elliptic curve
1 Basics
An elliptic curve over a field K is a projective nonsingular
algebraic curve
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 geometry
sense, and has no relation
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 numbers
ℂ.
Using the Riemann-Roch theorem for curves, one can show that every elliptic curve E is the zero set of a Weierstrass equation of the form
E:y2+a1xy+a3y=x3+a2x2+a4x+a6, |
for some ai∈K, where the polynomial on the right hand side has no double roots. When K has characteristic
other than 2 or 3, one can further simpify this Weierstrass equation into the form
E:y2=x3-27c4x-54c6. |
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 translate 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 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 dimensions, let alone two.
Figure 1 is a graph of the elliptic curve y2=x3-x.
Figure 2 shows the 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.
3 The Group Law
The points on an elliptic curve have a natural group structure, which makes the elliptic curve into an abelian variety
. There are many equivalent
ways to define this group structure; two of the most common are:
-
•
Every Weyl divisor
on E is linearly equivalent to a unique divisor of the form [P]-[O] for some P∈E, where O∈E 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 O∈E 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 multiple
intersection
). For any two points P,Q∈E, define their sum as:
-
(a)
Form the line between P and Q; let R be the third point on E that intersects this line;
-
(b)
Form the line between O and R; define P+Q to be the third point on E that intersects this line.
This addition
operation
yields a group operation
on the points of E having the base point O for identity
.
-
(a)
4 Elliptic Curves over ℂ
Over the complex numbers, the general correspondence between algebraic 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.
A lattice in ℂ is a subgroup
L of the additive group
ℂ which is generated by two elements ω1,ω2∈ℂ that are linearly independent
over ℝ.
Definition 2.
For any lattice L in ℂ, the Weierstrass pL–function of L is the function 𝔭L:ℂ⟶ℂ given by
𝔭L(z):= |
When the lattice is clear from context, it is customary to suppress it from the notation and simply write for the Weierstrass –function.
Properties of the Weierstrass –function:
-
•
is a meromorphic function with double poles at points in .
-
•
is constant on each coset of .
- •
The last property above implies that, for any , the point lies on the elliptic curve . Let be the map given by
(where denotes the point at infinity on ). Then is actually a bijection (!), and moreover the map is an isomorphism of Riemann surfaces as well as a group isomorphism (with the addition operation on inherited from , and the elliptic curve group operation on ).
We can go even further: it turns out that every elliptic curve over can be obtained in this way from some lattice . More precisely, the following is true:
Theorem 3.
-
1.
For every elliptic curve over , there is a unique lattice whose constants and satisfy and .
-
2.
Two elliptic curves and over are isomorphic if and only if their corresponding lattices and satisfy the equation for some scalar .
References
- 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 | Isogeny![]() |
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 | -function |