minimal model for an elliptic curve

Let K be a local fieldMathworldPlanetmath, completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with respect to a discrete valuationPlanetmathPlanetmath ν (for example, K could be p, the field of numbers, which is complete with respect to the valuationMathworldPlanetmathPlanetmath).

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


where a1,a2,a3,a4,a6 are constants in K. By a suitable change of variables, we may assume that ν(ai)0. As it is pointed out in entry, any other Weierstrass equation for E is obtained by a change of variables of the form


with u,r,s,tK and u0. Moreover, by PropositionPlanetmathPlanetmath 2 in the same entry, the discriminantsPlanetmathPlanetmathPlanetmathPlanetmath of both equations satisfy Δ=u12Δ, so they only differ by a 12th power of a non-zero number in K. Let us define a set:

S={ν(Δ):Δ is the discriminant of a Weierstrass eq. for E and ν(Δ)0}

Since ν is a discrete valuation, the set S is a set of non-negative integers, therefore it has a minimum value mS. Moreover, by the remark above, m satisfies 0m<12 and m is the unique number tS with 0t<12.


Let E/K be an elliptic curve over a local field K, complete with respect to a discrete valuation ν. A Weierstrass equation for E with discriminant Δ is said to be a minimal model for E (at ν) if ν(Δ)=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.


Let F be a number fieldMathworldPlanetmath and let ν be an infiniteMathworldPlanetmathPlanetmath 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 minimalPlanetmathPlanetmath at ν if the same model is minimal over Fν, the completion of F at ν. A Weierstrass equation for E/F is said to be minimal if it is minimal at ν for all places ν of F.

It can be shown that all elliptic curves over 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 ν for every ν).

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