Applications of spinor class fields: embeddings of orders and quaternionic lattices.
type: Item
instance of: scholarly article
title: Applications of spinor class fields: embeddings of orders and quaternionic lattices.
zbMATH Open document ID: 1060.11018
author: Luis Arenas-Carmona
publication date: 2004-08-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116092
review text: Let \(K\) be an algebraic number field, and let \(D\) be a non-quaternion central simple \(K\)-algebra which is locally either a matrix algebra or a skew-field at every finite place. For a maximal subfield \(L\supset K\) of \(D\) with ring of integers \({\mathcal{O}}_L\), let \(n\) denote the number of all conjugacy classes of maximal orders in \(D\), and \(n_L\) the number of conjugacy classes of those maximal orders which admit a suborder isomorphic to \({\mathcal{O}}_L\). The author proves the existence of an abelian extension \(\Sigma\) of \(K\) such that \(n:n_L=[\Sigma\cap L:K]\) holds for all \(L\). For a quaternion algebra \(D\) satisfying the Eichler condition, a similar result was proved by \textit{T. Chinburg} and \textit{E. Friedman} [J. Lond. Math. Soc., II. Ser. 60, No. 1, 33--44 (1999; Zbl 0940.11053)]. In case \(D\) is a matrix algebra, the formula \(n:n_L=[\Sigma\cap L:K]\) is due to \textit{C. Chevalley} [L'arithmétique dans les algèbres de matrices, Paris (1936; Zbl 0014.29006; JFM 62.0102.01)]. Here \(\Sigma\) is the Hilbert class field of \(K\).
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 11R54
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 2093465
zbMATH Keywords: spinor norm
zbMATH Keywords: spinor genus
zbMATH Keywords: class fields
zbMATH Keywords: skew-Hermitian forms
zbMATH Keywords: maximal orders
zbMATH Keywords: central simple algebras
MaRDI profile type: Publication
OpenAlex ID: W282106639
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DOI: 10.5802/AIF.1999
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2093465