An embedding theorem for Eichler orders
Let \(K\) be an algebraic number field with ring of integers \(R\), and let \(D\) be a central simple \(K\)-algebra. An old question of \textit{C. Chevalley} [L'arithmétique dans les algèbres de matrices, Paris (1936; Zbl 0014.29006; JFM 62.0102.01)] asks whether for a commutative \(R\)-order \(\Omega\subset D\) of maximal \(R\)-rank, every maximal order in \(D\) contains an isomorphic copy of \(\Omega\). While this is false in general, Chevalley gave a quantitive answer to this problem in terms of the Hilbert class field of \(K\) when \(D\) is a matrix algebra over \(K\). Recently, \textit{T. Chinburg} and \textit{E. Friedman} [J. Lond. Math. Soc., II. Ser. 60, No. 1, 33--44 (1999; Zbl 0940.11053)] settled the case where \(D\) is a quaternion algebra satisfying the Eichler condition. They gave a precise criterion for the property that every maximal order in \(D\) contains a conjugate of \(\Omega\). The authors extend this result to Eichler orders instead of maximal orders in \(D\). Since, in general, \(\Omega\) need not be contained in an Eichler order of a fixed (non-trivial) level, some minor modifications have to be made.
scientific article; zbMATH DE number 2091706