Equivariant Iwasawa Theory for Ritter-Weiss Modules and Applications

Abstract: We consider a finite, abelian, CM extension $H/F$ of a totally real number field $F$, and construct a $\mathbb{Z}p[[G(H\infty/F)]]-$module $\nabla_S^T(H_\infty)_p$, where $p>2$ is a prime and $H_\infty$ is the cyclotomic $\Bbb Z_p$-extension of $H$. This is the Iwasawa theoretic analogue of a module introduced by Ritter and Weiss in \cite{Ritter-Weiss} and studied further by Dasgupta and Kakde in \cite{Dasgupta-Kakde}. Our main result states that the $\Bbb Z_p[[G(H_\infty/F]]^-$-module $\nabla_S^T(H_\infty)_p$ is of projective dimension $1$, is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant $p$-adic $L$-function $\Theta_S^T(H_\infty/F)$. As a first application, we compute the Fitting ideal of an arithmetically interesting $\Bbb Z_p[[G(H_\infty/F)]]^-$-module $X_S^{T,-}$, which is a variant of the classical unramified Iwasawa module $X$ (the Galois group of the maximal abelian, unramified, pro-$p$ extension of $H_\infty$), extending earlier results of Greither-Kataoka-Kurihara \cite{Greither-Kataoka-Kurihara}. These are all instances of what is now called an Equivariant Main Conjecture in the Iwasawa theory of totally real number fields, and refine the classical main conjecture, proved by Wiles in \cite{wiles}. As a final application, we give a short, Iwasawa theoretic proof of the minus $p$-part of the far-reaching Equivariant Tamagawa Number Conjecture for the Artin motive $h_{H/F}$, for all primes $p>2$, a result also obtained, independently and with different (Euler system) methods, by Bullack-Burns-Daoud-Seo \cite{Bullach-Burns-Daoud-Seo} and Dasgupta-Kakde-Silliman \cite{Dasgupta-Kakde-Silliman-ETNC}.