On the classification of lattices over \(\mathbb Q(\sqrt{-3})\) which are even unimodular \(\mathbb Z\)-lattices of rank 32
type: Item
instance of: scholarly article
title: On the classification of lattices over \(\mathbb Q(\sqrt{-3})\) which are even unimodular \(\mathbb Z\)-lattices of rank 32
zbMATH Open document ID: 1367.11042
DOI: 10.1155/2013/837080
author: Andreas Henn
author: Michael Hentschel
author: Gabriele Nebe
author: Aloys Krieg
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2013-05-23T00:00:00Z
review text: Let \(\mathcal{O}=\mathbb{Z}[(1+\sqrt{-3})/2]\) be the ring of Eisenstein integers. A Hermitian (with respect to complex conjugation) \(\mathcal{O}\)-lattice \((\Lambda,h)\) of \(\mathcal{O}\)-rank \(r\) is called an Eisenstein lattice if it is positive definite and if its quadratic trace lattice \((\Lambda,q)\) of \(\mathbb{Z}\)-rank \(n=2r\) with \(q(x,y)=h(x,y)+\overline{h(x,y)}\) is an even unimodular lattice, so in particular, \(n\) must be a multiple of \(8\). The three last named authors classified Eisenstein lattices for \(n=8,\;16,\;24\) in [Abh. Math. Semin. Univ. Hamb. 80, No. 2, 183--192 (2010; Zbl 1229.11062)] using the Kneser-Niemeier classification of positive definite even unimodular \(\mathbb{Z}\)-lattices for these ranks and the fact, that any such lattice that is a trace lattice of an Eisenstein lattice must contain an automorphism with minimal polynomial \(X^2-X+1\). To treat the case \(n=32\) in this paper, this method is not feasible because there are simply too many positive definite even unimodular \(\mathbb{Z}\)-lattices in that case. Instead, the authors apply a generalization of Kneser's neighbor method to Hermitian lattices over \(\mathcal{O}\), starting from a pre-constructed set of Eisenstein lattices including such lattices that are orthogonal sums of Eisenstein lattices of smaller rank. Using a computer, they thus construct a list of representatives of Eisenstein lattices and use the mass formula to check that the list is indeed exhaustive. It turns out that up to isometry, there are \(80\) Eisenstein lattices of \(\mathcal{O}\)-rank \(16\). The paper contains a list of these lattices and of the orders of their automorphism groups.
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11H56
zbMATH DE Number: 6165373
zbMATH Keywords: even unimodular lattice
zbMATH Keywords: Hermitian lattice
zbMATH Keywords: Eisenstein lattice
zbMATH Keywords: Kneser's neighbor method
zbMATH Keywords: mass formula
zbMATH Keywords: automorphism of a lattice
Wikidata QID: Q58989332
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/837080
OpenAlex ID: W2029890711
cites work: On the classification of even unimodular lattices with a complex structure
cites work: On the classification of lattices over \(\mathbb Q(\sqrt{-3})\), which are even unimodular \(\mathbb Z\)-lattices
cites work: A mass formula for unimodular lattices with no roots
cites work: Classification of Hermitian forms with the neighbour method
cites work: Classification of Two Genera of 32-Dimensional Lattices of Rank 8 over the Hurwitz Order
cites work: Applications of coding theory to the construction of modular lattices
cites work: Even unimodular Gaussian lattices of rank 12
cites work: Q3870964
cites work: Réseaux unimodulaires quaternioniens en dimension ≤ 32
cites work: Q4219187
description: scientific article; zbMATH DE number 6165373