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LSpice
LSpice
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Let$\DeclareMathOperator\Br{Br}$Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\text{Br}(X)/\text{Br}(k)$$\Br(X)/{\Br(k)}$ being an infinite group? It is known that $\text{Br}_1(X)/\text{Br}(k)$$\Br_1(X)/{\Br(k)}$ is finite under these conditions, but what about $\text{Br}(X)/\text{Br}_1(X)$$\Br(X)/{\Br_1(X)}$ (or equivalently $\text{Br}(\overline{X})^\Gamma$$\Br(\overline{X})^\Gamma$ for $\Gamma=\text{Gal}(\overline{k}/k)$$\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!

Thank in advance, Victor

Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\text{Br}(X)/\text{Br}(k)$ being an infinite group? It is known that $\text{Br}_1(X)/\text{Br}(k)$ is finite under these conditions, but what about $\text{Br}(X)/\text{Br}_1(X)$ (or equivalently $\text{Br}(\overline{X})^\Gamma$ for $\Gamma=\text{Gal}(\overline{k}/k)$)?

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

Thank in advance, Victor

$\DeclareMathOperator\Br{Br}$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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Victor de Vries
Victor de Vries
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Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\text{Br}(X)/\text{Br}(k)$ being an infinite group? It is known that $\text{Br}_1(X)/\text{Br}(k)$ is finite under these conditions, but what about $\text{Br}(X)/\text{Br}_1(X)$ (or equivalently $\text{Br}(\overline{X})^\Gamma$ for $\Gamma=\text{Gal}(\overline{k}/k)$)?

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

Thank in advance, Victor

Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\text{Br}(X)/\text{Br}(k)$ being infinite group? It is known that $\text{Br}_1(X)/\text{Br}(k)$ is finite under these conditions, but what about $\text{Br}(X)/\text{Br}_1(X)$ (or equivalently $\text{Br}(\overline{X})^\Gamma$ for $\Gamma=\text{Gal}(\overline{k}/k)$?

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

Thank in advance, Victor

Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\text{Br}(X)/\text{Br}(k)$ being an infinite group? It is known that $\text{Br}_1(X)/\text{Br}(k)$ is finite under these conditions, but what about $\text{Br}(X)/\text{Br}_1(X)$ (or equivalently $\text{Br}(\overline{X})^\Gamma$ for $\Gamma=\text{Gal}(\overline{k}/k)$)?

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

Thank in advance, Victor

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Victor de Vries
Victor de Vries
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An example of a geometrically simply connected variety with infinite Brauer group (modulo constants)

Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\text{Br}(X)/\text{Br}(k)$ being infinite group? It is known that $\text{Br}_1(X)/\text{Br}(k)$ is finite under these conditions, but what about $\text{Br}(X)/\text{Br}_1(X)$ (or equivalently $\text{Br}(\overline{X})^\Gamma$ for $\Gamma=\text{Gal}(\overline{k}/k)$?

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

Thank in advance, Victor