Shapes of tame abelian number fields

Abstract

The shape of a number field $K$ of degree $n\ge 2$ is the similarity class of the lattice obtained by projecting ${\mathcal{O}}_K$ orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-$(n-1)$ lattices, ${\mathcal{S}}_{n-1}:={\mathrm{GL}}_{n-1}(\mathbb{Z})\backslash {\mathrm{GL}}_{n-1}(\mathbb{R})/{\mathrm{GO}}_{n-1}(\mathbb{R})$, where ${\mathrm{GO}}_{n-1}(\mathbb{R})$ is the group of orthogonal similitudes. Let $G$ be a fixed nontrivial finite abelian group, and let $K/\mathbb{Q}$ vary over tamely ramified Galois extensions with Galois group isomorphic to $G$. We study the variation of the shapes of such number fields in ${\mathcal{S}}_{n-1}$, where $n=|G|$.

arXiv commentversion 1: 32 pages