Successive minima of orders in number fields

Abstract

Orders in number fields provide interesting examples of lattices. We ask: what lattices arise from orders in number fields and how are they distributed? In the first chapter of this thesis, we prove that all nontrivial multiplicative constraints on successive minima of orders come from multiplication. Moreover, for infinitely many positive integers $n$ (including all $n < 18$), we explicitly determine all multiplicative constraints on successive minima of orders in degree $n$ number fields. We also prove analogous results for scrollar invariants of curves. Now suppose $3 \leq n \leq 5$ and let $G \subseteq S_n$. An order $\mathcal{O}$ of absolute discriminant $\Delta$ in a degree $n$ number field has $n$ successive minima $1 = \lambda_0 \leq \lambda_1 \leq \dots \leq \lambda_{n-1}$. In the next three chapters, we compute for many $G$ the distribution 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 $\Delta \rightarrow \infty$.