Park–Shahabi Rubin–Stark determinant and nonvanishing conjecture

Let $\mathcal{S}^{E,p}_{\infty}$ be the cyclic $\Lambda$-module generated by the tower of Rubin–Stark elements after localization and twisting, let $\Gamma_n=\operatorname{Gal}(F_n/F)$, and let $\mathfrak{d}_{n,i}$, $\mathfrak{u}_{i,j}$, $\tau(\chi)$, $\omega_n^{\epsilon}$, and $L(E/F^+,\chi,s)$ be as defined in the paper, with $\epsilon$ the sign of $(-1)^n$. Park–Shahabi conjecture. (i) There exists a generator $\Xi_1\wedge\cdots\wedge\Xi_g$ of $\mathcal{S}^{E,p}_{\infty}$ such that, for every $n\in\mathbb{Z}^+$, primitive character $\chi:\operatorname{Gal}(F_n/F)\to\boldsymbol{\mu}_{p^\infty}$, and positive integer $k$, the determinant identity displayed in the source holds. (ii) For all but finitely many characters $\chi$ of $\Gamma_{\mathrm{cyc}}$, one has $L(E/F^+,\chi,1)\ne0$. This conjecture predicts an explicit regulator determinant formula for the Rubin–Stark tower together with generic nonvanishing of the twisted complex $L$-values; the source gives no resolution status.