Lattice packings through division algebras

Abstract

In this text, we will show the existence of lattice packings in a family of dimensions by employing division algebras. This construction is a generalization of Venkatesh’s lattice packing result Venkatesh (Int Math Res Notices 2013(7): 1628–1642, 2013). In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. We employ a probabilistic argument to show the existence of lattices in certain dimensions with good packing densities. The approach improves the best known lower bounds on the lattice packing problem for certain dimensions.We work with a moduli space of lattices that are invariant under the action of a finite group, one that can be embedded inside a division algebra. To obtain our existence result, we prove a division algebra variant of the Siegel’s mean value theorem. In order to establish this, we describe a useful description of the Haar measure on our moduli space and a coarse fundamental domain to perform the integration.