## Algebra & Number Theory

### Elliptic nets and elliptic curves

Katherine Stange

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ant/1513882120

Digital Object Identifier
doi:10.2140/ant.2011.5.197

Mathematical Reviews number (MathSciNet)
MR2833790

Zentralblatt MATH identifier
1277.11063

#### Citation

Stange, Katherine. Elliptic nets and elliptic curves. Algebra Number Theory 5 (2011), no. 2, 197--229. doi:10.2140/ant.2011.5.197. https://projecteuclid.org/euclid.ant/1513882120

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