PlanetMath: minimal model for an elliptic curve

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minimal model for an elliptic curve

(Definition)

Let be a local field, complete with respect to a discrete valuation $\nu$ (for example, could be $\mathbb{Q}_p$ , the field of -adic numbers, which is complete with respect to the -adic valuation).

Let be an elliptic curve defined over given by a Weierstrass equation

$\displaystyle y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$

where

are constants in

. By a suitable change of variables, we may assume that $\nu(a_i)\geq 0$ . As it is pointed out in this entry, any other Weierstrass equation for

is obtained by a change of variables of the form

$\displaystyle x=u^2x'+r,\quad y=u^3y'+su^2x'+t$

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

th power of a non-zero number in

. Let us define a set:

$\displaystyle S=\{ \nu(\Delta) : \Delta$ is the discriminant of a Weierstrass eq. for $E$ and $\displaystyle \nu(\Delta)\geq 0\}$

Since $\nu$ is a discrete valuation, the set

is a set of non-negative integers, therefore it has a minimum value $m\in S$ . Moreover, by the remark above,

satisfies $0\leq m <12$ and

is the unique number $t\in S$ with $0\leq t < 12$ .

Definition 1 Let be an elliptic curve over a local field , complete with respect to a discrete valuation $\nu$ . A Weierstrass equation for with discriminant $\Delta$ is said to be a minimal model for (at $\nu$ ) if $\nu(\Delta)=m$ , the minimum of the set above.

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

Definition 2 Let be a number field and let $\nu$ be an infinite or finite place (archimedean or non-archimedean prime) of . Let be an elliptic curve over . A given Weierstrass model for is said to be minimal at $\nu$ if the same model is minimal over $F_\nu$ , the completion of at $\nu$ . A Weierstrass equation for is said to be minimal if it is minimal at $\nu$ for all places $\nu$ of .

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 which do not have a global minimal model (i.e. any given model is not minimal at $\nu$ for every $\nu$ ).

"minimal model for an elliptic curve" is owned by alozano.

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Other names:

minimal equation

Also defines:

minimal model

This object's parent.

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Cross-references: places, completion, minimal, Weierstrass model, prime, non-archimedean, archimedean, finite place, infinite, number field, integers, number, equations, discriminants, proposition, variables, Weierstrass equation, elliptic curve, field, discrete valuation, complete, local field
There are 2 references to this entry.

This is version 1 of minimal model for an elliptic curve, born on 2006-03-23.
Object id is 7763, canonical name is MinimalModelForAnEllipticCurve.
Accessed 1817 times total.

Classification:

AMS MSC:	14H52 (Algebraic geometry :: Curves :: Elliptic curves)
	11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields)
	11G07 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over local fields)

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