On the Mordell-Gruber Spectrum

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 κμ of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality κμ1 < κμ2 when the μi are finite and supp(μ1) supp(μ2). 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.