Introduction

An ideal lattice is a pair (I, b), where I is an ideal of a number field, and b is a lattice, satisfying an invariance relation (see §1 for the precise definition). Ideal lattices naturally occur in many parts of number theory, but also in other areas. They have been studied in special cases, but, as yet, not much in general. In the special case of integral ideal lattices, the survey paper Bayer-Fluckiger (1999) collects and slightly extends the known results.

The first part of the paper (see §2) concerns integral ideal lattices, and states some classification problems. In §3, a more general notion of ideal lattices is introduced, as well as some examples in which this notion occurs.

The aim of §4 is to define twisted embeddings, generalising the canonical embedding of a number field. This section, as well as the subsequent one, is devoted to positive definite ideal lattices with respect to the canonical involution of the real étale algebra generated by the number field. These are also called Arakelov divisors of the number field. The aim of §5 is to study invariants of ideals and also of the number field derived from Hermite type invariants of the sphere packings associated to ideal lattices. This again gives rise to several open questions.

Definitions, notation and basic facts

A lattice is a pair (L, b), where L is a free Z-module of finite rank, and b : L × L → R is a non-degenerate symmetric bilinear form.