Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds

Abstract

We show that if $M$ is an arithmetic hyperbolic $3$-manifold, the set $\mathbb{Q}L(M)$ of all rational multiples of lengths of closed geodesics of $M$ both determines and is determined by the commensurability class of $M$. This implies that the spectrum of the Laplace operator of $M$ determines the commensurability class of $M$. We also show that the zeta function of a number field with exactly one complex place determines the isomorphism class of the number field