Elliptic Curves and Points Associated To EDS
Ward proves that associated to any nonsingular EDS (Wn) is an elliptic curve E/Q and a point P ε E(Q) such that
Here ψn is the n division polynomial of E; the roots of ψn are the nonzero points of order n on E. There is a complicated formula for E and P in terms of W1, W2, W3, and W4.
There is an alternative definition of EDS that directly uses elliptic curves and yields a sequence which, up to sign, almost satisfies the EDS recursion. This definition starts with an elliptic curve E/Q given by a Weierstrass equation and a nontorsion point P ε E(Q). One writes the x-coordinates of the multiples of P as
Then the sequence (Dn) is also called an elliptic divisibility sequence. It is a divisibility sequence, and there exists an integer k so that the subsequence ( ±Dnk )n ≥ 1 (with an appropriate choice of signs) is an EDS in the earlier sense.
Read more about this topic: Elliptic Divisibility Sequence
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