### On Yui-Zagier conjecture

Keynote Speaker:Yang Tonghai

Abstract:

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, inspired by my earlier joint work with Hongbo Yin.

Brief Introduction to the Keynote Speaker:

Yang Tonghai is a tenured professor at the University of Wisconsin (USA) and a world-renowned mathematician. He received his Ph.D. from the University of Maryland in 1995 and then was appointed as a full professor at the University of Wisconsin in 2007. He has been funded by the National Science Foundation (NSF) for various times and is now a member of the American Mathematical Society. He was invited to address the 45-minute report at the International Congress of Chinese Mathematicians for three consecutive sessions (2001, 2004 and 2007). With his efforts, the Hometown Education Foundation was founded in 2004 in the United States, with the focus on the education in his hometown in China.

Inviter:

Lyu Guangshi, professor in School of Mathematics

Time:

9:00-10:00, November 14 (Thursday)

Location:

Lecture Hall 1044, Block B, Zhixin Building, Central Campus

Sponsored by: School of Mathematics, Shandong University