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Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that

--- $X(K)\neq\emptyset$,

--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,

--- the $p$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_p)$ is crystalline for every $i\in\mathbb{N}$,

and yet,

--- $X$ is not the generic fibre of any smooth proper $\mathbb{Z}_p$-scheme ?

My example of a Châtelet surface with these properties is simple enough, but can one do better ?

Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that

--- $X(K)\neq\emptyset$,

--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,

--- the $p$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_p)$ is crystalline for every $i\in\mathbb{N}$,

and yet,

--- $X$ is not the generic fibre of any smooth proper $\mathbb{Z}_p$-scheme ?

My example of a Châtelet surface with these properties is simple enough, but can one do better ?

Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that

--- $X(K)\neq\emptyset$,

--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,

--- the $p$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_p)$ is crystalline for every $i\in\mathbb{N}$,

and yet,

--- $X$ is not the generic fibre of any smooth $\mathbb{Z}_p$-scheme ?

My example of a Châtelet surface with these properties is simple enough, but can one do better ?

Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that

--- $X(K)\neq\emptyset$,

--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,

--- the $p$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_p)$ is crystalline for every $i\in\mathbb{N}$,

and yet,

--- $X$ is not the generic fibre of any smooth proper $\mathbb{Z}_p$-scheme ?

My example of a Châtelet surface with these properties is simple enough, but can one do better ?

Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that

--- $X(K)\neq\emptyset$,

--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,

--- the $p$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_p)$ is crystalline for every $i\in\mathbb{N}$,

and yet,

--- $X$ is not the generic fibre of any smooth proper flat $\mathbb{Z}_p$-scheme ?

My example of a Châtelet surface with these properties is simple enough, but can one do better ?

Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that

--- $X(K)\neq\emptyset$,

--- the $l$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,

--- the $p$-adic étale cohomology $H^i(X\times_K\bar K,\mathbb{Q}_p)$ is crystalline for every $i\in\mathbb{N}$,

and yet,

--- $X$ is not the generic fibre of any smooth $\mathbb{Z}_p$-scheme ?

My example of a Châtelet surface with these properties is simple enough, but can one do better ?