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Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. It is well-known that whether the jacobian $J(\mathcal{X})$ of a lifting $\mathcal{X}$ of $X$ to $W$ is isomorphic to the canonical lifting of the Jacobian of $X$, can be checked by looking at the de Rham cohomology $H^{1}(\mathcal{X})$ or $H^1_{cris}$ i.e. it happens when $Fil_{\mathcal{X}}=Fil_{can}$. Where $Fil_{\mathcal{X}}=\sigma(F^{1}_{Hodge})$ where $\sigma$: $H^1_dR(\mathcal{X}) \rightarrow H^{1}_{cris}(X/k)$ is the isomorphism between the deRham and crystalline cohomology.

Because it is known that the canonical lifting of an abelian variety is an abelian vareity of CM type, this observation prompts the following generalization in case where $X$ is not necessarily an ordinary curve: Is there a characterization of CM liftings of Jacobians which can be read from the deRham (or crystalline) cohomology of $\mathcal{X}$?

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``... the Jacobian of the canonical lifting...'' What is the canonical lifting to whose Jacobian you refer here? A curve, even if ordinary, does not have a canonical lift. And the canonical lift of a Jacobian is not necessarily a Jacobian. – inkspot May 2 '12 at 16:27
Thank you for your comment. You are absolutely right. I edited my question and now it is in it's correct form. I meant when the Jacobian of a lifting is isomorphic to the canonical lifting of the jacobian of our $X$. – Jack May 3 '12 at 10:28

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