Log-rigid class values as Gross–Stark logarithms

Let $F$ be a totally real field in which $p$ is inert. Let $\tauin X_p$ be a point attached to an oriented integral ideal as above, let $H$ be the narrow Hilbert class field of $F$, and let $u\in \mathcal O_H[1/p]_-^\times\otimes\mathbb Q$ be the associated Gross–Stark unit. The log-rigid class value $J_{E,\mathcal L}[\tau]\in F_p$ is defined by evaluation at $\tau$ and cap product with the positively oriented generator of $H_{n-1}(U_F,\mathbb Z)$. The log-rigid class value conjecture. One has

$$J_{E,\mathcal L}[\tau]=\log_p(u^{\sigma_{\mathfrak a}}),$$

where $\sigma_{\mathfrak a}\in\operatorname{Gal}(H/F)$ is the Frobenius associated to the class of $\mathfrak a$. This refines the trace identity proved using the rank-one Gross–Stark conjecture; it is proved in certain Galois situations, while the general equality remains conjectural.