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 $P=(x,y)\in E(K)$ be a point. Then for each $n$ $\geq 1$ we can write the $x$- and $y$-coordinates of the point $[n]P$ as\[ [n]P=\left( \frac{G_{n}(P)}{F_{n}^{2}(P)},\frac{H_{n}(P)}{F_{n}^{3}(P)}\right)\]where $F_{n}$, $G_{n}$, and $H_{n}\in K[x,y]$ are division polynomials of $E$. In this work we give explicit formulas for sequences\[(F_{n}(P))_{n\geq 0},\,(G_{n}(P))_{n\geq 0},\,\text{and}\,(H_{n}(P))_{n\geq 0}\]associated to an elliptic curve $E$ defined over $\mathbb{Q}$ 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.