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 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.


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) 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.


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