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

TL;DR

This paper explicitly computes the L-function of certain elliptic curves over characteristic 3 function fields, constructing high-dimensional lattices with improved sphere packing densities using the Mordell--Weil group and Néron-Tate height.

Contribution

It provides explicit lattice constructions from elliptic curves over function fields in characteristic 3, enhancing known sphere packing densities in specific dimensions.

Findings

Constructed lattices in dimensions 2·3^n with improved sphere packing densities.

Connected the L-function computation to the zeta function of superelliptic curves.

Achieved packing densities matching or exceeding the best known in certain dimensions.

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 N\'eron-Tate height on the Mordell--Weil group, we obtain lattices in dimension $2 \cdot 3^n$ for every $n \geq 1$, 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 $6$.