Mordell--Weil Lattices in Charactistic 2, III: A Mordell--Weil Lattice of Rank 128

Abstract

We analyze the $128$-dimensional Mordell--Weil lattice of a certain elliptic curve over the rational function field k(t), where k is a finite field of $2^{12}$ elements. By proving that the elliptic curve has trivial Tate--Šafarevič group and nonzero rational points of height $22$, we show that the lattice's density achieves the lower bound derived in our earlier work. This density is by a considerable factor the largest known for a sphere packing in dimension 128. We also determine the kissing number of the lattice, which is by a considerable factor the largest known for a lattice in this dimension.