Vol. 5, No. 2, 2011

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Elliptic nets and elliptic curves

Katherine Stange

Vol. 5 (2011), No. 2, 197–229
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 P1,,Pn 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