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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 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.

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(could be a comment but too long...)

That's quite a natural question. I am not sure it is possible to prove that the "conjecture" you state is true with the current technology (and to be sure I have no idea how to prove it), but my intuition would differ from yours in that I believe the conjecture to be true. Let me give a very rough "argument" why. Let us assume for the sake of the argument that your Galois representation satisfies a self-duality condition so that it is expected to be attached to an automorphic representation for an unitary group. Then because of your hypothesis that your Galois representation is unramified, the automorphic representation should be unramified at places $\mathfrak{p}$ above $p$, that is have invariant by maximal compact hyperspecial subgroups $K_\mathfrak p$. The Shimura variety for the unitary group with hyper special maximal level structure is conjectured (and actually, known) to have good reduction (Milne conjecture). Assume also that the representation $\pi_\infty$ is such that the Galois representation attached to $\pi$ appears in the étale cohomology of that Shimura variety (like for a modular eigenform of weight 2, whose representation appears in the cohomology of the modular curve and not of a Kuga-Sato variety over it). Then your Galois rep. appears in the cohomology of a variety with good reduction.

Admittedly this argument is not very convincing, because I have made very strong assumption on the Galois representation. However, as those assumptions seem (to me) quite orthogonal to the problem discussed, I believe this is a decent evidence in favor of the conjecture.

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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.

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