The GL2 Main Conjecture over CM fields

Let $g\in S_2(\Gamma_0(\mathfrak n))$ be a Hilbert modular newform over a totally real field $F$, and let $p$ be an odd prime unramified in $F$ and good ordinary for $g$. Let $M/F$ be a CM quadratic extension satisfying the source's splitting condition, and assume that the residual representation $\bar\rho_g$ is irreducible as a $G_M$-representation. Let $\Lambda_M$ be the relevant CM Iwasawa algebra, let $\mathfrak{X}_{\rm ord}(g/M_\infty)$ be the ordinary Selmer module, and let $\mathcal{L}_p(g/M)$ be the associated $(d+1)$-variable $p$-adic $L$-function. The GL2 Main Conjecture. The module $\mathfrak{X}_{\rm ord}(g/M_\infty)$ is $\Lambda_M$-torsion, with

$${\rm ch}_{\Lambda_M}\bigl(\mathfrak{X}_{\rm ord}(g/M_\infty)\bigr)=\bigl(\mathcal{L}_p(g/M)\bigr).$$

This predicts equality between the characteristic ideal of the ordinary Selmer module and the principal ideal generated by the $p$-adic $L$-function; the paper discusses divisibility results toward it.