Generalized Gross–Stark conjecture

Fix a subset $\Sigma\subseteq\Sigma_S(E)$ satisfying the source's condition, set $r=|\Sigma_S(\mathcal{K}_\infty)|$, $r'=|\Sigma|$, and $\Sigma'=\Sigma\setminus\Sigma_S(\mathcal{K}_\infty)$, so that $r\le r'<|S|$ and $|\Sigma'|=r'-r$. Let $\partial_\gamma^{\,r'-r}$ denote the indicated derivative and $\mathscr{L}^{\Sigma'}_{\gamma,S}$ the corresponding $\mathscr{L}$-invariant map. Generalized Gross–Stark conjecture. If no place in $\Sigma'$ is archimedean, then

$$\partial^{\,r'-r}_{\gamma}(\varepsilon^{\rm RS}_{\mathcal{K}_\infty,S,T})=\mathscr{L}^{\Sigma'}_{\gamma,S}(\varepsilon^{\Sigma_S(\mathcal{K}_\infty)}_{E/K,S\setminus\Sigma',T})$$

in $\mathbb{C}_p\cdot{\bigcap}^{r}_{\mathbb{Z}_p[\mathcal{G}_E]}\mathcal{O}_{E,S,p}^\times$. This gives a higher-order derivative formula for inverse-limit Rubin–Stark elements; the supplied text gives no resolution status.