Sequences associated to elliptic curves

Abstract

Let E be an elliptic curve defined over a field K (with char(K) ≠ 2) given by a Weierstrass equation and let P = (x, y) ∈ E(K) be a point. Then for each n ≥ 1 and some γ ∈ K∗we can write the x- and y-coordinates of the point [n]P as {equation presented} where φn, φn, ωn ∈ K[x, y], gcd(φn, ψn) = 1 and Fn(P) = γ1-n2ψn(P),Gn(P) = γ-2n2φn(P),Hn(P) = γ-3n2ωn(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 (Gn(P))n≥0and (Hn(P))n≥0of the suitably normalized division polynomials of E evaluated at a point P ∈ E(K). Then we give the general terms of the sequences (Gn(P))n≥0and (Hn(P))n≥0associated to Tate normal form of an elliptic curve. As an application of this we determine square and cube terms in these sequences.