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$\DeclareMathOperator\Br{Br}$Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\Br(X)/{\Br(k)}$ being an infinite group? It is known that $\Br_1(X)/{\Br(k)}$ is finite under these conditions, but what about $\Br(X)/{\Br_1(X)}$ (or equivalently $\Br(\overline{X})^\Gamma$ for $\Gamma=\operatorname{Gal}(\overline{k}/k)$)?

Would be amazing if anyone knows a concrete example of this or if someone knows how to prove that no such variety exists!

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This is an open problem; I personally suspect it cannot happen.

More generally let $X$ be a smooth projective variety over a field $k$ which is finitely generated over $\mathbb{Q}$. Then Skorobogatov and Zarhin ask in Question 2 of [1] whether $\mathrm{Br}(\bar{X})^\Gamma$ is in fact always finite.

It is known that the Tate conjecture implies finiteness of the $\ell$-primary part of this group for all primes $\ell$. You can read more about this in Section 16.1 of [2]. The real challenge is to therefore show that the $\ell$-primary part is actually trivial for all but finitely many primes $\ell$; this is closely related to the Tate conjecture but does formally follow from it.

[1] Skorobogatov, Alexei N.; Zarhin, Yuri G. A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces. J. Algebraic Geom. 17 (2008), no. 3, 481--502.

[2] Colliot-Thélène, Jean-Louis; Skorobogatov, Alexei N. The Brauer-Grothendieck group. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 71. Springer, Cham, [2021].

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    $\begingroup$ Note that it is a theorem of Cadoret-Hui-Tamagawa that for $k$ finitely generated over its prime subfield and $\mathrm{char}(k)=p>0$, the Tate conjecture for divisors for all $\ell\neq p$ is equivalent to the finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{k^{\mathrm{sep}}})^{\mathrm{Gal}(k^{\mathrm{sep}}/k)}$. The independence of $\ell$ in this generality is due to Pal, I believe. The $p$-primary part can be infinite though; see Proposition 5.4 in D'Addezio's "Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety". $\endgroup$
    – Oli Gregory
    Commented Nov 22, 2023 at 13:16
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    $\begingroup$ Thanks Oli for pointing this out. Nice to see these recent developments in positive characteristic! $\endgroup$
    – Daniel Loughran
    Commented Nov 22, 2023 at 13:54

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