A note on \(K3\) surfaces and sphere packings
type: Item
instance of: scholarly article
title: A note on \(K3\) surfaces and sphere packings
zbMATH Open document ID: 0982.14024
author: Tetsuji Shioda
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-03-21T00:00:00Z
review text: The main object of the paper under review is the Mordell-Weil lattice of an elliptic \(K3\) surface defined over an algebraically closed field \(k\) contained in the field of complex numbers. To be more precise, the focus is on a special class of such surfaces studied earlier by Inose and Kuwata; they are related to the Kummer surfaces \(\text{Km}(C_1\times C_2)\) where \(C_1\) and \(C_2\) are elliptic curves over \(k\). The reason is that some of such surfaces have high Mordell-Weil rank given by \(r=16+h\) where \(h=\text{rk Hom}(C_1,C_2)\). For the Mordell--Weil lattice of such a surface, assuming that \(C_1\) and \(C_2\) are not isomorphic, the author's main results give the precise value of the minimal norm (\(\mu =4\)) and expressions for the determinant and the center density depending on some parameters. This leads to lattices denser than the recorded ones in dimensions 16, 17, and 18 under the condition that surfaces with required parameters really exist.
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 11H31
Mathematics Subject Classification ID: 05B40
Mathematics Subject Classification ID: 14J27
zbMATH DE Number: 1538525
zbMATH Keywords: elliptic surface
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: Mordell-Weil lattice
zbMATH Keywords: minimal norm
zbMATH Keywords: determinant
zbMATH Keywords: center density
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: pjaa.76.68
OpenAlex ID: W2044627669
cites work: Q4520550
cites work: Q4023809
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cites work: Mordell-Weil Lattices and Sphere Packings
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DOI: 10.3792/PJAA.76.68
description: scientific article; zbMATH DE number 1538525