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](https://mathoverflow.net/questions/416/existence-of-smooth-models/10264#10264) of a Châtelet surface with these properties is simple enough, but can one do better ?



A nice variety without a smooth model
As Minhyong suggests, a curve $C$ (with $C(\mathbb{Q}\_p)\neq0$) which has bad reduction but whose jacobian $J$ has good reduction would do the affair. This works because the cohomology of $C$ is essentially the same as that of $J$, and because an abelian $\mathbb{Q}\_p$-variety has good reduction if and only if its $l$-adic étale cohomology is unramified for some (and hence for every) prime $l\neq p$ (Néron-Ogg-Shafarevich) or its $p$-adic étale cohomology is crystalline (Fontaine-Coleman-Iovita).


I asked Qing Liu for explicit examples. He suggested the curve
$$
y^2=(x^3+1)(x^3+ap^6)\qquad (a\in\mathbb{Z}\_p^\times)
$$
when $p\neq2,3$, and $y^2=(x^3+x+1)(x^3+a3^4x+b3^6)$, with $a,b\in\mathbb{Z}\_3^\times$, for $p=3$.


He refers to Proposition 10.3.44 in his book for computing the stable reduction of these $C$, and to Bosch-Lütkebohmert-Raynaud,
*Néron models,* Chapter 9, for showing that $J$ has good reduction.


I "accept" this answer as coming from Minhyong Kim and Qing Liu.


