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 Jacobian of 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 $\mathcal{X}$?
