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complex multiplication
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(Definition)
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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.
The following theorem implies that every endomorphism is also a group endomorphism:
Theorem 1 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  in a quadratic imaginary field  then we say that the elliptic curve E has complex multiplication by  (or complex multiplication by  ).
Note:
 always contains a subring isomorphic to
 , formed by the multiplication by n maps:
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.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.
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"complex multiplication" is owned by alozano.
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(view preamble)
Cross-references: complex number, points, curve, fix, multiplication, subring, contains, quadratic imaginary field, isomorphic, group homomorphism, group endomorphism, endomorphism, implies, maps, composition, addition, ring, structure, group, identity element, regular maps, elliptic curve
There are 12 references to this entry.
This is version 12 of complex multiplication, born on 2003-06-16, modified 2006-03-10.
Object id is 4367, canonical name is ComplexMultiplication.
Accessed 6339 times total.
Classification:
| AMS MSC: | 11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields) |
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Pending Errata and Addenda
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![$\displaystyle [n]\colon E \to E,\quad [n]P=n\cdot P$](http://images.planetmath.org:8080/cache/objects/4367/l2h/img22.png "$\displaystyle [n]\colon E \to E,\quad [n]P=n\cdot P$"){kind=small}
![$\displaystyle [i]\colon E \to E,\quad [i](x,y)=(-x,iy)$](http://images.planetmath.org:8080/cache/objects/4367/l2h/img31.png "$\displaystyle [i]\colon E \to E,\quad [i](x,y)=(-x,iy)$"){kind=small}