Elliptic nets and elliptic curves

Stange, Katherine

doi:10.2140/ant.2011.5.197

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Elliptic nets and elliptic curves

Katherine Stange

Vol. 5 (2011), No. 2, 197–229

DOI: 10.2140/ant.2011.5.197

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.

Keywords

elliptic net, elliptic curve, Laurentness, elliptic divisibility sequence, recurrence sequence

Mathematical Subject Classification 2000

Primary: 11G05, 11G07, 11B37

Secondary: 11B39, 14H52

Milestones

Received: 28 April 2010

Revised: 16 September 2010

Accepted: 17 October 2010

Published: 27 August 2011

Authors

Katherine Stange
Department of Mathematics Stanford University 450 Serra Mall, Building 380 Stanford, CA, 94305 United States
http://math.katestange.net

Vol. 5, No. 2, 2011