A generalization of Voronoi's reduction theory and its application

Abstract

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 generally, the theory is developed for forms that 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 [BF] and Bayer-Fluckiger and Nebe [BFN]. Moreover, we apply it to construct new best-known sphere coverings in dimensions $9,\dots ,15$.