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Two-dimensional complex tori with multiplication by $\sqrt{d}$

Wolfgang M. Ruppert
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Abstract

We give an elementary argument for the well known fact that the endomorphism algebra $End_Q(A)$ 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 $Q(\sqrt{d})\subseteq End_Q(A)$ where $Q(\sqrt{d})$ is real quadratic, is algebraic.

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Wolfgang M. Ruppert. 1998-08-05. Two-dimensional complex tori with multiplication by $\sqrt{d}$. https://arxiv.org/abs/math/9808020

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