MinimalModelForAnEllipticCurve 2006-03-23 17:09:29 2006-03-23 17:09:29 Definition Weierstrass equation of an elliptic curve minimal model % 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}} Let $K$ be a local field, complete with respect to a discrete valuation $\nu$ (for example, $K$ could be $\Rats_p$, the field of \PMlinkid{$p$-adic numbers}{PAdicIntegers}, which is complete with respect to the \PMlinkid{$p$-adic valuation}{PAdicValuation}). Let $E/K$ be an elliptic curve defined over $K$ given by a Weierstrass equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ where $a_1, a_2, a_3,a_4,a_6$ are constants in $K$. By a suitable change of variables, we may assume that $\nu(a_i)\geq 0$. As it is pointed out in \PMlinkid{this entry}{WeierstrassEquationOfAnEllipticCurve}, any other Weierstrass equation for $E$ is obtained by a change of variables 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, by Proposition 2 in the same entry, the discriminants of both equations satisfy $\Delta=u^{12}\Delta'$, so they only differ by a $12$th power of a non-zero number in $K$. Let us define a set: $$S=\{ \nu(\Delta) : \Delta \text{ is the discriminant of a Weierstrass eq. for $E$ and } \nu(\Delta)\geq 0\}$$ Since $\nu$ is a discrete valuation, the set $S$ is a set of non-negative integers, therefore it has a minimum value $m\in S$. Moreover, by the remark above, $m$ satisfies $0\leq m <12$ and $m$ is the unique number $t\in S$ with $0\leq t < 12$. \begin{defn} Let $E/K$ be an elliptic curve over a local field $K$, complete with respect to a discrete valuation $\nu$. A Weierstrass equation for $E$ with discriminant $\Delta$ is said to be a minimal model for $E$ (at $\nu$) if $\nu(\Delta)=m$, the minimum of the set $S$ above. \end{defn} It follows from the discussion above that every elliptic curve over a local field $K$ has a minimal model over $K$. \begin{defn} Let $F$ be a number field and let $\nu$ be an infinite or finite place (archimedean or non-archimedean prime) of $F$. Let $E/F$ be an elliptic curve over $F$. A given Weierstrass model for $E/F$ is said to be minimal at $\nu$ if the same model is minimal over $F_\nu$, the completion of $F$ at $\nu$. A Weierstrass equation for $E/F$ is said to be minimal if it is minimal at $\nu$ for all places $\nu$ of $F$. \end{defn} It can be shown that all elliptic curves over $\Rats$ have a global minimal model. However, this is not true over general number fields. There exist elliptic curves over a number field $F$ which do not have a global minimal model (i.e. any given model is not minimal at $\nu$ for every $\nu$).