Selectivity in Quaternion Algebras

Let A be a quaternion algebra over a number field K and assume that A satisfies the Eichler condition; that is, there exists an archimedean prime of K which does not ramify in A. Let Ω be a commutative, quadratic OK-order and let R ⊂ A be an order. We determine the isomorphism classes in the genus of R which admit an embedding of Ω. In particular, we show that the proportion of the genus of R admitting an embedding of Ω is either 0, 1/2 or 1. Additionally, conditions are developed which determine the orders Ω which can be embedded into an order in the genus of R in the case that R is locally a primitive order or an Eichler order at every finite non-dyadic prime of K. These results generalize work of Chinburg and Friedman (in which only maximal orders were considered) and Guo and Qin (in which only Eichler orders were considered).