Bruinier–Yang conjecture on arithmetic intersections of Hirzebruch–Zagier divisors

Let $K=F(\sqrt{\Delta})$ be a quartic non-biquadratic CM number field with real quadratic subfield $F$, let $\mathcal T_m$ be the integral Hirzebruch–Zagier divisor, and let $\mathcal{CM}(K)$ be the corresponding CM cycle on the Hilbert modular surface. For a CM type of $K$, let $\widetilde K$ be the reflex field, with real quadratic subfield $\widetilde F$ and discriminant $\widetilde D=d_{\widetilde F}$. Define

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

where

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

and

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

with $|\mathfrak p|$ the norm of $\mathfrak p$ and

$$\rho(\mathfrak a)=\#\{\mathfrak A\subset\mathcal O_{\widetilde K}:N_{\widetilde K/\widetilde F}\mathfrak A=\mathfrak a\}.$$

Bruinier–Yang conjecture. The arithmetic intersection satisfies

$$\mathcal T_m\cdot\mathcal{CM}(K)=\frac12 b_m,$$

or equivalently, for every prime $p$,

$$(\mathcal T_m\cdot\mathcal{CM}(K))_p=\frac12 b_m(p).$$

This conjecture gives an explicit formula for arithmetic intersections between Hirzebruch–Zagier divisors and CM cycles in terms of ideal-counting data in the reflex field. Its status is not resolved by the supplied source context.