Selective orders in central simple algebras and isospectral families of arithmetic manifolds

Abstract

Let k be a number field and B be a central simple algebra over k of dimension p 2 where p is prime. In the case that p = 2 we assume that B is not totally definite. In this paper we study sets of pairwise nonisomorphic maximal orders of B with the property that a $${\mathcal{O}_{k}}$$ -order of rank p embeds into either every maximal order in the set or into none at all. Such a set is called nonselective. We prove upper and lower bounds for the cardinality of a maximal nonselective set. This problem is motivated by the inverse spectral problem in differential geometry. In particular we use our results to clarify a theorem of Vigneras on the construction of isospectral nonisometric hyperbolic surfaces and 3-manifolds from orders in quaternion algebras. We conclude by giving an example of isospectral nonisometric hyperbolic surfaces which arise from a quaternion algebra exhibiting selectivity.