Vol. 5, No. 2, 2011

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 4, 815–1054
Issue 3, 545–813
Issue 2, 275–544
Issue 1, 1–274

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
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