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

TL;DR

This paper investigates the asymptotic behavior of successive minima of lattices from orders in low degree number fields, revealing piecewise linear patterns supported on polytopes as the discriminant grows.

Contribution

It provides the first detailed asymptotic analysis of lattice minima from orders in degree 3 to 5 number fields with specified Galois groups.

Findings

Successive minima asymptotics are piecewise linear functions.

Results are supported on finite unions of polytopes.

Asymptotic behavior depends on the Galois group of the field.

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 $3 \leq n \leq 5$ 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 $\Delta \rightarrow \infty$. 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.