On the Mordell-Gruber spectrum

TL;DR

This paper studies the Mordell constant for lattices from totally real fields, establishing inequalities and properties of the spectrum using algebraic invariants and dynamics of the diagonal group.

Contribution

It introduces the almost sure Mordell constant for lattice families, proves strict inequalities between measures, and develops algebraic tools to analyze lattice orbits.

Findings

Established strict inequality k_mu < k_nu for certain measures.

Extended the understanding of the Mordell-Gruber spectrum's accumulation points.

Developed algebraic invariants to characterize lattice orbit properties.

Abstract

We investigate the Mordell constant of certain families of lattices, in particular, of lattices arising from totally real fields. We define the almost sure value k_mu of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality k_mu < k_nu, when mu,nu are finite and the support of mu is strictly contained in the support of nu. In combination with known results regarding the dynamics of the diagonal group we obtain isolation results as well as information regarding accumulation points of the Mordell-Gruber spectrum, extending previous work of Gruber and Ramharter. One of the main tools we develop is the associated algebra, an algebraic invariant attached to the orbit of a lattice under a block group, which can be used to characterize closed and finite volume orbits.