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.