On the Mordell–Weil lattice of y^2 = x^3 + b x + t^{3^n + 1} in characteristic 3

Abstract

We study the elliptic curves given by y^2 = x^3 + b x +t^{3^n+1} over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u^3 + b u = v^{3^n + 1}. In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension 2 cdot 3^n for every n ge 1, which improve on the currently best known sphere packing densities in dimensions 162 (case n=4) and 486 (case n=5). For n=3, the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n=1 it has the same density as the lattice E_6 in dimension 6.