A nice variety without a smooth model

Question

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 ?

Answer 1

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.

Answer 2

Chandan, can you tell me the numbers of the theorems in Bosch-Lütkebohmert-Raynaud?