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
Famous quotes containing the words abstract and/or theory:
“What a cheerful rhyme! Clean not mean!
Been not seen! Not tired—expired!
We must now decide about place.
We decide that place is the big weeping face
And the other abstract lace of the race.”
—Allen Tate (1899–1979)
“The weakness of the man who, when his theory works out into a flagrant contradiction of the facts, concludes “So much the worse for the facts: let them be altered,” instead of “So much the worse for my theory.””
—George Bernard Shaw (1856–1950)