CM Iwasawa Main Conjecture (equality of analytic and algebraic ideals)

Prove the equality of ideals (F_{Σ}(ψ)) = (L_{Σ}(ψ)) in the Iwasawa algebra Λ, where F_{Σ}(ψ) is the characteristic power series of the ψ-isotypic Selmer module X_{Σ}^{(ψ)} attached to the maximal p-abelian Σ_p-ramified extension of K′_∞ over a p-ordinary CM quadratic extension K/F, and L_{Σ}(ψ) is the Katz p-adic L-function interpolating the algebraic parts of critical Hecke L-values for twists of the finite order Hecke character ψ of Δ = Gal(K′/K). Here Λ = R[![Γ_K]!] with R = W(\overline{\mathbb{F}}_p)[ψ], Γ_K = Gal(K_∞/K), and K_∞ is the compositum of the cyclotomic \mathbb{Z}_p-extension and the anticyclotomic \mathbb{Z}_p^{[F:\mathbb{Q}]}-extension of K.

Background

The paper studies the arithmetic of CM fields K over their maximal totally real subfields F, focusing on non-vanishing of Hecke L-values and applications to Iwasawa theory. In the Iwasawa-theoretic set-up, K′/K is a finite abelian extension containing K(μp) and disjoint from K∞. With Δ = Gal(K′/K), R = W(\overline{\mathbb{F}}p)[ψ], and Λ = R[![Γ_K]!], the module X{Σ}^{(ψ)} captures the Galois structure of the maximal p-abelian Σp-ramified extension MΣ of K′∞. Its characteristic ideal is generated by F{Σ}(ψ).

On the analytic side, the Katz p-adic L-function L_{Σ}(ψ) lies in Λ and interpolates algebraic parts of critical Hecke L-values related to ψ. The CM Iwasawa main conjecture predicts an equality between the ideal generated by L_{Σ}(ψ) and the characteristic ideal of X_{Σ}^{(ψ)}. In this paper, the authors prove a divisibility L_{Σ}(ψ) | F_{Σ}(ψ) under certain hypotheses, completing Hsieh's approach. The full equality remains the central conjectural statement in this context.