Sequences associated to elliptic curves

TL;DR

This paper explores sequences derived from division polynomials of elliptic curves, showing how they encode the curve's coefficients and analyzing their properties, including identifying perfect powers.

Contribution

It introduces a method to recover elliptic curve coefficients from sequences of normalized division polynomial values at a point.

Findings

Coefficients of elliptic curves can be expressed via sequences of division polynomial values.

Explicit formulas for sequences associated with Tate normal form are provided.

Conditions for terms in these sequences to be perfect squares or cubes are determined.

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.