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I have basic question: Suppose I have two abelian varieties $A,B$ over $\overline{\mathbb{F}_p}$. Let $M(A), M(B)$ denote the (covariant) Dieudonne modules (over $W$ the Witt vectors of $\overline{\mathbb{F}_p}$ ) of their respective p-divisible groups. Is it true that the natural map is an injection $ Hom(A,B) \otimes \mathbb{Z}_p \hookrightarrow Hom(M(A), M(B)) ,$ where the RHS is in the category of Dieudonne modules?

I also believe we have an injection $Hom(A, B) \otimes \widehat{\mathbb{Z}}^p \hookrightarrow Hom(T^p(A), T^p(B)),$ where $\widehat{\mathbb{Z}}^p = \prod_{l \neq p} \mathbb{Z}_l$, and $T^p(A) = \prod_{l \neq p} T_l(A)$, the product of all $l$-adic Tate modules away from $p$.

If these are both true, is there a good reference for these statemenst?

Thanks!

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    I'm just starting to get my feet wet with this stuff, so I could be wrong, but I think the problem is that in characteristic p, the Tate module T_p(A) is too small to capture all the endomorphisms.2011-12-21

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