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 κ_μ of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality κ 1<κ2 when the μ_i are finite and supp(μ₁) (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.