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 $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}$. In the case that the relative discriminant ideal of $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}$. We explicitly parameterize the isomorphism classes of orders in the genus of $\mathcal{R}$ which admit an embedding of $\hat{I}\text{c◯}$. In particular, we show that the proportion of the genus of $\mathcal{R}$ admitting an embedding of $\hat{I}\text{c◯}$ is either 0, 1/2 or 1. Analogous statements are proven for optimal embeddings.

Final version; to appear in the Journal of Number Theory