# Crystalline cohomology of abelian varieties

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