# Homs of abelian varieties and Tate/Dieudonne modules

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 can't remember how all these (anti)equivalences of categories go off the top of my head, but shouldn't you be able to rephrase that first injection as an injection into the $Hom(T_p(A), T_p(B))$, which seems more natural to me, or do you only get equivalences when taking contravariant Dieudonne modules? –  Matt Dec 21 '11 at 19:39
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. –  qewrrewer Dec 21 '11 at 22:22