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

TL;DR

This paper investigates the structure of maximal orders in central simple algebras over number fields, establishing bounds on nonselective sets and applying these results to construct and analyze isospectral hyperbolic manifolds.

Contribution

It provides bounds for nonselective sets of maximal orders and applies these findings to the inverse spectral problem, clarifying and extending results on isospectral hyperbolic manifolds.

Findings

Bounds for the size of nonselective sets of maximal orders

Application to constructing isospectral nonisometric hyperbolic surfaces

Example of hyperbolic surfaces from quaternion algebras with selectivity

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 Vign{\'e}ras 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.