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Question 0. Can a field of definitions (without automorphisms) of an (almost arbitrary) abelian variety of CM-type, originally defined over ${\mathbb{C}}$, be chosen to be a totally real number field or at least a real number field?

I specify. Let $L$ be a CM-field of degree $[L:{\mathbb{Q}}]=2n$, i.e., a totally imaginary quadratic extension of a totally real number field. Let $M$ be an order in $L$, for example, $M=O_L$, the ring of integers of $L$. Let $\Phi$ denote the set of embeddings of $L$ into ${\mathbb{C}}$. Let $\Psi$ be a CM-type for $L$, i.e., a subset $\Psi$ of $\Phi$ such that $\Phi$ is a disjoint union of $\Psi$ and $\overline{\Psi}$. Here $\overline{\Psi}=\{\bar\varphi\ |\ \varphi\in\Psi\}$. Then $|\Psi|=n$. We embed the order $M$ into ${\mathbb{C}}^n$ using $\Psi$, then $M$ is a lattice in ${\mathbb{C}}^n$. We set $A={\mathbb{C}}^n/M$. Then the complex torus $A$ is an abelian variety over ${\mathbb{C}}$, and we have an embedding $M\hookrightarrow{\mathrm{End}}\,A$.

Question 1. Is the field of moduli of $A$ with endomorphisms from $M$ a CM-field?

Question 2. Is the field of moduli of $A$ without endomorphisms totally real? If not, is it real, i.e., does it admit an embedding into ${\mathbb{R}}$?

Question 3. Can one choose a field of definition of $A$, with endomorphisms from $M$, to be a CM-field?

Question 4. Can one choose a field of definition of $A$ without endomorphisms to be totally real? If not, can it be chosen real, i.e., admitting an embedding into $\mathbb{R}$?

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