Mordell-Weil Lattices in Characteristic 2: I. Construction and First Properties

In a famous 1967 paper [ŠT], Šafarevič and Tate constructed elliptic curves of arbitrarily large rank r over rational function fields of positive characteristic. The group of rational points of such a curve yields a lattice in Euclidean space of dimension r via the canonical Néron-Tate height. It seems natural to investigate the structure of this lattice, but curiously it was not until about 1990 that such lattices were examined in earnest. Shioda [Sh2] then found, as part of his analysis of the Néron-Severi groups of Fermat surfaces, a family of elliptic curves in characteristic congruent to 5 mod 6 similar to those of [ŠT], whose associated lattices produce sphere packings that improve on the previous density records for many r in the range 80 ≤ r 1000. At about the same time, I found three such families with similar sphere-packing properties for large r, one of which turned out to be identical with Shioda’s. The other two families are in characteristic 2 and 3, and their lattices also recover previously known records in “small” dimensions r ≤ 32 in addition to improving records for larger r. Further investigation revealed that these curves also provide fascinating examples and test cases for the arithmetic of elliptic curves over function fields. This paper is the first of a series devoted to the characteristic-2 family, giving lattices whose rank r is a power of 2. (Oesterlé has already reported on this family briefly in [Oe].) We begin with a quick summary of lattices in Euclidean space and the associated sphere packings. We then introduce a family of hyperelliptic curves in characteristic 2, and use them to construct potentially constant elliptic curves of high rank analogous to those of [ŠT] in odd characteristic. By investigating the arithmetic of these curves we deduce lower bounds on the sphere-packing densities of their Mordell-Weil lattices and