bad reduction

1 Singular Cubic Curves

Let E be a cubic curve over a field K with Weierstrass equation f(x,y)=0, where:


which has a singular pointMathworldPlanetmathPlanetmath P=(x0,y0). This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to:


and so we can write the Taylor expansionMathworldPlanetmath of f(x,y) at (x0,y0) as follows:

f(x,y)-f(x0,y0) = λ1(x-x0)2+λ2(x-x0)(y-y0)+λ3(y-y0)2-(x-x0)3
= [(y-y0)-α(x-x0)][(y-y0)-β(x-x0)]-(x-x0)3

for some λiK and α,βK¯ (an algebraic closureMathworldPlanetmath of K).

Definition 1.

The singular point P is a node if αβ. In this case there are two different tangent lines to E at P, namely:


If α=β then we say that P is a cusp, and there is a unique tangent line at P.

Note: See the entry for elliptic curveMathworldPlanetmath for examples of cusps and nodes.

There is a very simple criterion to know whether a cubic curve in Weierstrass form is singular and to differentiate nodes from cusps:

Proposition 1.

Let E/K be given by a Weierstrass equation, and let Δ be the discriminantPlanetmathPlanetmathPlanetmathPlanetmath and c4 as in the definition of Δ. Then:

  1. 1.

    E is singular if and only if Δ=0,

  2. 2.

    E has a node if and only if Δ=0 and c40,

  3. 3.

    E has a cusp if and only if Δ=0=c4.


See [2], chapter III, PropositionPlanetmathPlanetmath 1.4, page 50. ∎

2 Reduction of Elliptic Curves

Let E/ be an elliptic curve (we could work over any number fieldMathworldPlanetmath K, but we choose for simplicity in the exposition). Assume that E has a minimal model with Weierstrass equation:


with coefficients in . Let p be a prime in . By reducing each of the coefficients ai modulo p we obtain the equation of a cubic curve E~ over the finite fieldMathworldPlanetmath 𝔽p (the field with p elements).

Definition 2.
  1. 1.

    If E~ is a non-singular curve then E~ is an elliptic curve over 𝔽p and we say that E has good reduction at p. Otherwise, we say that E has bad reduction at p.

  2. 2.

    If E~ has a cusp then we say that E has additive reduction at p.

  3. 3.

    If E~ has a node then we say that E has multiplicative reduction at p. If the slopes of the tangent lines (α and β as above) are in 𝔽p then the reductionPlanetmathPlanetmath is said to be split multiplicative (and non-split otherwise).

From Proposition 1 we deduce the following:

Corollary 1.

Let E/Q be an elliptic curve with coefficients in Z. Let pZ be a prime. If E has bad reduction at p then pΔ.


  1. 1.

    E1:y2=x3+35x+5 has good reduction at p=7.

  2. 2.

    However E1 has bad reduction at p=5, and the reduction is additivePlanetmathPlanetmath (since modulo 5 we can write the equation as [(y-0)-0(x-0)]2-x3 and the slope is 0).

  3. 3.

    The elliptic curve E2:y2=x3-x2+35 has bad multiplicative reduction at 5 and 7. The reduction at 5 is split, while the reduction at 7 is non-split. Indeed, modulo 5 we could write the equation as [(y-0)-2(x-0)][(y-0)+2(x-0)]-x3, being the slopes 2 and -2. However, for p=7 the slopes are not in 𝔽7 (-1 is not in 𝔽7).


  • 1 James Milne, Elliptic Curves, course notes.
  • 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
  • 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
  • 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
Title bad reduction
Canonical name BadReduction
Date of creation 2013-03-22 13:49:21
Last modified on 2013-03-22 13:49:21
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 12
Author alozano (2414)
Entry type Definition
Classification msc 14H52
Related topic EllipticCurve
Related topic JInvariant
Related topic HassesBoundForEllipticCurvesOverFiniteFields
Related topic TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve
Related topic ArithmeticOfEllipticCurves
Related topic SingularPointsOfPlaneCurve
Defines bad reduction
Defines good reduction
Defines cusp
Defines node
Defines multiplicative reduction
Defines additive reduction