Sphere Packing Densities of Sublattices of the Mordell-Weil Lattices of two Families of Elliptic Curves

TL;DR

This paper investigates the sphere packing densities of specific sublattices within Mordell-Weil lattices of elliptic curves over fields of positive characteristic, providing explicit bounds and volume calculations.

Contribution

It introduces explicit bounds on sphere packing densities for sublattices of Mordell-Weil lattices of two elliptic curve families, a novel analysis in positive characteristic.

Findings

Derived lower bounds on sphere packing densities

Computed volumes of fundamental domains

Established minimal norms for sublattices

Abstract

In this paper, we examine certain maximal rank sublattices of the Mordell-Weil lattices of two families of elliptic curves over fields of characteristic $p > 0$. We compute explicit lower bounds on the densest sphere packings of these sublattices by finding lower bounds on the minimal norms of the sublattices and explicitly computing the volumes of their fundamental domains.