The distribution of lattices arising from orders in low degree number fields

Abstract

Orders in number fields provide natural examples of lattices. We ask: what can the successive minima of lattices arising from orders in number fields be? Given an order $\mathcal{O}$ of absolute discriminant $\Delta$ in a degree $n$ number field, let $1={\lambda }_0,\dots ,{\lambda }_{n-1}$ denote the successive minima. For $3\le n\le 5$ and many groups $G\subseteq S_n$, we compute asymptotics of the points $({\log }_{\Delta }{\lambda }_1,\dots ,{\log }_{\Delta }{\lambda }_{n-1})\in {\mathbb{R}}^{n-1}$ as $\mathcal{O}$ ranges across orders in degree $n$ fields with Galois group $G$ as $\Delta \to \infty$. In many cases, we find that the asymptotics, normalized appropriately, are given by a piecewise linear expression and are supported on a finite union of polytopes.

This is an updated version of a previous manuscript. Comments always welcome!