Sequences associated to elliptic curves

Abstract

Let E be an elliptic curve defined over a field K (with char(K)<is not equal to> 2) given by a Weierstrass equation and let P = (x, y) is an element of E(K) be a point. Then for each n >= 1 and some gamma is an element of K* we can write the x- and y-coordinates of the point [n]P as [n]P = (phi(n)(P)/psi(2)(n)(P), omega(n)(P)/psi(3)(n)(P)) = (gamma(2)G(n)(P)/F-n(2)(P), gamma H-3(n)(P)/F-n(3)(P))where phi(n), psi(n), omega n is an element of K[x, y], gcd(phi(n), psi(n)) = 1 andF-n(P) =gamma(1-n2)psi(n)(P), G(n)(P) = gamma(-2n2) phi(n)(P), H-n(P) = gamma(-3n2)omega(n)(P)are suitably normalized division polynomials of E. In this work we show the coefficients of the elliptic curve E can be defined in terms of the sequences of values (G(n)(P))(n >= 0) and (H-n(P))(n >= 0) of the suitably normalized division polynomials of E evaluated at a point P is an element of E(K). Then we give the general terms of the sequences (G(n)(P))(n >= 0) and (H-n(P))(n >= 0) associated to Tate normal form of an elliptic curve. As an application of this we determine square and cube terms in these sequences.