Rings of Integers in Number Fields and Root Lattices

Abstract

This paper investigates whether a root lattice can be similar to the lattice $$\mathcal{O}$$ of all integer elements of a number field K endowed with the inner product $$(x,y)$$ := $${\text{Trac}}{{{\text{e}}}_{{K{\text{/}}\mathbb{Q}}}}(x \cdot \theta (y))$$ , where θ is an involution of the field K. For each of the following three properties (1), (2), (3), a classification of all the pairs K, θ with this property is obtained: (1) $$\mathcal{O}$$ is a root lattice; (2) $$\mathcal{O}$$ is similar to an even root lattice; (3) $$\mathcal{O}$$ is similar to the lattice $${{\mathbb{Z}}^{{[K:\mathbb{Q}]}}}$$ . The necessary conditions for similarity of $$\mathcal{O}$$ to a root lattice of other types are also obtained. It is proved that $$\mathcal{O}$$ cannot be similar to a positive definite even unimodular lattice of rank ≤48, in particular, to the Leech lattice.