Elliptic Stark Conjecture

Let $(\kappa,\lambda,\mu)\in \mathcal{W}_\textup{\bf f}^{\rm cl}$ satisfy ${\rm w}(\kappa)=2$ and ${\rm w}(\lambda)={\rm w}(\mu)=1$. Assume that $\textup{\bf f}_\kappa$ is the $p$-ordinary $p$-stabilization of a newform attached to an elliptic curve $E_{/\mathbb{Q}}$, and let $\rho_g$ and $\rho_h$ be the Galois representations attached to $\mathbf{g}_\lambda$ and $\textup{\bf h}_\mu$. Suppose that $\operatorname{ord}_{s=1}L(f\otimes g\otimes h,s)=2$. Elliptic Stark Conjecture. One has

$$\mathcal{L}_p^\mathbf{g}(\textup{\bf f}\otimes\mathbf{g}\otimes\textup{\bf h})(\kappa,\lambda,\mu)=\frac{\operatorname{Reg}_{g_\alpha}(E,\rho_g\otimes\rho_h)}{\log_p(u_{g_\alpha})}.$$

Here $\operatorname{Reg}_{g_\alpha}(E,\rho_g\otimes\rho_h)$ is a $2\times 2$ $p$-adic regulator on the $\rho_g\otimes\rho_h$-isotypic component of the Stark points of $E$ over the compositum of the number fields cut out by $\rho_g$ and $\rho_h$, and $u_{g_\alpha}$ is a Gross--Stark unit. The conjecture was formulated in the cited work and proved in a variety of cases.