Algebra & Number Theory

Elliptic nets and elliptic curves

Katherine Stange

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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.

Article information

Algebra Number Theory, Volume 5, Number 2 (2011), 197-229.

Received: 28 April 2010
Revised: 16 September 2010
Accepted: 17 October 2010
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52] 11G07: Elliptic curves over local fields [See also 14G20, 14H52] 11B37: Recurrences {For applications to special functions, see 33-XX}
Secondary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

elliptic net elliptic curve Laurentness elliptic divisibility sequence recurrence sequence


Stange, Katherine. Elliptic nets and elliptic curves. Algebra Number Theory 5 (2011), no. 2, 197--229. doi:10.2140/ant.2011.5.197.

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