Elliptic nets and elliptic curves

TL;DR

This paper introduces elliptic nets as higher-dimensional analogues of elliptic divisibility sequences, establishing their properties, relations to elliptic curves, and integrality results over arbitrary fields.

Contribution

It extends elliptic divisibility sequences to higher dimensions, providing explicit bijections with elliptic curves and points, and proves Laurentness and integrality properties.

Findings

Established a bijection between elliptic nets and elliptic curves with points

Proved Laurentness and integrality of elliptic nets

Generalized elliptic divisibility sequences to higher dimensions

Abstract

An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P_1, ..., P_n are points on E defined over K. To this information we associate an n-dimensional array of values in K satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set of elliptic curves with specified points. We also obtain Laurentness/integrality results for elliptic nets.