Bruinier–Yang arithmetic intersection conjecture for Hirzebruch–Zagier divisors

Let $F$ be a real quadratic field, let $K=F(\sqrt{\Delta})$ be a quartic non-biquadratic CM field, and let $\mathcal M$ be the moduli stack of principally polarized abelian surfaces with real multiplication by $\mathcal O_F$. Let $\mathcal T_m$ be the integral Hirzebruch–Zagier divisor in $\mathcal M$, and let $\mathcal{CM}(K)$ be the direct image in $\mathcal M$ of the CM moduli stack. For a CM type $\Phi$ of $K$, let $\tilde K$ be the reflex field, let $\tilde F=\mathbb Q(\sqrt{\tilde D})$ be its real quadratic subfield, and write $d_{\tilde F}=\tilde D$. Define

$$b_m=\sum_p b_m(p)\log p,$$

where

$$b_m(p)\log p=\sum_{\mathfrak p\mid p}\sum_{\substack{t=\frac{n+m\sqrt{\tilde D}}{2D}\in d_{\tilde K/\tilde F}^{-1}\\ |n|<m\sqrt{\tilde D}}}B_t(\mathfrak p),$$

and

$$B_t(\mathfrak p)=\begin{cases}0&\text{if }\mathfrak p\text{ is split in }\tilde K,\\(\operatorname{ord}_{\mathfrak p}t_n+1)\rho(t d_{\tilde K/\tilde F}\mathfrak p^{-1})\log|\mathfrak p|&\text{if }\mathfrak p\text{ is not split in }\tilde K,\end{cases}$$

with $\rho(\mathfrak a)=\#\{\mathfrak A\subset\mathcal O_{\tilde K}:N_{\tilde K/\tilde F}\mathfrak A=\mathfrak a\}$. Bruinier–Yang arithmetic intersection conjecture. The arithmetic intersection number satisfies

$$\mathcal T_m\mathbin{.}\mathcal{CM}(K)=\frac12b_m,$$

or, equivalently, for every prime $p$,

$$(\mathcal T_m\mathbin{.}\mathcal{CM}(K))_p=\frac12b_m(p).$$

The formula gives an explicit arithmetic intersection number for properly intersecting Hirzebruch–Zagier divisors and CM cycles on Hilbert modular surfaces. It was first stated by Bruinier and Yang; its resolution is not determined by the supplied source context.