The algebra $\mathrm{End}\_{\mathbb{Q}}(A)=\mathrm{End}(A)\otimes\mathbb{Q}$ of endomorphisms of an abelian variety (defined over $\mathbb{C}$) is well understood and in particular the following is true : if $A$ is simple, the latter is a skew field and its center is (possibly a quadratic extension of) a totally real number field.


I was wondering if the same would hold if $A$ was replaced with $T$ a simple complex torus (is the center of the algebra of the $\mathbb{Q}$-endomorphisms a quadratic extension of a totally real number field?). The proof of the above assertion (which can be found for instance in the book by Birkenhake-Lange) uses the fact that a rational polarization on the abelian variety induces first a Rosati involution on the algebra $\mathrm{End}\_{\mathbb{Q}}(A)$ and then a rational polarization on the latter. If $T$ is merely a complex torus, we could try to use real polarization and real Rosati involution but I'm not sure it's enough to get the same conclusion.


Does someone have an idea on this question? or a counterexample?



Endomorphism of complex tori
The answer to your question is "no". Oort and Zarhin (*Endomorphism algebras of complex tori*, Math. Ann. **303** (1995), 11–29) showed that any finite-dimensional division algebra over $\mathbb Q$ arises as $\mathrm{End}\_{\mathbb{Q}}(T)$ for a simple complex torus $T$.


