An Embedding Theorem for Quaternion Algebras

Abstract

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.

The finite subgroups of maximal arithmetic kleinian groups

“…splits in k(^)/k [13], p. 78. This proves (2). We note that by replacing x by x 3 for some integer j relatively prime to n, we may assume that C is mapped to Cn under the A;-isomorphism A;(C) ^ A;(Cn)-…”

“…Condition (1) in Theorem A, which is the only one involving Z>, is analyzed in [2]. In Theorem 3.3 below we give simple necessary and sufficient conditions under which an embeddability obstruction vanishes, so that (1) can be replaced by an elementary criterion (1') that only involves k and Ram(B).…”

“…Now suppose Cn ^ A;. Condition (2) implies that A;(Cn)A embeds in B [13], p. 78. Let C be the image of Cn under this embedding, and let d be the order of C in B*/k*.…”

“…Let us first fix a dihedral subgroup H C B*/^* of order 2n and examine the effect on d and w of varying the choices allowed in Lemma 2.3. When n > 2, there is a unique cyclic subgroup Cn C H of order n. We saw in the proof of the previous lemma that the coset dk* 2 = (x -i{x)) 2 When n = 2 there is an additional choice involved in Lemma 2.3, as Cn = C2 C H is not unique. We can replace x by ^/ or xy, leading to d being replaced by w or -dw.…”

“…CHINBURG, E. FRIEDMAN While our interest in the subject of finite subgroups of arithmetic Kleinian groups originated in the study of hyperbolic 3-manifolds, it turned out that the subject is linked to interesting aspects of indefinite quaternion algebras over a number field. Thus, the results of this paper required a prior study [2] of the embeddability of a commutative order into a maximal order. This uncovered, in certain exceptional cases, the existence of a global obstruction affecting exactly half of the maximal orders.…”