Sequences associated to elliptic curves

Abstract

Let $E$ be an elliptic curve defined over a field $K$ (with $char(K)\neq 2$) given by a Weierstrass equation and let $P=(x,y)\in E(K)$ be a point. Then for each $n$ $\geq 1$ and some $\gamma \in K^{\ast }$ we can write the $x$- and $y$-coordinates of the point $[n]P$ as \begin{equation*} \lbrack n]P=\left( \frac{\phi_n(P)}{\psi_n^2(P)}, \frac{\omega_n(P) }{\psi_n^3(P)} \right) =\left( \frac{\gamma^2 G_n(P)}{F_n^2(P)}, \frac{\gamma^3 H_n(P)}{F_n^3(P)}\right) \end{equation*} where $\phi_n,\psi_n,\omega_n \in K[x,y]$, $\gcd (\phi_n,\psi_n^2)=1$ and \begin{equation*} F_n(P) = \gamma^{1-n^2}\psi_n(P), G_n(P) = \gamma ^{-2n^{2}}\phi_{n}(P),H_{n}(P) = \gamma ^{-3n^{2}} \omega_n(P) \end{equation*} 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 $(G_{n}(P))_{n\geq 0}$ and $(H_{n}(P))_{n\geq 0}$ of the suitably normalized division polynomials of $E$ evaluated at a point $P \in E(K)$. Then we give the general terms of the sequences $(G_{n}(P))_{n\geq 0}$ and $(H_{n}(P))_{n\geq 0}$ associated to Tate normal form of an elliptic curve. As an application of this we determine square and cube terms in these sequences.