The three-variable Iwasawa–Greenberg main conjecture for Rankin–Selberg families

Let ${\mathbb{I}}_{\mathcal F,\mathcal G}$ be the coefficient algebra of the Hida families $\mathcal F$ and $\mathcal G$, let $\Gamma_{\mathrm{cyc}}$ be the cyclotomic Galois group, and let $\mathrm{Sel}^{\Sigma_0}_{\mathbf{\underline{\mathcal F}},\mathcal G}(\mathbb{Q})$ be the discrete non-primitive Selmer group associated with the dominant family $\mathcal F$. Write $\theta^{\Sigma_0}_{\pmb{4},3}$ for the corresponding non-primitive three-variable Rankin–Selberg $p$-adic $L$-function.

Iwasawa–Greenberg main conjecture. The ${\mathbb{I}}_{\mathcal F,\mathcal G}[[\Gamma_{\mathrm{cyc}}]]$-module $\mathrm{Sel}^{\Sigma_0}_{\mathbf{\underline{\mathcal F}},\mathcal G}(\mathbb{Q})^\vee$ is torsion. Furthermore, the divisors satisfy

$$\operatorname{Div}\left(\mathrm{Sel}^{\Sigma_0}_{\rho_{\pmb{4},3}}(\mathbb{Q})^\vee\right)=\operatorname{Div}\left(\theta^{\Sigma_0}_{\pmb{4},3}\right).$$

This is the three-variable Rankin–Selberg Iwasawa–Greenberg main conjecture: it predicts that the characteristic divisor of the dual Selmer group agrees with that of the associated $p$-adic $L$-function. Its resolution is not established in the supplied text.