Selectivity in quaternion algebras

We prove an integral version of the classical Albert–Brauer–Hasse–Noether theorem regarding quaternion algebras over number fields. Let K be a number field with ring of integers OK, and let A be a quaternion algebra over K satisfying the Eichler condition. Let Ω be a commutative, quadratic OK-order and let R⊂A be an order of full rank. Assume that there exists an embedding of Ω into R. We describe a number of criteria which imply that every order in the genus of R admits an embedding of Ω. In the case that the relative discriminant ideal of Ω is coprime to the level of R and the level of R is coprime to the discriminant of A, we give necessary and sufficient conditions for an order in the genus of R to admit an embedding of Ω. We explicitly parameterize the isomorphism classes of orders 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. Analogous statements are proven for optimal embeddings.