An Integral Equivariant Refinement of the Iwasawa Main Conjecture for Totally Real Fields
Rusiru Gambheera
TL;DR
The work delivers an integral, equivariant refinement of the Iwasawa main conjecture for totally real fields by constructing the modified Ritter-Weiss module $\Omega_{S,S'}^T(H)$ to fuse ETNC with Ritter-Weiss techniques. It proves an explicit formula for the Fitting ideal of the minus-part of the $S$-modified Iwasawa module, expressed via a Stickelberger-type $p$-adic $L$-function $\Theta$ and local correction factors, and then passes to the Iwasawa tower to obtain a fully integral description of $Fitt_{\mathbb{Z}_p[[\mathcal G]]^-}(X_S^{T,-})$ in terms of $\Theta$ and local data. The approach unifies ETNC reformulations with classical Ritter-Weiss machinery, yields generalized Burns-Kurihara-Sano and Kurihara-type predictions, and recovers Wiles’ main conjecture upon tensoring with $\mathbb{Q}_p$, while also proposing conjectural Fitting-ideal descriptions for $X^-$ and related modules. Overall, the paper provides a concrete, computable integral framework for the equivariant Iwasawa theory of CM/totally real field towers and strengthens connections between ETNC, Iwasawa theory, and explicit $p$-adic $L$-functions.
Abstract
For an abelian, CM extension $H/F$ of a totally real number field $F$, we improve upon the reformulation of the Equivariant Tamagawa Number Conjecture for the Artin motive $h_{H/F}$ by Atsuta-Kataoka in \cite{Atsuta-Kataoka-ETNC} and extend the results proved in \cite{Bullach-Burns-Daoud-Seo}, \cite{Dasgupta-Kakde-Silliman-ETNC}, \cite{gambheera-popescu} and \cite{Dasgupta-Kakde} on conjectures by Burns-Kurihara-Sano \cite{Burns-Kurihara-Sano} and Kurihara \cite{Kurihara}. Then, we consider the $\mathbb{Z}_p[[Gal(H_\infty/F)]]-$module $X_S^{T}$ where $p>2$ is a prime and $H_{\infty}$ is the cyclotomic $\mathbb{Z}_p-$ extension of $H$. This is a generalization of the classical unramified Iwasawa module $X$. By taking the projective limits of the results proved at finite layers of the Iwasawa tower, as our main result, extending the earlier results of Gambheera-Popescu in \cite{gampheera-popescu-RW}, we calculate the Fitting ideal of $X_S^{T,-}$ for non-empty $T$, which is an integral equivariant refinement of the Iwasawa main conjecture for totally real fields proved by Wiles. We also give a conjectural answer to the Fitting ideal of the module $X^-$.