Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8

We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the formerly best known given by the lattice A8*. Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense.Comment: 13 pages; (v2) major revision: proof of rigidity corrected, full discussion of E8-case included, src of (v3) contains MAGMA program, (v4) some correction

Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus

We compare two rational polyhedral admissible decompositions of the cone of positive definite quadratic forms: the perfect cone decomposition and the 2nd Voronoi decomposition. We determine which cones belong to both the decompositions, thus providing a positive answer to a conjecture of V. Alexeev and A. Brunyate. As an application, we compare the two associated toroidal compactifications of the moduli space of principal polarized abelian varieties: the perfect cone compactification and the 2nd Voronoi compactification.Comment: 27 pages, 2 figures, final version, to appear in Mathematische Annale

A generalization of Voronoi's reduction theory and its application

We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.Comment: 31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke Math.