An elliptic curve over a field is a projective nonsingular algebraic curve over of genus 1 together with a point of defined over . The word “genus” is taken here in the algebraic geometry sense, and has no relation with the topological notion of genus (defined as , where is the Euler characteristic) except when the field of definition is the complex numbers .
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 with , and with ).
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 ) are very difficult to graph in three dimensions, let alone two.
Figure 1 is a graph of the elliptic curve .
Figure 2 shows the graph of :
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:
Let denote the base point. Then one can show that every line joining two points on intersects a unique third point of (after properly accounting for tangent lines as a multiple intersection). For any two points , define their sum as:
Form the line between and ; let be the third point on that intersects this line;
Form the line between and ; define to be the third point on that intersects this line.
4 Elliptic Curves over
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 . The starting point for this investigation is the Weierstrass –function, which we define here.
For any lattice in , the Weierstrass –function of is the function given by
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:
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:
For every elliptic curve over , there is a unique lattice whose constants and satisfy and .
Two elliptic curves and over are isomorphic if and only if their corresponding lattices and satisfy the equation for some scalar .
- 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.
|Date of creation||2013-03-22 12:03:02|
|Last modified on||2013-03-22 12:03:02|
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