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Mordell-Weil lattices in characteristic 2. III: A Mordell-Weil lattice of rank 128. (English) Zbl 1040.11041

Summary: 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-Shafarevich group and nonzero rational points of height 22, we show that the lattices 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.
For Parts I, II, see Noam D. Elkies [Int. Math. Res. Not. 1994, 343–361 (1994; Zbl 0813.52017), Invent. Math. 128, 1–8 (1997; Zbl 0897.11023)].

MSC:

11G05 Elliptic curves over global fields
11H06 Lattices and convex bodies (number-theoretic aspects)

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