Stability of ideal lattices from quadratic number fields

TL;DR

This paper investigates the stability properties of ideal lattices derived from real quadratic number fields, providing explicit conditions for stability and demonstrating that semi-stability occurs with positive probability.

Contribution

It introduces explicit criteria for the stability of ideal lattices of trace type from real quadratic fields and establishes the prevalence of semi-stable cases.

Findings

Infinite families of semi-stable and unstable ideal lattices identified.

Explicit conditions for stability based on the ideal's canonical basis.

Semi-stability occurs with positive probability for trace type lattices.

Abstract

We study semi-stable ideal lattices coming from real quadratic number fields. Specifically, we demonstrate infinite families of semi-stable and unstable ideal lattices of trace type, establishing explicit conditions on the canonical basis of an ideal that ensure stability; in particular, our result implies that an ideal lattice of trace type coming from a real quadratic field is semi-stable with positive probability. We also briefly discuss the connection between stability and well-roundedness of Euclidean lattices.