On a Hilbert modular surface over $\mathbb Z$, there are two families of arithmetic cycles. One family consists of the Hirzebruch-Zagier divisors $\mathcal T_m$ of codimension $1$, indexed by positive integers $m$, and another consists of the CM cycles $CM(K)$ of codimension 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 formula 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 of the consequences is a generalization of the Chowla-Selberg formula, and another is a conjecture of Lauter on Igusa invariants.