On well-rounded ideal lattices

TL;DR

This paper explores the relationship between well-rounded and ideal lattices, proving that only cyclotomic fields produce well-rounded lattices from number fields and identifying infinitely many quadratic fields with such ideals.

Contribution

It establishes a link between well-rounded and ideal lattices, showing that only cyclotomic fields yield well-rounded lattices and demonstrating the existence of infinitely many quadratic fields with such ideals.

Findings

Only cyclotomic fields produce well-rounded lattices from full rings of integers.

Infinitely many quadratic fields contain ideals that generate well-rounded lattices in the plane.

The study connects lattice properties with algebraic number theory structures.

Abstract

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.