Algebraicity conjecture for Heegner-cycle vanishing and central L-function derivatives

Let $G\in S_{2\kappa}^{new}(N)$ be a normalized newform, let $F_G$ be the totally real field generated by its eigenvalues, and let $g\in S^{new}_{\frac{1}{2}+\kappa,\tilde{\rho}_L}$ correspond to $G$ under the Shimura correspondence. Let $f\in H_{3/2-\kappa,\bar{\tilde{\rho}}_L}(F_G)$ satisfy

$$\xi_{3/2-\kappa}(f)=\|g\|^{-2}g.$$

For a fundamental discriminant $\Delta$ and an admissible $r$, write $Z_{\Delta,r,\kappa}(f)$ for the associated Heegner cycle and $c^+(|\Delta|,\mu_r)$ for the corresponding Fourier coefficient. The vanishing–algebraicity conjecture. The following statements are equivalent: $L'(G,\chi_\Delta,\kappa)=0$; $Z_{\Delta,r,\kappa}(f)$ vanishes in $\operatorname{CH}^\kappa(\mathcal{Y})$; and $c^+(|\Delta|,\mu_r)\in F_G$. For $\kappa=1$, this is known by work of Gross and Zagier, Borcherds, and Bruinier and Ono; the implication from vanishing of the Heegner cycle to vanishing of the derivative follows from the paper's theorem. The higher-weight equivalence remains conjectural and concerns the algebraicity of Fourier coefficients and the order of vanishing of twisted $L$-functions.