Perfect Lattices over Imaginary Quadratic Number Fields

TL;DR

This paper extends Voronoi theory to imaginary quadratic number fields with higher class number, providing algorithms for classifying perfect Hermitian forms and applications to lattice group generators.

Contribution

It adapts Voronoi theory to imaginary quadratic fields with class number > 1, including characterizations and algorithms for perfect Hermitian forms.

Findings

Characterization of extreme Hermitian forms

Algorithm for enumerating perfect Hermitian forms in dimensions 2 and 3

Method to determine generators of the general linear group of an $ ext{O}_K$-lattice

Abstract

We present an adaptation of Voronoi theory for imaginary quadratic number fields of class number greater than 1. This includes a characterisation of extreme Hermitian forms which is analogous to the classic characterisation of extreme quadratic forms as well as a version of Voronoi's famous algorithm which may be used to enumerate all perfect Hermitian forms for a given imaginary quadratic number field in dimensions 2 and 3. We also present an application of the algorithm which allows to determine generators of the general linear group of an $\O_K$-lattice.