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

$W_n = \psi_n(P)\qquad\text{for all}~n \ge 1.$

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

$x(nP) = \frac{A_n}{D_n^2} \quad \text{with}~\gcd(A_n,D_n)=1~\text{and}~D_n \ge 1.$

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.

Read more about this topic: Elliptic Divisibility Sequence

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