Narrow Results

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 $(x,y):={\mathrm{\text{Trace}}}_{K/\mathbb{Q}}(x\cdot 𝜃(y))$, where $𝜃$ is an involution of $K$. We classify all pairs $K$, $𝜃$ 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 $K$, $𝜃$ 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 $\le 48$, in particular, $\mathcal{𝒪}$ is not similar to the Leech lattice. In Appendix B, we give a general cyclicity criterion for the primary components of the discriminant group of $\mathcal{𝒪}$.