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

Counting isospectral manifolds

Given a simple Lie group $H$ of real rank at least $2$ we show that the maximum cardinality of a set of isospectral non-isometric $H$-locally symmetric spaces of volume at most $x$ grows at least as fast as $x^{c\log x/ (\log\log x)^2}$ where $c = c(H)$ is a positive constant. In contrast with the real rank $1$ case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [BL]. Our proof uses Sunada's method, results of [BL], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.Comment: 9 pages; v2: one reference added, this is a final version; v3 includes small corrections to the published versio