2012/07/26 by Uri Shapira, Shapira, Uri, Barak Weiss +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra over a field #Algebraic number #Combinatorics #Computer science #Constant (computer programming) #Diagonal #Discrete mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Finite group #Geometry #Group (periodic table) #Homogeneous #Homogeneous space #Invariant (physics) #Lattice (music) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Number Theory (math.NT) #Orbit (dynamics) #Physics #Pure mathematics #Quantum mechanics #Spectrum (functional analysis) #math.DS #math.NT
paper · pdf · doi:10.48550/arxiv.1207.6343
36 pages
arxiv created 2012/07/26 · openalex publication_date 2012/07/26 · arxiv updated 2012/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 kmu of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality kmu < knu, 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.