CM Iwasawa main conjecture over CM fields

Establish the equality of ideals (F_{Σ}(ψ)) = (L_{Σ}(ψ)) in the Iwasawa algebra Λ for a p-ordinary CM quadratic extension K/F, where F_{Σ}(ψ) is the characteristic ideal of the Σ_p-ramified Iwasawa module X_{Σ}^{(ψ)} attached to the finite-order Hecke character ψ of Δ = Gal(K′/K) and L_{Σ}(ψ) is the Katz p-adic L-function interpolating the algebraic parts of critical Hecke L-values for twists of ψ by p-adic characters of Γ_K. Precisely, prove that the characteristic ideal of X_{Σ}^{(ψ)} equals the ideal generated by L_{Σ}(ψ) in Λ = R[[Γ_K]], with R = W(\overline{\mathbb{F}}_p)[ψ], after the setup specified in Section 6.1.

Background

The paper defines, for a p-ordinary CM quadratic extension K/F and a p-adic CM type Σp, the Iwasawa module X{Σ} as the Galois group of the maximal p-abelian Σp-ramified extension of K′∞, where K′/K is a finite abelian extension disjoint from K_∞, and Λ = R[[ΓK]] with R = W(\overline{\mathbb{F}}_p)[ψ]. The characteristic ideal of its ψ-isotypic quotient X{Σ}^{(ψ)} is denoted F_{Σ}(ψ).

On the analytic side, the Katz p-adic L-function L_{Σ}(ψ) in Λ interpolates algebraic parts of critical Hecke L-values associated to twists of ψ by certain p-adic characters of ΓK. The CM Iwasawa main conjecture asserts that these two ideals coincide. The paper proves the divisibility L{Σ}(ψ) | F_{Σ}(ψ) under explicit hypotheses, thus addressing one direction toward the conjecture but leaving the full equality open.