The Iwasawa main conjecture for CM fields: equality of the Fitting ideal of Selmer modules with Katz's analytic $p$-adic $L$-function

For a CM field $K$ satisfying the $p$-ordinary splitting condition at an odd prime $p$, the Iwasawa main conjecture predicts an exact equality, inside the multivariable Iwasawa algebra $O[[\Gamma_\infty]]$ attached to the compositum $K^{(p\infty)}$ of all $\mathbb{Z}_p$-extensions of $K$, between the Fitting ideal of the extended Selmer group $\widetilde{H}^2_f(G_{K,S}, T_\infty, \Delta_\Sigma)$ and the ideal generated by Katz's analytic $p$-adic $L$-function $L^{\mathrm{Katz},\iota}_{p,\chi}$; equivalently, the characteristic ideal of the Iwasawa module $X^\chi_\Sigma$ (the $\chi$-part of the Galois group of the maximal abelian pro-$p$ extension of $K^{\ker(\chi)}K^{(p\infty)}$ unramified outside $\Sigma$) is generated by $(L^{\mathrm{Katz}}_{p,\chi})^\iota$. The conjecture goes back to Katz (1978) and Hida–Tilouine (1994) and is recorded as Conjecture 2.13 of Büyükboduk–Sakamoto (IMRN 2023). It is known unconditionally in full only for imaginary quadratic $K$ (Rubin's Euler system of elliptic units) and, under hypotheses, for anticyclotomic characters (Hida–Tilouine, Hida); for general CM fields only one divisibility is known, via Hsieh's Eisenstein congruence method on $U(2,1)$, whose proof was completed in 2025 by Burungale–He–Tian–Ye after a 2018 gap. The problem asks for a proof or refutation of the equality in the stated generality — the missing converse divisibility, together with the removal of auxiliary hypotheses. A resolution would equate the analytic and algebraic sides of the Iwasawa theory of CM fields and unlock currently conditional applications: $p$-adic Birch–Swinnerton-Dyer-type corollaries for CM abelian varieties, exceptional-zero leading-term formulas, and $p$-adic Artin $L$-function main conjectures.

Background

Let $K$ be a CM field, i.e. a totally imaginary quadratic extension $K/F$ of a totally real field $F$ of degree $d$; the non-trivial automorphism of $K/F$ is complex conjugation. Fix an odd prime $p$ satisfying the $p$-ordinary hypothesis: every prime of $F$ above $p$ splits in $K$. A $p$-adic CM type is a choice $\Sigma \subset S_p(K)$ of one prime of $K$ above each prime of $F$ above $p$, arising from a CM type of $K$; it plays the role for $p$-adic interpolation that the choice of embedding $\overline{\mathbb{Q}}_p \hookrightarrow \mathbb{C}$ plays for classical $L$-values. Let $K^{(p\infty)}$ be the compositum of all $\mathbb{Z}_p$-extensions of $K$ and $\Gamma_\infty = \mathrm{Gal}(K^{(p\infty)}/K)$; when Leopoldt's conjecture holds for $K$ this group has $\mathbb{Z}_p$-rank $d+1$, the (conjecturally zero) Leopoldt defect adding further variables otherwise. The completed group ring $\Lambda = O[[\Gamma_\infty]]$, with $O$ the ring of integers of a sufficiently large finite extension of $\mathbb{Q}_p$, is the multivariable Iwasawa algebra over which the conjecture lives. On the analytic side, Katz (1978) constructed a multivariable $p$-adic $L$-function: for each finite-order ray class character $\chi$ of $K$ of conductor prime to $p$ and order prime to $p$ (a character of a ray class group of $K$, i.e. of the fractional ideals coprime to a fixed modulus modulo the principal ideals congruent to $1$ along that modulus), the element $L^{\mathrm{Katz}}_{p,\chi} \in \mathrm{Frac}(\Lambda)$ interpolates the algebraic parts of Hecke $L$-values $L(\chi\xi, n)$ at integers $n$, for the finite-order twists $\xi$ factoring through $\Gamma_\infty$ (Hecke $L$-functions are the $L$-series of algebraic characters of the idèle class group of $K$). On the algebraic side, let $L = K^{\ker(\chi)}$ and let $M_{L,\Sigma}$ be the maximal abelian pro-$p$ extension of $LK^{(p\infty)}$ unramified outside the primes above $\Sigma$; the Iwasawa module $X^\chi_\Sigma$ is the maximal $\chi$-isotypic direct summand of $\mathrm{Gal}(M_{L,\Sigma}/K^{(p\infty)})$, a finitely generated torsion $\Lambda$-module (torsionness relies on the weak $\Sigma$-Leopoldt theorem of Hida–Tilouine). Equivalently, by Proposition 2.12 of Büyükboduk–Sakamoto (2023), $X^\chi_\Sigma$ is pseudo-isomorphic to the second cohomology $\widetilde{H}^2_f(G_{K,S}, T_\infty, \Delta_\Sigma)$ of Nekovář's Selmer complex — the cohomological refinement of the classical Selmer group that prescribes local conditions at each prime — for the representation $T_\infty = O(1) \otimes \chi^{-1} \otimes_O O[[\Gamma_\infty]]^\iota$ (the free rank-one $\Lambda$-module on which $G_K$ acts diagonally: through the inverse $\gamma \mapsto \gamma^{-1}$ of the tautological character $G_K \twoheadrightarrow \Gamma_\infty \hookrightarrow \Lambda^\times$ on the group-ring factor $O[[\Gamma_\infty]]^\iota$, twisted by $\chi^{-1}$ and the $p$-adic cyclotomic character), with local conditions $\Delta_\Sigma$ at the primes above $\Sigma$ (and $S$ a sufficiently large finite set of primes of $K$ containing those above $p$ and the primes ramified in the extensions considered). The Fitting ideal of this Selmer module (the ideal generated by the maximal minors of a finite presentation matrix of the module) — a finer invariant than the characteristic ideal (the invariant of a finitely generated torsion $\Lambda$-module, well defined up to pseudo-isomorphism, that records the module's order at each height-one prime of $\Lambda$) when $\Lambda$ is not a regular ring — forms the left-hand side of the conjecture. The Iwasawa main conjecture for $K$ asserts that the characteristic ideal of $X^\chi_\Sigma$ is generated by the twist $(L^{\mathrm{Katz}}_{p,\chi})^\iota$ by the involution $\iota$ of $\Lambda$ induced by inversion on $\Gamma_\infty$; Büyükboduk–Sakamoto (2023) reformulate it (Conjecture 2.13) as the equality of the Fitting ideal of $\widetilde{H}^2_f(G_{K,S}, T_\infty, \Delta_\Sigma)$ with the ideal generated by $L^{\mathrm{Katz},\iota}_{p,\chi}$, as an equality of ideals after extension of scalars to $\mathbb{Z}_p^{\mathrm{ur}}[[\Gamma_\infty]]$, and defines the algebraic Katz $p$-adic $L$-function via that Fitting ideal (Definition 2.14). This is the CM-field analogue of the classical Iwasawa main conjecture over $\mathbb{Q}$, proved for totally real fields by Wiles (1990). It is the central open problem of Büyükboduk–Sakamoto (2023), which assumes it — together with the $\Sigma$-Leopoldt conjecture (injectivity of the diagonal map from the $\Sigma$-units of $K$ into the product of the local units at the primes above $\Sigma$, a strengthening of Leopoldt's conjecture that is itself open in general, though it follows from the $p$-adic Schanuel conjecture and a weak form is a theorem of Hida–Tilouine) and the non-vanishing of Gross' $p$-adic regulator (a determinant built from the $p$-adic logarithms of the $\Sigma$-units of $K$, defined by Federer–Gross; its non-vanishing is known when $K/\mathbb{Q}$ is abelian or when $F$ has a single prime above $p$, but is open in general) — as standing hypotheses for its results on $\mathcal{L}$-invariants and leading terms.

Problem Statement

Let $K$ be a CM field with maximal totally real subfield $F$, let $p$ be an odd prime such that every prime of $F$ above $p$ splits in $K$, and fix a $p$-adic CM type $\Sigma \subset S_p(K)$. Let $K^{(p\infty)}$ denote the compositum of all $\mathbb{Z}_p$-extensions of $K$, let $\Gamma_\infty = \mathrm{Gal}(K^{(p\infty)}/K)$, and let $\Lambda = O[[\Gamma_\infty]]$ be the Iwasawa algebra with coefficients in the ring of integers $O$ of a sufficiently large finite extension of $\mathbb{Q}_p$. For a finite-order ray class character $\chi$ of $K$ of conductor prime to $p$ and order prime to $p$, let $L^{\mathrm{Katz}}_{p,\chi} \in \mathrm{Frac}(\Lambda)$ be Katz's analytic $p$-adic $L$-function and let $\iota$ be the involution of $\Lambda$ induced by inversion on $\Gamma_\infty$. On the algebraic side let $L = K^{\ker(\chi)}$, let $M_{L,\Sigma}$ be the maximal abelian pro-$p$ extension of $LK^{(p\infty)}$ unramified outside the primes above $\Sigma$, let $X^\chi_\Sigma$ be the maximal $\chi$-isotypic direct summand of $\mathrm{Gal}(M_{L,\Sigma}/K^{(p\infty)})$, and let $\widetilde{H}^2_f(G_{K,S}, T_\infty, \Delta_\Sigma)$ be the second cohomology of Nekovář's Selmer complex — the cohomological refinement of the classical Selmer group that prescribes local conditions at each prime — for $T_\infty = O(1) \otimes \chi^{-1} \otimes_O O[[\Gamma_\infty]]^\iota$, on which $G_K$ acts diagonally, the group-ring factor $O[[\Gamma_\infty]]^\iota$ carrying the inverse of the tautological character $G_K \twoheadrightarrow \Gamma_\infty \hookrightarrow \Lambda^\times$, with local conditions $\Delta_\Sigma$ at the primes above $\Sigma$, where $S$ is a sufficiently large finite set of primes of $K$ containing those above $p$ and the ramified primes; this module is pseudo-isomorphic to $X^\chi_\Sigma$ by Proposition 2.12 of Büyükboduk–Sakamoto (2023). The Iwasawa main conjecture for $K$ (Conjecture 2.13 of Büyükboduk–Sakamoto (2023), equivalent to the classical Katz–Hida–Tilouine form) asserts

$$\mathrm{Fitt}_{\Lambda}\big(\widetilde{H}^2_f(G_{K,S}, T_\infty, \Delta_\Sigma)\big)\cdot \mathbb{Z}_p^{\mathrm{ur}}[[\Gamma_\infty]] \;=\; \big(L^{\mathrm{Katz},\iota}_{p,\chi}\big)\cdot \mathbb{Z}_p^{\mathrm{ur}}[[\Gamma_\infty]],$$

i.e. the characteristic ideal of $X^\chi_\Sigma$ is generated by $(L^{\mathrm{Katz}}_{p,\chi})^\iota$. The problem: prove, or refute, this equality for all CM fields satisfying the stated $p$-splitting hypothesis, all $p$-adic CM types $\Sigma$, and all such characters $\chi$. In the direction of proof, this means establishing the converse divisibility to the known Eisenstein-congruence divisibility (Hsieh, completed by Burungale–He–Tian–Ye) and removing the auxiliary hypotheses under which partial results are currently known (Hsieh's congruence conditions such as $p \nmid 6h_K^- D_F \#\Delta$, and, where used, assumptions such as $\Sigma$-Leopoldt or non-vanishing of Gross' $p$-adic regulator).

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