Construction of a Hermitian lattice without a basis of minimal vectors
type: Item
instance of: scholarly article
title: Construction of a Hermitian lattice without a basis of minimal vectors
zbMATH Open document ID: 1296.11024
author: Poo-Sung Park
published in: Proceedings of the Japan Academy. Series A
publication date: 2012-08-23T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1336394841
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11H50
zbMATH DE Number: 6071044
zbMATH Keywords: Hermitian lattice
zbMATH Keywords: minimal vector
zbMATH Keywords: unimodular matrix
zbMATH Keywords: quadratic field
MaRDI profile type: Publication
OpenAlex ID: W2072423123
cites work: A lattice without a basis of minimal vectors
cites work: Q4375605
cites work: Hermitian lattices without a basis of minimal vectors
cites work: On a lower bound for the class number of an imaginary quadratic field
DOI: 10.3792/PJAA.88.75
review text: It is known that infinitely many imaginary quadratic fields allow Hermitian lattices which are generated by minimal vectors but have no basis of minimal vectors. \textit{B. M. Kim} and the author found in [Proc. Am. Math. Soc. 136, No. 9, 3041--3044 (2008; Zbl 1165.11040)] two types of binary free Hermitian lattices which are generated by their minimal vectors, but are not generated by any 3 minimal vectors. It was a variant of Conway-Sloane's answer. However it has not been analyzed how the Hermitian lattices were constructed yet. In this article, the authors show the structure of those lattices and other lattices of the same property have been constructed. The main idea of this article is to convert a binary diagonal lattice with non-principal coefficient ideals into a binary free lattice. Then, four vectors generate the lattice, but three of them cannot generate it. The main results are as follows. Let \(m\) be a positive square-free integer such that \( m = a^{2}-b^{2}\) with \(a\) odd and \(b\) even satisfying \( 0< 2b <a.\) Choose two integers \(A\) and \(B\) such that \( aA +2bB =1\). Then the binary unimodular Hermitian lattice \(L\) with Gram matrix NEWLINE\[NEWLINE\left( \begin{matrix} a(1+B^{2}) & B+(1+B^{2})(-b+\sqrt{-m}) \\ B+(1+B^{2})(-b-\sqrt{-m}) & a(1+B^{2})+A \\ \end{matrix} \right)NEWLINE\]NEWLINE over \( Q(\sqrt{-m})\) has minimal vectors of squared length \(a\), is generated by its \(8\) minimal vectors, but is not generated by any 3 minimal vectors. Further, if \( m\) is a positive squarefree integer such that \(m = (2a)^{2}-(2b+1)^{2}\) with \( 0< 2b+1 <a,\) choose two integers \(A\) and \(B\) such that \( aA +(2b+1)B =1\). Then the binary unimodular Hermitian lattice \(L\) with Gram matrix NEWLINE\[NEWLINE\left( \begin{matrix} a(1+B^{2}) & B+(1+B^{2})(-b-1+\omega) \\ B+(1+B^{2})(-b-1+\overline{\omega}) & a(1+B^{2})+A \\ \end{matrix} \right)NEWLINE\]NEWLINE over \( Q(\sqrt{-m})\) has minimal vectors of squared length \(a\), is generated by its…
description: scientific article; zbMATH DE number 6071044