WeierstrassEquationOfAnEllipticCurve 2006-03-23 15:37:51 2006-03-23 16:49:23 Definition elliptic curve Weierstrass equation % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem*{thm}{Theorem} \newtheorem*{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem*{lemma}{Lemma} \newtheorem*{cor}{Corollary} \theoremstyle{definition} \newtheorem{exa}{Example} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Rats}{\mathbb{Q}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\Cl}{\operatorname{Cl}} Recall that an {\it elliptic curve} over a field $K$ is a projective nonsingular curve $E$ defined over $K$ of genus $1$ together with a point $O\in E$ defined over $K$. \begin{defn} Let $K$ be an arbitrary field. A Weierstrass equation for an elliptic curve $E/K$ is an equation of the form: $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ where $a_1, a_2, a_3,a_4,a_6$ are constants in $K$. \end{defn} All elliptic curves have a Weierstrass model in $\mathbb{P}^2(K)$, the projective plane over $K$. This is a simple application of the \PMlinkid{Riemann Roch theorem for curves}{RiemannRochTheorem}: \begin{thm} Let $E$ be an elliptic curve defined over a field $K$. Then there exists rational functions $x,y\in K(E)$ such that the map $\psi:E\to \mathbb{P}^2(K)$ sending $P$ to $[x(P),y(P),1]$ is an isomorphism of $E/K$ to the projective curve given by $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ where $a_1, a_2, a_3,a_4,a_6$ are constants in $K$. \end{thm} Moreover, the following proposition specifies any possible change of variables. \begin{prop} \label{change} Let $E/K$ be an elliptic curve given by a Weierstrass model of the form: $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ with $a_i\in K$. Then: \begin{enumerate} \item The only change of variables $(x,y)\mapsto (x',y')$ preserving the projective point $[0,1,0]$ and which also result in a Weierstrass equation, are of the form: $$x=u^2x'+r,\quad y=u^3y'+su^2x'+t$$ with $u,r,s,t\in K$ and $u\neq 0$. \item Any two Weierstrass equations for $E/K$ differ by a change of variables of the form given in $(1)$. \end{enumerate} \end{prop} Once we have one Weierstrass model for a given elliptic curve $E/K$, and as long as the characteristic of $K$ is not $2$ or $3$, there exists a change of variables (of the form given in the previous proposition) which simplifies the model considerably. \begin{cor} Let $K$ be a field of characteristic different from $2$ or $3$. Let $E$ be an elliptic curve defined over $K$. Then there exists a Weierstrass model for $E$ of the form: $$y^2=x^3+Ax+B$$ where $A,B$ are elements of $K$. \end{cor} Finally, remember that the $j$-invariant of an elliptic curve is invariant under isomorphism, but the discriminant depends on the model chosen. \begin{prop} Let $E/K$ be an elliptic curve and let $$ E_1: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, \quad E_2: y'^2+a_1x'y'+a_3y'=x'^3+a_2x'^2+a_4x'+a_6$$ be two distinct Weierstrass models for $E/K$. Then (by Prop. \ref{change}) there exists a change of variables $(x,y)\mapsto (x',y')$ of the form: $$x=u^2x'+r,\quad y=u^3y'+su^2x'+t$$ with $u,r,s,t\in K$ and $u\neq 0$. Moreover, $j(E_1)=j(E_2)$, i.e. the $j$ invariants are equal ($j(E)$ is defined in \PMlinkid{this entry}{JInvariant}) and $\Delta(E_1)=u^{12}\Delta(E_2)$, where $\Delta(E_i)$ is the discriminant (as defined in \PMlinkid{here}{JInvariant}). \end{prop}