Let be an elliptic curve. The endomorphism ring of , denoted , is the set of all regular maps such that , where is the identity element for the group structure of . Note that this is indeed a ring under addition () and composition of maps.
Let be elliptic curves, and let be a regular map such that . Then is also a group homomorphism, i.e.
[Proof: See , Theorem 4.8, page 75]
If is isomorphic (as a ring) to an order (http://planetmath.org/OrderInAnAlgebra) in a quadratic imaginary field then we say that the elliptic curve E has complex multiplication by (or complex multiplication by ).
and, in general, these are all the maps in the endomorphism ring of .
Example: Fix . Let be the elliptic curve defined by
then this curve has complex multiplication by (more concretely by ). Besides the multiplication by maps, contains a genuine new element:
(the name complex multiplication comes from the fact that we are “multiplying” the points in the curve by a complex number, in this case).
- 1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
- 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.
|Date of creation||2013-03-22 13:41:35|
|Last modified on||2013-03-22 13:41:35|
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