Hida–Tilouine Iwasawa Main Conjecture for CM fields

Let $M_\Sigma$ be the maximal abelian pro-$p$ extension of $Lk(p^\infty)$ that is $\Sigma$-ramified, let $X_\Sigma:=\operatorname{Gal}(M_\Sigma/Lk(p^\infty))$, and let $X_\Sigma^\chi:=e_\chi(\mathbb Z_p^{\rm ur}\otimes_{\mathbb Z_p}X_\Sigma)$. Let $L_{p,\Sigma}^{\chi}$ be Katz's $p$-adic $L$-function. Iwasawa Main Conjecture.

$$\operatorname{char}(X_\Sigma^\chi)=(L_{p,\Sigma}^{\chi,\iota}).$$

The conjecture identifies the characteristic ideal of the relevant Iwasawa module with Katz's $p$-adic $L$-function and is central to the paper's implications for exceptional zeros. The supplied context gives no resolution evidence.