TITLE: Motivic generalisation of Neron-Ogg-Shaferevich criterion
QUESTION [15 upvotes]: Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory tells us its p-adic cohomology is crystalline).
My question is to what extent it's possible to find a converse to this statement. More precisely, I have yet to see a counterexample to the following "conjecture" (though I still suspect it's wrong).
"Conjecture": Let $K$ be a number field, $p$ and $l$ primes, and $V$ a geometric (say, coming from the variety $Y$) $l$-adic representation of $G_K$ that is unramified/crystalline at $\mathfrak{p}|p$. Then there exists a smooth proper variety $X$ such that $X$ has good reduction at $\mathfrak{p}$ and $V$ can be cut out of the cohomology of $X$.
From googling around, the things I know so far are (at least for $l \not= p$):

If $Y$ is an abelian variety, the classical Neron-Ogg-Shafarevich condition means that $Y$ itself is a witness to the conjecture.
We can take torsors for abelian varieties with no $K$-rational points, and these can have the same representations, but fail to have good reduction (in this paper http://arxiv.org/abs/math/0605326 of Dalawat).
There exist curves which have bad reduction, but whose Jacobians have good reduction.

If anyone knows any more about this story I'd be interested to hear. Ultimately I guess it would be nice to have a definition for when a motive is unramified/has good reduction, and cohomologically this surely has to mean unramified/crystalline, but it would be nice if this could always be realised "geometrically".
Thanks,
Tom.

REPLY [4 votes]: Unfortunately I don't know much about motives in genereal, but this might be relevant to your question. One result of my thesis, that I am currently writing, is to prove Neron-Ogg-Shafarevich for 1-motives. The proof is not particularly difficult and it ultimately reduces to the corresponding results for the components of the 1-motive. I will describe below what good reduction means in this particular case.
A 1-motive $M = [u\colon Y\to G]$ over a scheme $S$ consists of a group scheme $Y$, which is locally etale isomorphic to $\mathbb{Z}^r$, a group scheme $G$ which is an extension of an abelian scheme $A$ by a torus $T$ and a homomorphism $u\colon Y\to G$. If $S$ is the spectrum of a field $K$, this means that $Y$ is a free finitely-generated $\mathbb{Z}$-module with a continuous action of the absolute Galois group $\Gamma_K$ and that $u$ is a $\Gamma_K$-equivariant homomorphism $u\colon Y\to G(\bar K)$.
If $R$ is a complete discrete valuation ring with a fraction field $K$ we say that a 1-motive $M$ over $K$ has good reduction if there exists a 1-motive $\widetilde{M}$ over $R$ whose generic fiber is isomorphic to $M$. This is equivalent to the following:

$G$ has good reduction $\widetilde{G}$ over $R$, which is equivalent to saying that both $A$ and $T$ have good reduction;
The action of $\Gamma_K$ on $Y$ is unramified;
The image of $u(Y)$ is contained in the set of those points in $G(K')$ which can be reduced, where $K'/K$ is some finite field extension. Equivalently, $u(Y)$ is contained in the maximal compact subgroup of $G(K')$;

With this definition, the criterion of Neron-Ogg-Shafarevich is as follows: Let $l,p$ be primes, with $l\neq p$. A 1-motive $M/\mathbb{Q}$ has good reduction mod p if and only if the Tate module $T_l(M)$ is unramified at $p$. For general number fields replace $p$ by a prime ideal.
If you want to learn more about reduction of 1-motives you can look at M. Raynaud's paper 1-Motifs et Monodromie Géométrique.