Kurihara–et al. Iwasawa main conjecture for Rubin–Stark elements

Let $L_\infty/K$ be the relevant $p$-adic extension, let $\Lambda=\mathds{Z}_p\llbracket\operatorname{Gal}(L_\infty/K)\rrbracket$, let $\varepsilon$ be the specified idempotent, and let $Q(\Lambda)$ be the total quotient ring. Fix a $\Lambda\varepsilon$-order $\mathfrak{A}$ in $Q(\Lambda)\varepsilon$. Let $\pi_\infty$ be the canonical injective map from the determinant module to the inverse limit of exterior-power biduals, and let $\eta_{L_\infty/K,S}^w$ be the inverse system of Rubin–Stark elements. Kurihara et al. conjecture. There exists an $\mathfrak{A}$-basis

$$\mathfrak{z}_{L_\infty/K,S}^w\in\mathfrak{A}\otimes_\Lambda{\rm \det}_\Lambda^{-1}(\mathrm{R}\Gamma(\mathcal{O}_{K,S},\mathds{Z}_p(1)_{L_\infty/K}))$$

such that

$$\pi_\infty(\mathfrak{z}_{L_\infty/K,S}^w)=\eta_{L_\infty/K,S}^w.$$

This is an Iwasawa-theoretic integrality statement relating determinant modules to inverse systems of Rubin–Stark elements. It is presented as a conjectural formulation in the source and its general status is open.