Root lattices in number fields

Abstract

We explore whether a root lattice may be similar to the lattice $\mathcal{O}$ of integers of a number field $K$ endowed with the inner product $(x,y):={\mathrm{Trace}}_{K/\mathbb{Q}}(x\cdot \theta (y))$, where $\theta$ is an involution of $K$. We classify all pairs $K$, $\theta$ such that $\mathcal{O}$ is similar to either an even root lattice or the root lattice ${\mathbb{Z}}^{[K:\mathbb{Q}]}$. We also classify all pairs $K$, $\theta$ such that $\mathcal{O}$ is a root lattice. In addition to this, we show that $\mathcal{O}$ is never similar to a positive-definite even unimodular lattice of rank $⩽48$, in particular, $\mathcal{O}$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $\mathcal{O}$.

23 pages. Introduction rewritten, Proposition 1 added, minor corrections in the formulation and proof of Theorem 5 implemented