deRham cohomoloy of CM liftings of Jacobians - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:57:35Z http://mathoverflow.net/feeds/question/95755 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95755/derham-cohomoloy-of-cm-liftings-of-jacobians deRham cohomoloy of CM liftings of Jacobians Jack 2012-05-02T11:57:45Z 2012-05-03T10:27:04Z <p>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 $k$, <em>ordinary</em>, 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 <code>$H^1_{cris}$</code> i.e. it happens when <code>$Fil_{\mathcal{X}}=Fil_{can}$</code>. Where $Fil_{\mathcal{X}}=\sigma(F^{1}_{Hodge})$ where $\sigma$: <code>$H^1_dR(\mathcal{X}) \rightarrow H^{1}_{cris}(X/k)$</code> is the isomorphism between the deRham and crystalline cohomology.</p> <p>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 $\mathcal{X}$? </p>