ is the famous $j$-invariants and $\tau_i$ are CM points of discriminants $d_i <0$. This was a test case for the well-known Gross-Zagier formula. They gave two proofs for the formula, algebraic one and analytic ones. Algebraic idea have been extended by Goren, Lauter, Viray, Howard and myself and others to the cases $d_1$ and $d_2$ not relatively prime and also to Hilbert modular surfaces. Analytic proof have been extended to Shimura varieties of orthogonal and unitary type using Borcherds’ regularized theta liftings, by Schofar, Bruinier, Kudla, myself, and others. In 1990s, Yui and Zagier made a similar but more subtle and surprising conjectural formula for norm of the difference of CM values of some Weber functions of level 48.
In this talk, we will describe this conjectural formula and its proof using the so-called Big CM formula discovered by Bruinier, Kudla, and myself. This is joint work with Yingkun Li.