The structure of some minus class groups, and Chinburg's third conjecture for abelian fields
type: Item
instance of: scholarly article
title: The structure of some minus class groups, and Chinburg's third conjecture for abelian fields
zbMATH Open document ID: 0919.11072
DOI: 10.1007/PL00004647
author: Cornelius Greither
published in: Mathematische Zeitschrift
publication date: 1999-03-08T00:00:00Z
review text: This paper, in some way, is a ``multiplicative continuation'' of the author's work on Chinburg's second invariant \(\Omega(L/\mathbb{Q},2)\) for real abelian fields \(L\) of odd prime power conductor \(l^m\) [\textit{C. Greither}, J. Reine Angew. Math. 479, 1-37 (1996; Zbl 0856.11051)]. In fact, for the same class of fields it is shown that also the third Chinburg class \(\Omega_3=\Omega(L/\mathbb{Q},3)\) vanishes (actually, \(l=2\) is not forbidden anymore). The restriction of the absolute ramification of \(L\) clears away possible difficulties with the Tate class involved in the definition of the \(\Omega\)-invariants and arising from the global and local fundamental classes. There are some very nice and new ideas presented here in the multiplicative situation, which lead to even stronger results. Namely, the paper starts off from a splitting of \(\Omega_3\) into \(\Omega^+Cl(\mathbb{Z} G_+)\) and \(\Omega^-\in CL(\mathbb{Z} G_-)\) with \(G\) denoting the Galois group of a (non-necessarily real) abelian extension \(L/\mathbb{Q}\) with conductor \(l^m\). The splitting is induced by complex conjugation, \(c\) say, and indeed, \(\Omega^+=\Omega(L^+/\mathbb{Q},3)\), \(\mathbb{Z} G_+=\mathbb{Z} G/\langle c\rangle\). The first part of the paper concerns \(cl_L^-\) and \(\Omega^-\) under the restriction that \(L\) is imaginary and \(l\) is such that \(c\in\text{Gal}(\mathbb{Q} (\zeta_l)/\mathbb{Q})_p\), the decomposition subgroups of the primes \(p\) dividing \(l-1\). The discussion is based on a result of Schoof, by which \(cl_L^-\) has projective dimension \(\leq 1\) over \(\mathbb{Z} G_-\), and on Iwasawa theory by means of which the Fitting ideal of \(cl_L^-\) is determined and (almost) realized as a Stickelberger ideal. On account of a Tate sequence \(E_S \rightarrowtail A\to B\twoheadrightarrow\Delta S\) the two objects, \(\Omega^-\) and \(cl_L^-\), get related by \(\Omega^-=(E_S^-)-(\Delta S_-)\), \((E_S^-)+(cl_L^-)=(\mu_L)+\Delta S_-)\) in \(K_0(\mathbb{Z} G_-)\)…
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 1229584
zbMATH Keywords: Chinburg's invariant
zbMATH Keywords: third Chinburg class
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Fitting ideal
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: abelian field extensions
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
Wikidata QID: Q122898435
full work available at URL: pl00004647
OpenAlex ID: W2063555005
description: scientific article; zbMATH DE number 1229584