Sequences associated to elliptic curves with non-cyclic torsion subgroup

Authors : Betül GEZER

Pages : 1458-1470

Doi:10.15672/hujms.464130

View : 28 | Download : 8

Publication Date : 2020-08-06

Article Type : Research Paper

Abstract :Let $E$ be an elliptic curve defined over $K$ given by a Weierstrass equation and let $P=insert ignore into journalissuearticles values(x,y);\in Einsert ignore into journalissuearticles values(K);$ be a point. Then for each $n$ $\geq 1$ we can write the $x$- and $y$-coordinates of the point $[n]P$ as \[ [n]P=\leftinsert ignore into journalissuearticles values( \frac{G_{n}insert ignore into journalissuearticles values(P);}{F_{n}^{2}insert ignore into journalissuearticles values(P);},\frac{H_{n}insert ignore into journalissuearticles values(P);}{F_{n}^{3}insert ignore into journalissuearticles values(P);}\right);\] 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 \[insert ignore into journalissuearticles values(F_{n}insert ignore into journalissuearticles values(P););_{n\geq 0},\insert ignore into journalissuearticles values(G_{n}insert ignore into journalissuearticles values(P););_{n\geq 0},\,\text{and}\insert ignore into journalissuearticles values(H_{n}insert ignore into journalissuearticles values(P););_{n\geq 0}\] 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.
Keywords : Elliptic curves, division polynomials, elliptic divisibility sequences