Abstract Theory of Endomorphisms
The ring of endomorphisms of an elliptic curves can be of one of three forms:the integers Z; an order in an imaginary quadratic number field; or an order in a definite quaternion algebra over Q.
When the field of definition is a finite field, there are always non-trivial endomorphisms of an elliptic curve, coming from the Frobenius map, so the complex multiplication case is in a sense typical (and the terminology isn't often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the Hodge conjecture.
Read more about this topic: Complex Multiplication
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