Ideal class groups of CM-fields with non-cyclic Galois action
type: Item
instance of: scholarly article
title: Ideal class groups of CM-fields with non-cyclic Galois action
zbMATH Open document ID: 1276.11178
author: Masato Kurihara
author: Takashi Miura
published in: Tokyo Journal of Mathematics
publication date: 2013-03-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1358951328
review text: Let \(L/k\) be a finite abelian CM-extension of number fields with Galois group \(G\). Let \(\mu_L\) denote the roots of unity in \(L\) and \(cl_L\) the class group of \(L\). Then Brumer's conjecture asserts that \[ \mathrm{Ann}_{\mathbb{Z}[G]} (\mu_L) \theta_S \subseteq \mathrm{Ann}_{ \mathbb Z[G]} (cl_L), \] where \(S\) is a finite set of places of \(L\) containing all archimedean places and all that ramify in \(L/k\); here, \(\theta_S\) denotes the Stickelberger element which is defined via values of Artin L-series at zero. It is natural to ask if the stronger statement (SB) \[ \mathrm{Ann}_{\mathbb Z[G]} (\mu_L )\theta_S \subseteq \mathrm{Fitt}_{\mathbb Z[G]} (cl_L) \] might be true (here, \(\mathrm{Fitt}_{\mathbb Z[G]} (cl_L)\) denotes the (zeroth) Fitting ideal of the class group). It is shown by \textit{C. Greither} and the first author [Math. Z. 260, No. 4, 905--930 (2008; Zbl 1159.11042)] that (SB) does not hold in general. However, the dual version (DSB) of (SB), where \(cl_L\) is replaced with its Pontryagin dual, seems to be more likely to hold. For instance, its \(p\)-part (for odd \(p\)) is implied by the (appropriate special case of the) equivariant Tamagawa number conjecture if the \(p\)-part of the roots of unity in \(L\) is cohomologically trivial by a result of \textit{C. Greither} [Compos. Math. 143, No. 6, 1399--1426 (2007; Zbl 1135.11059)]. The first author [Tokyo J. Math. 34, No. 2, 407--428 (2011; Zbl 1270.11117)] has shown the existence of abelian CM-extensions for which neither (SB) nor (DSB) hold. However, explicit numerical examples for which (DSB) does not hold were not given. In the paper under review the authors give explicit conditions under which (DSB) does not hold and come up with explicit numerical examples for which neither (SB) nor (DSB) hold. In contrast to the aforementioned article of the first author the approach is not of Iwasawa-theoretic nature; the finite abelian extensions are studied directly. The authors deal with the…
reviewed by: Andreas Nickel
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 6146579
zbMATH Keywords: Brumer's conjecture
zbMATH Keywords: Fitting ideal
zbMATH Keywords: class groups
zbMATH Keywords: L-values
MaRDI profile type: Publication
OpenAlex ID: W1989053128
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DOI: 10.3836/TJM/1358951328
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description: scientific article; zbMATH DE number 6146579