Equivariant Main Conjecture, Fitting ideals and annihilators in Iwasawa theory
type: Item
instance of: scholarly article
title: Equivariant Main Conjecture, Fitting ideals and annihilators in Iwasawa theory
zbMATH Open document ID: 1098.11054
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2006-03-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/52736
review text: This article produces annihilators for so-called higher class groups, by exploiting the Equivariant Main Conjecture as proposed and proved by Ritter and Weiss. (The setting is as follows: \(K/k\) is an abelian extension of totally real fields, \(G\) is the Galois group and \(p\) is a fixed odd prime.) In fact more is proved: the author determines in Theorem 4.3 the Fitting ideal of the \({\mathbb Z}_p[G]\)-module \(C(S,K,m):=H^2(G_S(K),{\mathbb Z}_p(m))\) for \(m>0\) positive and even, where \(S\) denotes the set of places that are over \(p\) or ramify in \(K/k\). From the Voevodsky-Rost theorem (see e.g. Theorem 70 in \textit{C. Weibel's} survey [Handbook of \(K\)-theory, Volume 2. Berlin: Springer, 139--190 (2005; Zbl 1097.19003)], we know that \(K_{2m-2}(O_K)\{p\}\) is isomorphic to \(C(K,m):=H^2(G_{S_p}(K),\mathbb{Z}_p(m))\) where \(S_p\) denotes the places over \(p\). Since \(C(K,m)\) maps injectively to \(C(S,K,m)\), it follows that the Fitting ideal of \(C(S,K,m)\) annihilates the group \(K_{2m-2}(O_K)\{p\}\). This is a refined version of the Coates-Sinnott conjecture. We should point out that the proof of the Equivariant Main Conjecture makes serious use of the nullity of the \(\mu\)-invariant of \(K\) at \(p\). The author offers an interesting independent approach to the so-called envelope \(Y_\infty\) of the Iwasawa module in question and the resulting short exact sequence \(0 \to X_\infty \to Z_\infty \to z_\infty \to 0\), which leads to the definition of the invariant \(\mho_S\) of Ritter and Weiss and to the formulation of the Equivariant Main Conjecture. (The torsion module \(Z_\infty\) is an explicit quotient of \(Y_\infty\) and of finite projective dimension over \(\mathbb{Z}_p[G_\infty]\), where \(G_\infty\) is the Galois group of \(K_\infty\) over \(k\); the module \(z_\infty\) is an explicit quotient of \(\Delta G_\infty\).) This approach uses earlier work of the author [``Formations de classes et modules d'Iwasawa'', Number theory, Proc. Journ. A…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 5016579
zbMATH Keywords: Main conjecture
zbMATH Keywords: Fitting ideals
zbMATH Keywords: cohomology
Wikidata QID: Q122998953
author: Thong Nguyen Quang Do
MaRDI profile type: Publication
OpenAlex ID: W2073036269
cites work: On the equivariant Tamagawa number conjecture for Tate motives
cites work: Equivariant Weierstrass preparation and values of \(L\)-functions at negative integers
cites work: Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs sur un corps abélien
cites work: An analogue of Stickelberger's theorem for the higher K-groups
cites work: Values of abelian \(L\)-functions at negative integers over totally real fields
cites work: The structure of some minus class groups, and Chinburg's third conjecture for abelian fields
cites work: Some cases of Brumer's conjecture for abelian CM extensions of totally real fields
cites work: Computing Fitting ideals of Iwasawa modules
cites work: Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters
cites work: Q4549331
cites work: On \(\mathbb Z_{\ell}\)-extensions of algebraic number fields
cites work: Q3358834
cites work: Q4279437
cites work: Iwasawa theory and Fitting ideals
cites work: On the structure of ideal class groups of CM-fields
cites work: Twisted \(S\)-units, \(p\)-adic class number formulas, and the Lichtenbaum conjectures
cites work: On Fitting ideals of certain étale \(K\)-groups
cites work: Class fields of abelian extensions of \(\mathbb Q\)
cites work: Q3331297
cites work: On \(\mathbb{Z}_ p\)-torsion of some Galois modules
cites work: Analogues supérieurs du noyau sauvage
cites work: Q4935965
cites work: Q4195048
cites work: The lifted root number conjecture and Iwasawa theory
cites work: Toward equivariant Iwasawa theory
cites work: Two-primary algebraic 𝐾-theory of rings of integers in number fields
cites work: Über gewisse Galoiscohomologiegruppen
cites work: Q4175668
cites work: Algebraic K-groups as Galois modules
cites work: Relative $K_{0}$, annihilators, Fitting ideals and Stickelberger phenomena
cites work: The Stark conjectures on Artin \(L\)-functions at \(s=0\). Lecture notes of a course in Orsay edited by Dominique Bernardi and Norbert Schappacher.
cites work: The Iwasawa conjecture for totally real fields
cites work: On a conjecture of Brumer
DOI: 10.5802/JTNB.512
Recommended article: Equivariant Iwasawa theory and non-Abelian Stark-type conjectures
Recommended article: On the Coates-Sinnott Conjecture
Recommended article: Equivariant Iwasawa theory: an example
Recommended article: On the Coates-Sinnott conjecture for CM fields
Recommended article: The refined Coates-Sinnott conjecture for characteristic \(p\) global fields
Recommended article: L-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups
Recommended article: On the equivariant Tamagawa number conjecture in tame CM-extensions, II
Recommended article: On the Coates-Sinnott conjecture.
Recommended article: On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture
Recommended article: Stickelberger elements, Fitting ideals of class groups of CM-fields, and dualisation
description: scientific article; zbMATH DE number 5016579