Mordell-Weil lattices in characteristic 2. II: The Leech lattice as a Mordell-Weil lattice
type: Item
instance of: scholarly article
title: Mordell-Weil lattices in characteristic 2. II: The Leech lattice as a Mordell-Weil lattice
zbMATH Open document ID: 0897.11023
author: Noam D. Elkies
published in: Inventiones Mathematicae
publication date: 1998-06-16T00:00:00Z
review text: As in part I [Int. Math. Res. Not. 1994, No. 8, 343-361 (1994; Zbl 0813.52017)], the author studies the Mordell-Weil lattice \(L\) of a constant elliptic curve over the function field of a hyperelliptic curve \(C\), both attaining Weil's upper bound on the number of rational points over the finite field with \(q^2\) elements. He now takes \(q=64\) and \(C\) of genus \(g=6\). Then, by the general theory, \(L\) is an even lattice of rank \(n=4g=24\), and the product of \(\det L\) with the order of the Tate-Shafarevich group is \(2^n\). The author shows that here the group is trivial, any nonzero vector of \(L\) has at least norm 8, and all norms are multiples of 4. This proves that \(L\) is similar to Leech's lattice. Finally, the 196560 minimal vectors are found, and this enumeration is used to complete the proof in an alternative way. The author also announces a subsequent paper studying Mordell-Weil lattices of 2-power ranks up to \(n=32\).
Mathematics Subject Classification ID: 11H31
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 1026476
zbMATH Keywords: Leech lattice
zbMATH Keywords: Mordell-Weil lattice
zbMATH Keywords: elliptic curves
zbMATH Keywords: function field of a hyperelliptic curve
zbMATH Keywords: minimal vectors
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
full work available at URL: s002220050133
OpenAlex ID: W2073111855
DOI: 10.1007/S002220050133
description: scientific article; zbMATH DE number 1026476