Two-variable Iwasawa main conjectures for elliptic curves over imaginary quadratic fields

Let $E/{\mathbb Q}$ be an elliptic curve of conductor $N$, let $p>2$ be a prime at which $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field in which $p={\mathfrak p}\bar{{\mathfrak p}}$ splits. Assume that the residue representation $\bar{\rho}_E|_{G_K}:G_K\to\operatorname{Aut}(E[p])$ is irreducible. Let $K_\infty=K_\infty^+K_\infty^-$ and $\Lambda_K={\mathbb Z}_p[[\operatorname{Gal}(K_\infty/K)]]$, with $\Lambda_K^{\mathrm{ur}}=\Lambda_K\widehat\otimes_{{\mathbb Z}_p}{\mathbb Z}_p^{\mathrm{ur}}$. The modules $H^1_{\mathcal F_{\mathrm{ord}}}(K,T_pE\otimes\Lambda_K^\vee)$ and $H^1_{\mathcal F_{\mathrm{Gr}}}(K,T_pE\otimes\Lambda_K^\vee)$ use, respectively, ordinary local conditions at both primes above $p$, and relaxed conditions at ${\mathfrak p}$ with strict conditions at $\bar{{\mathfrak p}}$. Two-variable Iwasawa main conjectures. The Pontryagin duals of both Selmer groups are $\Lambda_K$-torsion and

$$\operatorname{Char}_{\Lambda_K}\bigl(H^1_{\mathcal F_{\mathrm{ord}}}(K,T_pE\otimes\Lambda_K^\vee)^\vee\bigr)=({\mathcal L}_p^{\mathrm{PR}}(E/K)),$$

while

$$\operatorname{Char}_{\Lambda_K}\bigl(H^1_{\mathcal F_{\mathrm{Gr}}}(K,T_pE\otimes\Lambda_K^\vee)^\vee\bigr)\Lambda_K^{\mathrm{ur}}=({\mathcal L}_p^{\mathrm{Gr}}(E/K)).$$

These are the two-variable Iwasawa main conjectures relating characteristic ideals of ordinary and Greenberg Selmer groups to the corresponding two-variable $p$-adic $L$-functions. The paper proves divisibilities, and under an additional condition proves the conjectured equalities.