This question is related to my previous question, 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 ageneric$\overline{\mathbb{Q}}$-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.