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Equivariant Iwasawa Theory for Ritter-Weiss Modules and Applications

Rusiru Gambheera, Cristian D. Popescu

TL;DR

This work develops an equivariant Iwasawa-theoretic framework for Ritter--Weiss modules $ abla_S^T(H_ fty)_p$ attached to CM extensions, proving that their minus-part is quadratically presented with projective dimension $1$ and that its $0$-th Fitting ideal is generated by the equivariant $p$-adic $L$-function $ heta_S^T(H_ abla/F)$. The main result yields a precise Equivariant Main Conjecture (EMC) relating these Fitting ideals to $p$-adic $L$-values and enables explicit control of the minus part of the unramified Iwasawa module via Fitting-ideal computations. As applications, the authors compute the $(-)$-part Fitting ideal of $X_S^{T,-}$, refining previous work, and provide a short, Euler-system-free Iwasawa-theoretic proof of the minus part of the Equivariant Tamagawa Number Conjecture for the Artin motive $h_{H/F}$ (away from 2). The paper also develops and applies shifted Fitting ideals, and performs detailed computations of Fitting ideals for local components $Y_v$, yielding intrinsic descriptions that underpin the ETNC conclusions and connect main conjectures to explicit module-theoretic data.

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 $Θ_S^T(H_\infty/F)$. 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}.

Equivariant Iwasawa Theory for Ritter-Weiss Modules and Applications

TL;DR

This work develops an equivariant Iwasawa-theoretic framework for Ritter--Weiss modules attached to CM extensions, proving that their minus-part is quadratically presented with projective dimension and that its -th Fitting ideal is generated by the equivariant -adic -function . The main result yields a precise Equivariant Main Conjecture (EMC) relating these Fitting ideals to -adic -values and enables explicit control of the minus part of the unramified Iwasawa module via Fitting-ideal computations. As applications, the authors compute the -part Fitting ideal of , refining previous work, and provide a short, Euler-system-free Iwasawa-theoretic proof of the minus part of the Equivariant Tamagawa Number Conjecture for the Artin motive (away from 2). The paper also develops and applies shifted Fitting ideals, and performs detailed computations of Fitting ideals for local components , yielding intrinsic descriptions that underpin the ETNC conclusions and connect main conjectures to explicit module-theoretic data.

Abstract

We consider a finite, abelian, CM extension of a totally real number field , and construct a module , where is a prime and is the cyclotomic -extension of . 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 -module is of projective dimension , is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant -adic -function . As a first application, we compute the Fitting ideal of an arithmetically interesting -module , which is a variant of the classical unramified Iwasawa module (the Galois group of the maximal abelian, unramified, pro- extension of ), 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 -part of the far-reaching Equivariant Tamagawa Number Conjecture for the Artin motive , for all primes , 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}.

Paper Structure

This paper contains 14 sections, 44 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Algebraic Preliminaries
  3. Fitting ideals and projective limits
  4. Shifted Fitting ideals
  5. Quadratically presented modules and their associated transposed modules.
  6. The Ritter-Weiss modules
  7. Definitions and Main Properties
  8. A link between the Ritter--Weiss and the Selmer modules
  9. Transition maps for Ritter--Weiss modules
  10. An Equivariant Main Conjecture for the Iwasawa--Ritter--Weiss module
  11. The Iwasawa theoretic set up.
  12. The Iwasawa--Ritter--Weiss Module.
  13. On the Equivariant Tamagawa Number Conjecture
  14. Shifted Fitting Ideal Computations

Key Result

Theorem 1.1

For the data $(H/F, S, T, p)$ as above, let $S_\infty$ be the set of infinite places in $F$, $S_{ram}$ the set of primes in $F$ that ramify in $H_\infty/F$ and $S_p$ the set of primes in $F$ above $p$. Suppose that the disjoint sets $S$ and $T$ satisfy the following additional properties: Then, we have an equality of $\Bbb Z_p[[\mathcal{G}]]^-$--ideals Moreover, there is a $k\in\Bbb Z_{>0}$, suc

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Bley-Popescu
  • Theorem 2.1: Greither-Kurihara
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.2
  • ...and 90 more