Two-dimensional complex tori with multiplication by √d

الأصل · EN

We give an elementary argument for the well known fact that the endomorphism algebra EndQ(A) of a simple complex abelian surface A can neither be an imaginary quadratic field nor a definite quaternion algebra. Another consequence of our argument is that a two-dimensional complex torus T with Q(√d) EndQ(A) where Q(√d) is real quadratic, is algebraic.