# complex multiplication

Let $E$ be an elliptic curve^{}. The endomorphism ring^{} of $E$,
denoted $\mathrm{End}(E)$, is the set of all regular maps $\varphi :E\to E$ such that $\varphi (O)=O$, where $O\in E$ is the
identity element^{} for the group structure of $E$. Note that this is
indeed a ring under addition ($(\varphi +\psi )(P)=\varphi (P)+\psi (P)$) and composition of maps.

The following theorem implies that every endomorphism^{} is also a
group endomorphism^{}:

###### Theorem 1

Let ${E}_{\mathrm{1}}\mathrm{,}{E}_{\mathrm{2}}$ be elliptic curves, and let $\varphi \mathrm{:}{E}_{\mathrm{1}}\mathrm{\to}{E}_{\mathrm{2}}$ be a regular map such that $\varphi \mathit{}\mathrm{(}{O}_{{E}_{\mathrm{1}}}\mathrm{)}\mathrm{=}{O}_{{E}_{\mathrm{2}}}$. Then $\varphi $ is also a group homomorphism, i.e.

$$\forall P,Q\in {E}_{1},\varphi (P{+}_{{E}_{1}}Q)=\varphi (P){+}_{{E}_{2}}\varphi (Q).$$ |

[Proof: See [2], Theorem 4.8, page 75]

If $\mathrm{End}(E)$ is isomorphic (as a ring) to an order (http://planetmath.org/OrderInAnAlgebra) $R$ in a quadratic imaginary field $K$ then we say that the elliptic curve E has complex multiplication by $K$ (or complex multiplication by $R$).

Note: $\mathrm{End}(E)$ always contains a subring isomorphic to $\mathbb{Z}$, formed by the multiplication by n maps:

$$[n]:E\to E,[n]P=n\cdot P$$ |

and, in general, these are all the maps in the endomorphism ring of $E$.

Example: Fix $d\in \mathbb{Z}$. Let $E$ be the elliptic curve defined by

$${y}^{2}={x}^{3}-dx$$ |

then this curve has complex multiplication by $\mathbb{Q}(i)$ (more concretely by $\mathbb{Z}(i)$). Besides the multiplication by $n$ maps, $\mathrm{End}(E)$ contains a genuine new element:

$$[i]:E\to E,[i](x,y)=(-x,iy)$$ |

(the name complex multiplication comes from the fact that we
are “multiplying” the points in the curve by a complex number^{}, $i$
in this case).

## References

- 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.

Title | complex multiplication |

Canonical name | ComplexMultiplication |

Date of creation | 2013-03-22 13:41:35 |

Last modified on | 2013-03-22 13:41:35 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 15 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 11G05 |

Related topic | EllipticCurve |

Related topic | KroneckerWeberTheorem |

Related topic | OrderInAnAlgebra |

Related topic | ArithmeticOfEllipticCurves |

Defines | complex multiplication |

Defines | endomorphism ring |