minimal model for an elliptic curve

Let $K$ be a local field, complete with respect to a discrete valuation $\nu$ (for example, $K$ could be $\mathbb{Q}_{p}$ , the field of http://planetmath.org/node/PAdicIntegers $p$ -adic numbers, which is complete with respect to the http://planetmath.org/node/PAdicValuation $p$ -adic valuation).

Let $E/K$ be an elliptic curve defined over $K$ given by a Weierstrass equation

$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$

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 http://planetmath.org/node/WeierstrassEquationOfAnEllipticCurvethis entry, any other Weierstrass equation for $E$ is obtained by a change of variables of the form

$x=u^{2}x^{\prime}+r,\quad y=u^{3}y^{\prime}+su^{2}x^{\prime}+t$

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 $\Delta=u^{12}\Delta^{\prime}$ , 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 $0\leq t<12$ .

Definition.

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.

It follows from the discussion above that every elliptic curve over a local field $K$ has a minimal model over $K$ .

Definition.

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 $F$ .

It can be shown that all elliptic curves over $\mathbb{Q}$ 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 $\nu$ ).

Title	minimal model for an elliptic curve
Canonical name	MinimalModelForAnEllipticCurve
Date of creation	2013-03-22 15:48:03
Last modified on	2013-03-22 15:48:03
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Definition
Classification	msc 14H52
Classification	msc 11G05
Classification	msc 11G07
Synonym	minimal equation
Defines	minimal model