# Tate sequences and Fitting ideals of Iwasawa modules

@article{Greither2016TateSA,
title={Tate sequences and Fitting ideals of Iwasawa modules},
author={Cornelius Greither and Masato Kurihara},
journal={St Petersburg Mathematical Journal},
year={2016},
volume={27},
pages={941-965}
}
• Published 30 September 2016
• Mathematics
• St Petersburg Mathematical Journal
We consider abelian CM extensions L/k of a totally real field k, and we essentially determine the Fitting ideal of the dualized Iwasawa module studied by the second author [Ku3] in the case that only places above p ramify. In doing so we recover and generalise results of loc. cit. Remarkably, our explicit description of the Fitting ideal, apart from the contribution of the usual Stickelberger element Θ at infinity, only depends on the group structure of the Galois group Gal(L/k) and not on the…
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## References

SHOWING 1-10 OF 11 REFERENCES
Ideal Class Groups of CM-fields with Non-cyclic Galois Action
• Mathematics
• 2012
Suppose that L/k is a finite and abelian extension such that k is a totally real base field and L is a CM-field. We regard the ideal class group ClL of L as a Gal(L/k)-module. As a sequel of the
On Stronger Versions of Brumer's Conjecture
Let k be a totally real number field and L a CM-field such that L/k is finite and abelian. In this paper, we study a stronger version of Brumer’s conjecture that the Stickelberger element times the
On the structure of ideal class groups of CM-fields.
For a CM-field K which is abelian over a totally real number field k and a prime number p, we show that the structure of the χ-component AχK of the p-component of the class group ofK is determined by
Stickelberger ideals and Fitting ideals of class groups for abelian number fields
• Mathematics
• 2011
In this paper, we determine completely the initial Fitting ideal of the minus part of the ideal class group of an abelian number field over Q up to the 2-component. This answers an open question of
Computing Fitting ideals of Iwasawa modules
Abstract.This paper determines, in an equivariant sense, the Fitting ideals of several Iwasawa modules including the most canonical one. The connection between the modules themselves, which are
Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture
Abstract We assume the validity of the equivariant Tamagawa number conjecture for a certain motive attached to an abelian extension K/k of number fields, and we calculate the Fitting ideal of the
Stickelberger elements, Fitting ideals of class groups of CM-fields, and dualisation
• Mathematics
• 2008
In this paper, we systematically construct abelian extensions of CM-fields over a totally real field whose Stickelberger elements are not in the Fitting ideals of the class groups. Our evidence
Fitting Ideals of Class Groups of Real Fields with Prime Power Conductor
• Mathematics, Physics
• 1998
Abstract For a totally real field of prime power conductor, we determine the Fitting ideal over the Galois group ring of the ideal class group and of the narrow ideal class group.
Class fields of abelian extensions of Q
• Mathematics
• 1984
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 0. Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Iwasawa conjecture for totally real fields
Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the