Mordell–Weil Lattices in Characteristic 2, III: A Mordell–Weil Lattice of Rank 128
Noam D. Elkies · Experimental Mathematics · 2001
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 212 elements. By proving that the elliptic curve has trivial Tate–Šafarevič up 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.