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.