Elliptic nets and elliptic curves

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 $P_1,\dots ,P_n$ 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.