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Mikhail Borovoi
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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)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.

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.

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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Yuri Zarhin
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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.