Elliptic Divisibility Sequence - Elliptic Curves and Points Associated To EDS

Elliptic Curves and Points Associated To EDS

Ward proves that associated to any nonsingular EDS (W_n) is an elliptic curve E/Q and a point P ε E(Q) such that

Here ψ_n is the n division polynomial of E; the roots of ψ_n 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 (D_n) is also called an elliptic divisibility sequence. It is a divisibility sequence, and there exists an integer k so that the subsequence ( ±D_nk )_{n ≥ 1} (with an appropriate choice of signs) is an EDS in the earlier sense.