An Integral Equivariant Refinement of the Iwasawa Main Conjecture for Totally Real Fields

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 $X^-$.