Mordell-Weil lattices in characteristic 2. II: The Leech lattice as a Mordell-Weil lattice
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 \(n=32\).
scientific article; zbMATH DE number 1026476