Selectivity in Quaternion Algebras

Abstract

We prove an integral version of the classical Albert-Brauer-Hasse-Noether theorem regarding quaternion algebras over number fields. Let $\mathfrak A$ be a quaternion algebra over a number field $K$ and assume that $\mathfrak A$ satisfies the Eichler condition; that is, there exists an archimedean prime of $K$ which does not ramify in $\mathfrak A$. Let $\Omega$ be a commutative, quadratic $\mathcal{O}_K$-order and let $\mathcal{R}\subset \mathfrak A$ be an order of full rank. Assume that there exists an embedding of $\Omega$ into $\mathcal R$. We describe a number of criteria which, if satisfied, imply that every order in the genus of $\mathcal R$ admits an embedding of $\Omega$. In the case that the relative discriminant ideal of $\Omega$ is coprime to the level of $\mathcal R$ and the level of $\mathcal R$ is coprime to the discriminant of $\mathfrak A$, we give necessary and sufficient conditions for an order in the genus of $\mathcal R$ to admit an embedding of $\Omega$. We explicitly parameterize the isomorphism classes of orders in the genus of $\mathcal R$ which admit an embedding of $\Omega$. In particular, we show that the proportion of the genus of $\mathcal{R}$ admitting an embedding of $\Omega$ is either 0, 1/2 or 1. Analogous statements are proven for optimal embeddings.