Iwasawa's main conjecture for definite and indefinite Rankin–Selberg eigenvarieties

Let $\mathbf{V}$ be a datum with $\mathfrak{m}=\emptyset$, let $\xi=(\xi^n,\xi^{n+1})$ be a pair of $\Sigma^+_p\setminus\lozenge^+$-trivial dominant hermitian weights, let $\gamma$ be as specified in the source, and let $\mathrm{J}<\mathrm{I}^0$ be a subgroup. Assume that $\gamma$ satisfies (G1), and also (G2) when $\mathbf{V}$ is indefinite, and fix a $\xi$-distinction when $\xi$ is interlacing. Let $\mathscr{E}'$ be an irreducible component of the normal locus of $\operatorname{Spec}\mathscr{E}_\mathrm{J}(\xi,\mathbf{V})_\gamma$. Iwasawa's main conjecture. If $\mathbf{V}$ is definite and $\boldsymbol{\lambda}_\mathrm{J}(\mathbf{V})$ is nonzero on $\mathscr{E}'$, then $\mathrm{H}^1_f(F,\mathscr{R}_\mathrm{J}(\xi,\mathbf{V})_\gamma)$ vanishes over $\mathscr{E}'$, $\mathscr{X}_\mathrm{J}(\xi,\mathbf{V})_\gamma$ is torsion over $\mathscr{E}'$, and

$$2\operatorname{char}_{\mathscr{E}'}\left(\mathscr{D}_\mathrm{J}(\xi,\mathbf{V})_\gamma^\mathsf{H}/\boldsymbol{\lambda}_\mathrm{J}(\mathbf{V})\right)=\operatorname{char}_{\mathscr{E}'}\left(\mathscr{X}_\mathrm{J}(\xi,\mathbf{V})_\gamma\right).$$

If $\mathbf{V}$ is indefinite and $\boldsymbol{\kappa}_\mathrm{J}(\mathbf{V})$ is nonzero on $\mathscr{E}'$, then both $\mathrm{H}^1_f(F,\mathscr{R}_\mathrm{J}(\xi,\mathbf{V})_\gamma)$ and $\mathrm{H}^1_f(F,\mathscr{D}_\mathrm{J}(\xi,\mathbf{V})_\gamma)^\mathsf{H}$ have generic rank one over $\mathscr{E}'$, $\mathscr{X}_\mathrm{J}(\xi,\mathbf{V})_\gamma$ has generic rank one over $\mathscr{E}'$, and

2\operatorname{char}_{\mathscr{E}'}\left(\mathrm{H}^1_f(F,\mathscr{D}_\mathrm{J}(\xi,\mathbf{V})_\gamma)^\mathsf{H}/\boldsymbol{\kappa}_\mathrm{J}(\mathbf{V})\right)=\operatorname{char}_{\mathscr{E}'}\left(\mathscr{X}_\mathrm{J}(\xi,\mathbf{V})_\gamma^\operatorname{tor}\right).

Here \mathscr{M}^\operatorname{tor} denotes the maximal torsion submodule over the normal locus. This is the eigenvariety analogue of Iwasawa's main conjecture, relating Bessel periods or cohomology classes to Selmer groups. The source states the conjectural equalities and rank or torsion assertions but gives no resolution status.