This question is related to [my previous question](https://mathoverflow.net/q/247301/4149), to which I got a partial answer.


Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta\_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta\_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta\_8]$ denote the ring of integers of $L$.



> 
> **Question 1.** Does there exist an abelian variety $A$ over $\overline{\mathbb{Q}}$ of dimension 4 (or of any other dimension) with
>  ${\mathrm{End\,}} A\simeq \Lambda$ ?
> 
> 
> 


I expect the answer "yes", but have no idea how to prove this.


More generally, let $(G,X)$ be a Shimura datum.
For any $x\in X$, let $h\_x\colon \mathbb{S}\to G\_\mathbb{R}$ be the corresponding homomorphism.
We define the *Mumford-Tate group* $\mathrm{MT}(x)$ to be the smallest $\mathbb{Q}$-subgroup of $G$ containing $\mathrm{im\,} h\_x$ over $\mathbb{R}$,
and we define $\mathrm{MT}(X)$ to be the smallest $\mathbb{Q}$-subgroup of $G$ containing $\mathrm{im\,} h\_x$ over $\mathbb{R}$ *for all* $x\in X$.
We say that $x\in X$ is *generic* if $\mathrm{MT}(x)=\mathrm{MT}(X)$.


Let $K\subset G(\mathbb{A}^f)$ be an open compact subgroup. Set $\Gamma=G(\mathbb{Q})\cap K$ (the intersection is taken in $G(\mathbb{A}\_f)\,$). 
We say that $\Gamma$ is a congruence subgroup of $G(\mathbb{Q})$.
Set $\mathrm{Sh}\_\Gamma=X/\Gamma$, it is a connected Shimura variety.
We say that $x\Gamma\in X/\Gamma$ is generic if $x\in X$ is generic.


On the other hand, $\mathrm{Sh}\_\Gamma$ has a (mysterious) canonical model over $\overline{\mathbb{Q}}$, so the notion of a $\overline{\mathbb{Q}}$-point of $\mathrm{Sh}\_\Gamma$ is defined.



> 
> **Question 2.** Does $\mathrm{Sh}\_\Gamma$ always have a *generic* $\overline{\mathbb{Q}}$-point?
> 
> 
> 


The positive answer to Question 2, even for Shimura varieties of abelian type, would imply the positive answer to Question 1.
Indeed, Shimura constructed an abelian scheme over a Shimura variety with prescribed automorphism ring of a *generic* fiber. 
If we find a generic $\overline{\mathbb{Q}}$-point of the base, then the fiber over this point will be a desired abelian variety over $\overline{\mathbb{Q}}$ with prescribed automorphism ring.



Does every Shimura variety contain a generic point defined over a number field?
The Answer to Question 1 is YES. It follows from Serre's variant of Hilbert's irreducibility theorem (for infinite Galois extensions) combined with the Tate conjecture on homomorphisms of abelian varieties in char 0 (proven by Faltings); see Prop. 1.3 and Cor. 1.5 of [1995 Compositio paper by Rutger Noot (vol 97, Oort Festschrift)](http://archive.numdam.org/ARCHIVE/CM/CM_1995__97_1-2/CM_1995__97_1-2_161_0/CM_1995__97_1-2_161_0.pdf) for details. 


The main idea is as follows. If $A$ is an abelian variety over a finitely generated field $K$ of char 0 and $\ell$ is a prime then there is an abelian variety $A\_0$ over a number field $K\_0$ that is a ``specialization/reduction" of $A$ and such that a canonical (up to a "conjugation") isomorphism of Tate modules $T\_{\ell}(A)$ and $T\_{\ell}(A\_0)$ induces an isomorphism between the corresponding $\ell$-adic images of the absolute Galois groups of $K$ and $K\_0$
in the automorphism groups of $T\_{\ell}(A)$ and $T\_{\ell}(A\_0)$ respectively. Since $End(A)$ embeds into $End(A\_0)$, one obtains
 that the endomorphism rings of $A$ and $A\_0$ are isomorphic.


