Applications of spinor class fields: embeddings of orders and quaternionic lattices.
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 \(K\).
scientific article; zbMATH DE number 2093465
scientific article