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An elliptic curve whose endomorphism ring is larger than $$\Z$$ is said to have complex multiplication (often abbreviated to CM). In this case, for curves defined over fields of characteristic zero, the endomorphism ring is isomorphic to an order in an imaginary quadratic field. The discriminant of this order is the CM discriminant.

An elliptic curve whose geometric endomorphism ring is larger than $$\Z$$ is said to have potential complex multiplication (potential CM).

The property of having CM over an extension depends only on the $j$-invariant of the curve. For an elliptic curve $E$ defined over a number field $K$ a necessary condition is that $j(E)$ is an algebraic integer, and there are only finitely many CM $j$-invariants in any specific number field. In particular, for an elliptic curve defined over the field of rational numbers, the $j$-invariant must be an integer, and it is known that there are precisely 13 CM $j$-invariants, associated to the 13 imaginary quadratic orders of class number $1$:

$$\begin{array}{c|ccccccccccccc} j & -12288000 & 54000 & 0 & 287496 & 1728 & 16581375 & -3375 & 8000 & -32768 & -884736 & -884736000 & -147197952000 & -262537412640768000\\ \text{CM discriminant} &-27 & -12 & -3 & -16 & -4 & -28 & -7 & -8 & -11 & -19 & -43 & -67 & -163 \end{array}$$

This was proved in the 1970s and is closely related to the problem of finding all imaginary quadratic fields of class number one, since these are precisely the values $j(\tau)$ where $j$ is the classical modular $j$-function and $\tau$ is an imaginary quadratic algebraic integer such that the ring $\Z[\tau]$ has unique factorization.

This is a special case of an abelian variety with complex multiplication.

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• Last edited by John Cremona on 2020-12-01 07:07:50
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