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 $3 \leq n \leq 5$ 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.