Elliptic Nets and Elliptic Curves
نویسنده
چکیده
Elliptic divisibility sequences are integer recurrence sequences, each of which is 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 of K satisfying a complicated nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. All elliptic nets arise from elliptic curves in this manner. In this paper we describe properties of elliptic nets, and in particular we prove an explicit bijection between the set of elliptic nets and the set of elliptic curves with specified points.
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