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 ?

  • 1
    $\begingroup$ Pedantic point: You should require that X is not the generic fiber of a smooth proper Z_p-scheme, or make some similar change, because X is always the generic fiber of itself, viewed as a Z_p-scheme rather than a Q_p-scheme. $\endgroup$ – JBorger Jan 2 '10 at 18:02
  • $\begingroup$ When X is viewed as a Z_p-scheme, is it smooth ? But you are right, I want X to have "good reduction" in the sense of being the generic fibre of a smooth proper Z_p-scheme. Notice that every abelian Q_p-variety is the generic fibre of a smooth Z_p-scheme (for example its néronian model) but it is the generic fibre of a smooth proper Z_p-scheme only when the néronian model is an abelian scheme. $\endgroup$ – Chandan Singh Dalawat Jan 3 '10 at 5:14
  • $\begingroup$ Hi Chandan! I'm a bit too lazy for a proper discussion, but were you somehow excluding a curve with tree-like reduction from your considerations? I'm sure you know that example better than I do. $\endgroup$ – Minhyong Kim Jan 3 '10 at 10:55
  • $\begingroup$ @Minhyong : Ah! You mean a curve of genus >1 which has bad reduction but whose jacobian has good reduction. Thanks for reminding me. Auld lang syne! $\endgroup$ – Chandan Singh Dalawat Jan 4 '10 at 8:20

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.


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

  • $\begingroup$ I don't have a copy of the book, but Liu was referring to Raynaud's results relating the Néron model of $J$ to the Picard scheme of a regular proper model of $C$. $\endgroup$ – Chandan Singh Dalawat Jan 10 '10 at 9:19
  • $\begingroup$ How does one calculate defining equations for tha Picard scheme or the Néron model? $\endgroup$ – TKe Jan 10 '10 at 11:44
  • 1
    $\begingroup$ The redution of $J$: Chapter 9, Example 9.2/8, page 246. $\endgroup$ – Qing Liu Jan 25 '10 at 23:18

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