elliptic curve

1 Basics

An elliptic curveMathworldPlanetmath over a field K is a projective nonsingularPlanetmathPlanetmath algebraic curveMathworldPlanetmath E over K of genus 1 together with a point O of E defined over K. The word “genus” is taken here in the algebraic geometryMathworldPlanetmathPlanetmath sense, and has no relationMathworldPlanetmathPlanetmathPlanetmath with the topological notion of genus (defined as 1-χ/2, where χ is the Euler characteristic) except when the field of definition K is the complex numbersMathworldPlanetmathPlanetmath .

Using the Riemann-Roch theorem for curves, one can show that every elliptic curve E is the zero setMathworldPlanetmath of a Weierstrass equation of the form


for some aiK, where the polynomialMathworldPlanetmathPlanetmathPlanetmath on the right hand side has no double roots. When K has characteristicPlanetmathPlanetmath other than 2 or 3, one can further simpify this Weierstrass equation into the form


The extremely strange numbering of the coefficients is an artifact of the process by which the above equations are derived. Also, note that these equation are for affine curves; to translateMathworldPlanetmath them to projective curves, one has to homogenize the equations (replace x with X/Z, and y with Y/Z).

2 Examples

We present here some pictures of elliptic curves over the field of real numbers. These pictures are in some sense not representative of most of the elliptic curves that people work with, since many of the interesting cases tend to be of elliptic curves over algebraically closed fields. However, curves over the complex numbers (or, even worse, over algebraically closed fields in characteristic p) are very difficult to graph in three dimensionsPlanetmathPlanetmathPlanetmath, let alone two.

Figure 1 is a graph of the elliptic curve y2=x3-x.

Figure 1: Graph of y2=x(x-1)(x+1)

Figure 2 shows the graph of y2=x3-x+1:

Figure 2: Graph of y2=x3-x+1

Finally, Figures 3 and 4 are examples of algebraic curves that are not elliptic curves. Both of these curves have singularities at the origin.

Figure 3: Graph of y2=x2(x+1). Has two tangents at the origin.
Figure 4: Graph of y2=x3. Has a cusp at the origin.

3 The Group Law

The points on an elliptic curve have a natural group structureMathworldPlanetmath, which makes the elliptic curve into an abelian varietyMathworldPlanetmath. There are many equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ways to define this group structure; two of the most common are:

  • Every Weyl divisorMathworldPlanetmathPlanetmathPlanetmath on E is linearly equivalent to a unique divisor of the form [P]-[O] for some PE, where OE is the base point. The divisor class group of E then yields a group structure on the points of E, by way of this correspondence.

  • Let OE denote the base point. Then one can show that every line joining two points on E intersects a unique third point of E (after properly accounting for tangent lines as a multipleMathworldPlanetmath intersectionMathworldPlanetmath). For any two points P,QE, define their sum as:

    1. (a)

      Form the line between P and Q; let R be the third point on E that intersects this line;

    2. (b)

      Form the line between O and R; define P+Q to be the third point on E that intersects this line.

    This additionPlanetmathPlanetmath operationMathworldPlanetmath yields a group operationMathworldPlanetmath on the points of E having the base point O for identityPlanetmathPlanetmathPlanetmathPlanetmath.

4 Elliptic Curves over

Over the complex numbers, the general correspondence between algebraicMathworldPlanetmath and analytic theory specializes in the elliptic curves case to yield some very useful insights into the structure of elliptic curves over . The starting point for this investigation is the Weierstrass 𝔭function, which we define here.

Definition 1.

A latticeMathworldPlanetmathPlanetmath in is a subgroupMathworldPlanetmathPlanetmath L of the additive groupMathworldPlanetmath which is generated by two elements ω1,ω2 that are linearly independentMathworldPlanetmath over .

Definition 2.

For any lattice L in , the Weierstrass pL–function of L is the function 𝔭L: given by


When the lattice L is clear from context, it is customary to suppress it from the notation and simply write 𝔭 for the Weierstrass 𝔭–function.

Properties of the Weierstrass p–function:

  • 𝔭(z) is a meromorphic function with double poles at points in L.

  • 𝔭(z) is constant on each coset of /L.

  • 𝔭(z) satisfies the differential equation


    where the constants g2 and g3 are given by

    g2 := 60ωL{0}1ω4
    g3 := 140ωL{0}1ω6

The last property above implies that, for any z/L, the point (𝔭(z),𝔭(z)) lies on the elliptic curve E:y2=4x3-g2x-g3. Let ϕ:/LE be the map given by


(where denotes the point at infinity on E). Then ϕ is actually a bijection (!), and moreover the map ϕ:/LE is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of Riemann surfaces as well as a group isomorphism (with the addition operation on /L inherited from , and the elliptic curve group operation on E).

We can go even further: it turns out that every elliptic curve E over can be obtained in this way from some lattice L. More precisely, the following is true:

Theorem 3.
  1. 1.

    For every elliptic curve E:y2=4x3-bx-c over , there is a unique lattice L whose constants g2 and g3 satisfy b=g2 and c=g3.

  2. 2.

    Two elliptic curves E and E over are isomorphic if and only if their corresponding lattices L and L satisfy the equation L=αL for some scalar α.


  • 1 Dale Husemoller, Elliptic Curves. Springer–Verlag, New York, 1997.
  • 2 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
  • 3 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer–Verlag, New York, 1986.
Title elliptic curve
Canonical name EllipticCurve
Date of creation 2013-03-22 12:03:02
Last modified on 2013-03-22 12:03:02
Owner djao (24)
Last modified by djao (24)
Numerical id 32
Author djao (24)
Entry type Definition
Classification msc 11G07
Classification msc 11G05
Classification msc 14H52
Related topic IsogenyMathworldPlanetmath
Related topic ComplexMultiplication
Related topic RankOfAnEllipticCurve
Related topic HeightFunction
Related topic LSeriesOfAnEllipticCurve
Related topic BirchAndSwinnertonDyerConjecture
Related topic JInvariant
Related topic MordellWeilTheorem
Related topic ConductorOfAnEllipticCurve
Defines 𝔭-function