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I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power structures, period rings and the de Rham-Witt complex. Before looking into these things, it would be nice to have an idea of what the cohomology that you construct at the end looks like.

The l-adic cohomology of abelian varieties has a simple description in terms of the Tate module. My question is: is there something similar for crystalline cohomology of abelian varieties?

More precisely, let $X$ be an abelian scheme over $\mathbb{Z}_p$. Is there a concrete description of $H^1(X_0/\mathbb{Z}_p)$? (or just $H^1(X_0/\mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$?) I think that this should consist of three things: a $\mathbb{Z}_p$-module $M$, a filtration on $M \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$ (which in the case of an abelian variety has only one term which is neither 0 nor everything) and a Frobenius-linear morphism $M \to M$.

I believe that the answer has something to do with Dieudonné modules, but I don't know what they are either.

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The $\ell$-adic Tate mod. of an ab. var. $A$ over field $k$ of char. distinct from is not the etale cohom. (with $\mathbf{Z}_ {\ell}$ coefficients) of the $A$ in general (unless $k = k_s$), but rather of $A_{k_s}$ (with such coefficients). The etale coh. of $A$ itself is rather more complicated! Just as you probably learned about Tate modules and $\ell$-adic Galois representations before etale cohomology, you should learn about Dieudonne modules and their relation with $p$-divisible groups before learning crystalline cohomology. –  BCnrd Apr 5 '10 at 12:25
Indeed, I meant the l-adic cohomology of the abelian variety over $k_s$. –  Martin Orr Apr 6 '10 at 21:40
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up vote 6 down vote accepted

To add a bit more to Brian's comment: the crystalline cohomology of an abelian variety (over a finite field of characteristic p, say) is canonically isomorphic to the Dieudonné module of the p-divisible group of the abelian variety (which is a finite free module over the Witt vectors of the field with a semi-linear Frobenius). If you start with an abelian scheme over the Witt vectors of this field then the crystalline cohomology of the special fibre is canonically isomorphic to the algebraic de Rham cohomology of the thing upstairs, hence receives a Hodge filtration also.

A good starting place to understand Dieudonné modules is Demazure's 'Lectures on p-divisible groups', which appears in Springer LNM. In particular, he gives a nice description of the analogy with Tate modules (and the relation between the various Frobenii that appear). For a general picture of crystalline cohomology, and the various structures that can be placed on it, I would look at Illusie's survey in the Motives volumes (this is a little out of date now, but gives a good description of the basic theory).

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