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# complex multiplication

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

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

###### Theorem 1.

Let $E_{1},E_{2}$ be elliptic curves, and let $\phi\colon E_{1}\to E_{2}$ be a regular map such that $\phi(O_{{E_{1}}})=O_{{E_{2}}}$. Then $\phi$ is also a group homomorphism, i.e.

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

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

If $\operatorname{End}(E)$ is isomorphic (as a ring) to an order $R$ 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: $\operatorname{End}(E)$ always contains a subring isomorphic to $\mathbb{Z}$, formed by the multiplication by n maps:

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

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, $\operatorname{End}(E)$ contains a genuine new element:

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

(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.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.

## Mathematics Subject Classification

11G05*no label found*

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