2010/05/28 by Linowitz, Benjamin
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1005.5326
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 Ω be a commutative, quadratic OK-order and let R⊂ \mathfrak A be an order of full rank. Assume that there exists an embedding of Ω 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 Ω. In the case that the relative discriminant ideal of Ω 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 Ω. We explicitly parameterize the isomorphism classes of orders in the genus of \mathcal 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.