Counting problems for geodesics on arithmetic hyperbolic surfaces

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra all contain a fixed (finite) set of nonnegative real numbers

Selectivity in Quaternion Algebras

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.Comment: Final version; to appear in the Journal of Number Theor