Sequences associated to elliptic curves with non-cyclic torsion subgroup

Abstract

Let E be an elliptic curve defined over K given by a Weierstrass equation and let P = (x, y) is an element of E(K) be a point. Then for each n >= 1 we can write the x- and y-coordinates of the point [n]P as[n]P = (G(n)(P)/F-n(2)(P), H-n(P)/F-n(3)(P))where F-n, G(n), and H-n is an element of K[x, y] are division polynomials of E. In this work we give explicit formulas for sequences(F-n(P))(n >= 0), (G(n)(P))(n >= 0), and (H-n(P))(n >= 0)associated to an elliptic curve E defined over Q with non-cyclic torsion subgroup. As applications we give similar formulas for elliptic divisibility sequences associated to elliptic curves with non-cyclic torsion subgroup and determine square terms in these sequences.