Sphere Packings Centered at S-Units of Algebraic Tori

摘要

The paper is inspired by a series of articles introducing several new explicitconstructions of dense sphere packings in R(sup n) which have good asymptotic properties while n tends to infinity. The authors present here a generalization of these constructions which seems quite natural. The main idea is to consider the group of S-units of an algebraic torus defined over a global field and to embed the torsion-free component of this finitely generated group into R(sup n) with the help of the logarithmic map (in the case of a number ground field), or the divisorial map (in the case of a function field). (For the trivial torus T = G(sub m) it is nothing but a construction of (R/Ts). The resulting lattice in R(sup n) generates a sphere packing whose parameters can be estimated while investigating arithmetical properties of algebraic tori. The key points are the (generalized) Dirichlet unit theorem valid for S-units of algebraic tori and some class number relations.