Among all even integral lattices in n-dimensional Euclidean spaces there are some particularly beautiful examples with large symmetry groups and other remarkable properties. Foremost among these are the Leech lattice of rank 24 and the E 8 lattice of rank 8. Also notable are some infinite families of lattices of increasing rank such as the Barnes Wall lattices of rank 2 f +1 .
Elkies and Shioda found that lattices such as E 8 and the Leech lattice may be reconstructed using the Mordell Weil groups of rational points on certain constant elliptic curves over global function fields (the quadratic form comes from the height pairing). They also found new dense n-dimensional sphere packings and some infinite families of Mordell Weil lattices with large groups acting on them. Furthermore Elkies showed that the Barnes Wall lattices may be realised (after scaling) as sublattices of index -|iii| in certain Mordell Weil lattices coming from hyperelliptic curves in characteristic 2, where iii is the Shafarevich Tate group.
Thompson [Th] showed how given a certain type of rational representation of a finite group one obtains an invariant lattice, unique up to scaling, which is even and unimodular with an appropriate choice of invariant positive definite quadratic form. The E 8 and Leech lattices may be realised in this way. Gross [Gr] generalised this method in such a way as to be able to produce various infinite families of interesting lattices. These include the Barnes Wall lattices, some of Elkies' Mordell Weil lattices and some lattices with finite symplectic group actions including those previously found by Gow [Go]. We show that (after scaling) these symplectic group lattices may be realised (as Gross suspected) as sublattices of index -|iii$| in certain Mordell Weil lattices coming from hyperelliptic curves in odd characteristic p. Here iii$ is the p-part of the Shafarevich Tate group. As a consequence we obtain estimates for the short vectors of the symplectic group lattices. The main result is Proposition 6 (and its corollary) in Section 8.
We now proceed to discuss all this in a little more detail. Let p be an odd prime number and q= p f a power of p such that either f is even or article no. 0154