The CM Iwasawa main conjecture for Katz's $p$-adic $L$-function

Let $K/F$ be a $p$-ordinary CM quadratic extension for an odd prime $p$, and let $h_K^-$ be its relative class number. Fix embeddings $\iota_\infty:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and $\iota_p:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_p$. Let $\Sigma$ be a $p$-ordinary CM type of $K$, with associated $p$-adic CM type $\Sigma_p$. Let $K_\infty$ be the compositum of the cyclotomic $\mathbb{Z}_p$-extension and the anticyclotomic $\mathbb{Z}_p^{[F:\mathbb{Q}]}$-extension of $K$, and put $\Gamma_K=\operatorname{Gal}(K_\infty/K)$. Let $K'$ be a finite abelian extension of $K$ containing $K(\mu_p)$ and disjoint from $K_\infty$, put $\Delta=\operatorname{Gal}(K'/K)$ and $K'_\infty=K_\infty K'$, and let $\psi:\Delta\to\overline{\mathbb{Z}}_p^\times$ be a finite-order Hecke character over $K$. Set $R=\overline{\mathbb{Z}}_p^{\mathrm{un}}[\psi]$ and $\Lambda=R[[\Gamma_K]]$. Let $M_\Sigma$ be the maximal $p$-abelian $\Sigma_p$-ramified extension of $K'_\infty$, and define

$$X_\Sigma=\operatorname{Gal}(M_\Sigma/K'_\infty)\otimes_{\mathbb{Z}_p[\Delta][[\Gamma_K]]}R[\Delta][[\Gamma_K]].$$

Let $X_\Sigma^{(\psi)}$ be the maximal $\psi$-isotypic quotient of $X_\Sigma$, and let $F_\Sigma(\psi)\in\Lambda$ be its characteristic power series. Let $L_\Sigma(\psi)\in\Lambda$ be the associated Katz $p$-adic $L$-function. The CM Iwasawa main conjecture. The ideals generated by the algebraic and analytic characteristic elements are equal:

$$(F_\Sigma(\psi))=(L_\Sigma(\psi)).$$

This conjecture asserts that the characteristic ideal of the relevant Iwasawa module agrees with the ideal generated by Katz's $p$-adic $L$-function, linking the arithmetic of the maximal $p$-abelian $\Sigma_p$-ramified extension to interpolated critical Hecke $L$-values. Its resolution is not specified in the supplied source.