Gross–Zagier algebraicity conjecture for higher Green's functions at CM points

Let $d_1$ and $d_2$ be discriminants such that one of them is fundamental when $k$ is even. For any CM points $z_1$ and $z_2$ of discriminants $d_1$ and $d_2$, let $H$ be the composite of the ring class fields $H_1$ and $H_2$ over $\mathds Q(\sqrt{d_1})$ and $\mathds Q(\sqrt{d_2})$ corresponding to $z_1$ and $z_2$, respectively. Let $E=\mathds Q(\sqrt{d_1},\sqrt{d_2})$. Gross–Zagier's algebraicity conjecture. There exists $\kappa$ in $\mathbb N$ and $\alpha$ in $H$, depending only on $d_1,d_2,k$ and $f$, such that

$$(d_1d_2)^{(k-1)/2}G^f_k(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|.$$

Moreover, for any $\sigma\in\operatorname{Gal}(H/E)$,

$$\sigma(\alpha(z_1,z_2))=\alpha(\sigma(z_1),\sigma(z_2)).$$

The conjecture is now a theorem of Li, with earlier conditional and related results by Gross, Kohnen and Zagier, Zhang, Mellit, Viazovska, Zhou, and Bruinier–Ehlen–Yang.