# lifting abelian varieties

Hello,

Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$. This abelian variety is endowed with a principal polarization and an action by $\mathcal{O}_E$, the ring of integers of an unramified extension $E/\mathbb{Q}_p$ of degree $2$. This means there is a morphism $\iota : \mathcal{O}_E \longrightarrow End(A)\otimes \mathbb{Z}_p$

Let $M$ be its Dieudonné module. It comes with an $\mathcal{O}_E$-action and a bilinear antisymmetric form $<,>:M\times M \longrightarrow W$ which are compatible.

Here is my question : I consider the problem of lifting $A$ with its structure to an abelian scheme over $W$ (the ring of Witt vectors of $k$). Is this problem equivalent to finding a sub-module $M'\subset M$ which satisfies the following conditions :

(1) $M'$ is a direct factor of $M$

(2) $M'$ is stablized by the $\mathcal{O}_E$-action

(3) $< M',M'> = 0$

(4) $M'+pM=VM$

Second question :

Let $A_0$, $A_1$ two such abelian varieties over $k$ with additional structure, and $f:A_1 \longrightarrow A_0$ an isogeny, compatible with the $\mathcal{O}_E$-action. This gives an inclusion $M_1 \subset M_0$. I wish to lift $A_0$, $A_1$ and $f$ to the ring $W$

Is the lifting problem of $f$ equivalent to finding $M'_0$ and $M'_1$ as above, such that: $$M'_1 \subset M'_0$$

Thank you very much !

JS

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For any abelian variety $A$, the ring $End(A)$ is of finite type as a $\mathbb Z$-module. So a local ring such as ${\mathcal O}_E$ cannot act on $A$. Clearly you mean something else, but what? –  inkspot Mar 30 '12 at 21:39
Of course. What I meant was a morphism $\iota : \mathcal{O}_E \longrightarrow End(A) \otimes \mathbb{Z}_p$. I've corrected the question. –  Koskivirta Apr 4 '12 at 14:40