Complete classification of binary normal regular Hermitian lattices
type: Item
instance of: scholarly article
title: Complete classification of binary normal regular Hermitian lattices
zbMATH Open document ID: 1248.11030
author: Byeong Moon Kim
author: Ji Young Kim
author: Poo-Sung Park
published in: Journal of the Mathematical Society of Japan
publication date: 2011-09-27T00:00:00Z
review text: Let \({\mathcal O}\) be the ring of algebraic integers in an imaginary quadratic number field \(K={\mathbb Q}(\sqrt{-m})\), \(m\in {\mathbb Z}\), \(m>0\) and square free. An integral Hermitian lattice \((H,L)\) of rank \(n\) is a finitely generated \({\mathcal O}\)-module \(L\) inside \(K^n\) such that \(KL=K^n\), together with a Hermitian form \(H:L\times L\to {\mathcal O}\). Note that for such lattices, \(H(v,v)\in {\mathbb Z}\) for all \(v\in L\), and we say that \(x\in {\mathbb Z}\) is represented by \((H,L)\) if there exists \(v\in L\) with \(H(v,v)=x\). \((H,L)\) is called normal if the ideal in \({\mathcal O}\) generated by \(\{ H(v,v)\,|\,v\in L\}\) is equal to the ideal generated by \(\{ H(v,w)\,|\,v,w\in L\}\), otherwise it is called subnormal. In this paper, only positive definite integral Hermitian lattices are considered, i.e. lattices for which \(H(v,v)>0\) for all \(v\in L\setminus\{ 0\}\). Such a positive definite lattice is called regular if any \(x\in {\mathbb Z}\) that is represented by all localizations of \(L\) at all places of \(K\) is also represented globally. Regular quadratic or Hermitian lattices and their classification have been studied to quite some extent in the literature. The main result by the authors is the full classification of all positive definite binary normal regular Hermitian lattices over rings of algebraic integers in imaginary quadratic number fields up to similarity. There are altogether \(68\) such lattices, and they are described explicitly in terms of Gram matrices. Those among them that are universal, i.e. that represent all positive integers, are highlighted. Such lattices only exist for \(m\in \{ 1,2,3,5,6,7,10,11,15,19,23,31\}\). The authors also announce new results concerning the case of subnormal regular Hermitian lattices.
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11E41
zbMATH DE Number: 5950730
zbMATH Keywords: Hermitian lattice
zbMATH Keywords: regular lattice
zbMATH Keywords: normal lattice
zbMATH Keywords: universal lattice
zbMATH Keywords: imaginary quadratic number field
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06331001
OpenAlex ID: W2015628375
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DOI: 10.2969/JMSJ/06331001
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description: scientific article; zbMATH DE number 5950730