Quaternionic exercises in \(K\)-theory Galois module structure. II
type: Item
instance of: scholarly article
zbMATH Open document ID: 0983.11068
author: Ted Chinburg
author: Manfred Kolster
author: Victor P. Snaith
publication date: 2001-04-17T00:00:00Z
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1588579
zbMATH Keywords: Euler characteristic
zbMATH Keywords: higher Chinburg invariant
zbMATH Keywords: totally real quaternionic fields
zbMATH Keywords: root number class
zbMATH Keywords: special values of Artin \(L\)-functions
MaRDI profile type: Publication
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title: Quaternionic exercises in \(K\)-theory Galois module structure. II
review text: For Part I, see Algebraic \(K\)-theory, 2nd Great Lakes conf. 1996, Fields, Inst. Commun. 16, 4-29 (1997; Zbl 0886.11063).NEWLINENEWLINENEWLINEThe purpose of this paper is to investigate, for a certain infinite family of totally real quaternionic fields \(N/\mathbb{Q}\), the relationship between \(W_{N/\mathbb{Q}}\) and \(\Omega_1(N/\mathbb{Q},3)\) in \(Cl(\mathbb{Z}[Q_8])\). Recall that \(W_{N/\mathbb{Q}}\) is the Cassou-Noguès-Fröhlich root number class, and \(\Omega_1(N/\mathbb{Q},3)\) is the ``higher Chinburg invariant'' constructed by Chinburg-Kolster-Pappas-Snaith, namely the ``Euler characteristic'' arising from a canonical 2-extension NEWLINE\[NEWLINEK_3^{\text{ind}} (O_{N,S})\to A\to B\to K_2(O_{N/S})'.NEWLINE\]NEWLINE Here \(S\) is any equivariant finite set of primes containing the ramified primes, and the dash indicates a modified \(K_2\) (in order to deal with real places). It is of course conjectured that \(\Omega_1(N/\mathbb{Q},3)= W_{N/\mathbb{Q}}\) (\(=1\) in the present situation), and the main result of the authors is a partial computation of \(\Omega_1(N/\mathbb{Q},3)\) in terms of special values of Artin \(L\)-functions. Not surprisingly, the calculations are concentrated at the prime~2.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00018].
reviewed by: Thong Nguyen Quang Do
description: scientific article