Toward equivariant Iwasawa theory
type: Item
instance of: scholarly article
title: Toward equivariant Iwasawa theory
zbMATH Open document ID: 1014.11066
author: A. Weiss
author: Jürgen Ritter
published in: Manuscripta Mathematica
publication date: 2002-12-01T00:00:00Z
review text: Let \(K/k\) be a Galois extension of totally real number fields, \(K_\infty\) the cyclotomic \(\mathbb{Z}_\ell\)-extension of \(K\) (\(\ell\) being an odd prime), \(\Gamma= \text{Gal} (K_\infty/K)\), \(G_\infty= \text{Gal} (K_\infty/k)\), \(X_\infty\) the Galois group over \(K_\infty\) of the maximal Abelian \(\ell\)-extension of \(K_\infty\) which is unramified outside a finite set of primes containing the Archimedean primes, the primes above \(\ell\) and those which ramify in \(K/k\). One could say that the main goal of Iwasawa theory is the description of the Galois module structure of \(X_\infty\) in algebraic-analytical terms. The Usual Main Conjecture (UMC, or Wiles' theorem) does this over the Iwasawa algebra \(\mathbb{Z}_\ell \Gamma\). An Equivariant Main Conjecture (EMC) should do the same thing over the complete algebra \(\mathbb{Z}_p G_\infty\) and should provide a powerful tool towards important conjectures in Galois module theory such as the ``lifted root number conjecture'' or more generally, the ``equivariant Tamagawa number conjecture''. In their ``tripod'' paper [Mem. Am. Math. Soc. 748 (2002; Zbl 1002.11082)], the authors had proposed a formulation of such an EMC in terms of the localization exact sequence relating \(K_0\) and \(K_1\). More precisely, let \(\partial\) be the connecting map \(K_1 (Q\mathbb{Z}_\ell G_\infty)\to K_0T (\mathbb{Z}_\ell G_\infty)\), where \(Q(.)\) denotes the total quotient ring and \(K_0T(.)\) the Grothendieck group of the category of finitely generated torsion modules of finite projective dimension. In their ``tripod'' paper, the authors constructed: i) A Galois module invariant \(\mho_S\in K_0T (\mathbb{Z}_\ell G_\infty)\) attached to \(X_\infty\). The fancy notation is meant to recall that this construction is mimicking, at the infinite level, the ``lifted \(\Omega\)-construction'' at the finite level. ii) Assuming that \(G_\infty\) is Abelian, an analytical element \(\Theta_S\in K_1(Q \mathbb{Z}_…
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1837447
zbMATH Keywords: equivariant main conjecture
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Galois module structure
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: s00229-002-0306-8
OpenAlex ID: W2770107105
DOI: 10.1007/S00229-002-0306-8
description: scientific article; zbMATH DE number 1837447