Editing Complex multiplication (section)

==Abstract theory of endomorphisms==
The ring of endomorphisms of an elliptic curve can be of one of three forms: the integers '''Z'''; an [[Order (ring theory)|order]] in an [[imaginary quadratic number field]]; or an order in a definite [[quaternion algebra]] over '''Q'''.{{sfn|Silverman|1986|p=102}}

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 every such curve has ''complex multiplication'' (and the terminology is not 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]].