An Embedding Theorem for Quaternion Algebras

An integral version of a classical embedding theorem concerning quaternion algebras B over a number field k is proved. Assume that B satisfies the Eichler condition, that is, some infinite place of k is not ramified in B , and let Ω be an order in a quadratic extension of k . The maximal orders of B which admit an embedding of Ω are determined. Although most Ω embed into either all or none of the maximal orders of B , it turns out that some Ω are ‘selective’, in the sense that they embed into exactly half of the isomorphism types of maximal orders of B . As an application, the maximal arithmetic subgroups of B */ k * which contain a given element of B */ k * are determined.