A Bound on the Norm of Shortest Vectors in Lattices Arising from CM Number Fields

TL;DR

This paper establishes bounds on the lengths of shortest vectors in lattices derived from CM number fields using a modified quadratic form, with implications for computational lattice problems and modular forms.

Contribution

It introduces a weighted norm to bound minimal vector lengths in CM lattice families and provides a finite set for identifying minimal vectors across principal ideals.

Findings

Bound on the field norm of minimal vectors

Finite set of elements for minimal vectors in principal ideals

Application to Craig's Difference Lattice problem

Abstract

This paper partially addresses the problem of characterizing the lengths of vectors in a family of Euclidean lattices that arise from any CM number field. We define a modified quadratic form on these lattices, the weighted norm, that contains the standard field trace as a special case. Using this modified quadratic form, we obtain a bound on the field norm of any vector that has a minimal length in any of these lattices, in terms of a basis for the group of units of the ring of integers of the field. For any CM number field F, we prove that there exists a finite set of elements of F which allows one to find the set of minimal vectors in every principal ideal of the ring of integers of F. We interpret our result in terms of the asymptotic behavior of a Hilbert modular form, and consider some of the computational implications of our theorem. Additionally, we show how our result can be applied to the specific Craig's Difference Lattice problem, which asks us to find the minimal vectors in lattices arising from cyclotomic number fields.