Title: An arithmetic intersection formula on a Hilbert modular surface
<p>Speaker: Tonghai Yang


<p>Abstract: On a Hilbert modular surface over $\mathbb Z$, there are two
families of arithmetic cycles. One family is the
Hirzebruch-Zagier divisors
$\mathcal T_m$  of codim $1$---indexed by positive integers $m$, and another is
the CM cycles $\CM(K)$ of codim. 2---indexed by
quartic CM number fields $K$. When $K$ is not biquadratic, $\mathcal T_m$ and
$\CM(K)$ intersect properly, and a natural
question is what is the intersection number? In this talk, we present a
conjectural fromula for the intersection number of
Bruinier and myself. We give two partial results in this talk. If time permits,
I will also briefly describe two applications,
one  is a generalization of the Chowla-Selberg formula, and another is a
conjecture of Lauter on Igusa invariants.