Some cases of Brumer's conjecture for abelian CM extensions of totally real fields
type: Item
instance of: scholarly article
title: Some cases of Brumer's conjecture for abelian CM extensions of totally real fields
zbMATH Open document ID: 0965.11047
author: Cornelius Greither
published in: Mathematische Zeitschrift
publication date: 2001-08-05T00:00:00Z
review text: Brumer's conjecture on annihilators of class groups for abelian CM extensions \(K\) of a totally real field \(F\), is a direct generalization of Stickelberger's theorem over \(\mathbb{Q}\). The author proves this conjecture for so-called ``nice'' extensions (extensions of prime power conductor are nice, but they are not the only ones). More precisely, using Iwasawa theory, he calculates the Fitting ideal of the minus class group of \(K\). The main difficulty consists in ``avoiding the trivial zeros'' of \(p\)-adic \(L\)-functions, by extending a method of \textit{A. Wiles} [Ann. Math. (2) 131, 555-565 (1990; Zbl 0719.11082)]. The result can also be applied to Chinburg's third conjecture, as in a previous paper of the author [Math. Z. 229, 107-136 (1998; Zbl 0919.11072)].
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1441739
zbMATH Keywords: Brumer's conjecture
zbMATH Keywords: annihilators of class groups
zbMATH Keywords: abelian CM extensions
zbMATH Keywords: totally real field
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Fitting ideal
zbMATH Keywords: minus class group
zbMATH Keywords: \(p\)-adic \(L\)-functions
zbMATH Keywords: Chinburg's third conjecture
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: s002090050485
OpenAlex ID: W2050324785
Wikidata QID: Q123314046
DOI: 10.1007/S002090050485
Recommended article: Q3344069
Recommended article: Q4661783
Recommended article: A Brumer-Stark conjecture for non-abelian Galois extensions
Recommended article: On non-abelian Brumer and Brumer–Stark conjecture for monomial CM-extensions
Recommended article: Abelian extensions of fields of CM-type
Recommended article: On the Rubin-Stark conjecture for a special class of CM extensions of totally real number fields.
Recommended article: Tate conjecture for some abelian surfaces over totally real or CM number fields
Recommended article: On CM Abelian varieties over imaginary quadratic fields
Recommended article: Some applications of a Bombieri-type theorem in totally real number fields
Recommended article: On \({\mathbb{Z}}_ p\)-extensions of real abelian fields
description: scientific article; zbMATH DE number 1441739