elliptic curve (original) (raw)

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 ℂ.

2 Examples

We present here some pictures of elliptic curves over the field ℝ 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

4 Elliptic Curves over ℂ

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 ℂ. The starting point for this investigation is the Weierstrass 𝔭–function, which we define here.

Definition 1.
Definition 2.

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

𝔭L⁢(z):=1z2+∑ω∈L∖{0}(1(z-ω)2-1ω2).

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:

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

Theorem 3.
    1. For every elliptic curve E:y2=4⁢x3-b⁢x-c over ℂ, there is a unique lattice L⊂ℂ whose constants g2 and g3 satisfy b=g2 and c=g3.
    1. Two elliptic curves E and E′ over ℂ are isomorphic if and only if their corresponding lattices L and L′ satisfy the equation L′=α⁢L for some scalar α∈ℂ.

References