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For $k$ a finite field , $A,A'$ an abelian varieties over $k$, $G$ the Galois group of $k$, $l$ a prime number different from the characteristic of $k$ . Tate has proved that:

$Q_l\otimes Hom_k(A,A')\rightarrow Hom_G(V_l(A),V_l(A'))$
is bijective , where $V_l(A)=Q_l\otimes_{Z_l}T_l(A)$ , $T_l(A)$ is the Tate module of $A$.

Now consider a Scheme $S$ over $F_p$, and Abelian schemes $A,A'$ over $S$ , is there any known result similar to Tate's theorem for this situation?

Thank you !

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Zarhin proved this when $S$ is a curve. – Felipe Voloch May 9 '12 at 13:23
Can you give me the reference ?Thank you! – TOM May 9 '12 at 13:27
Zarhin, Ju. G. Isogenies of abelian varieties over fields of finite characteristic. (Russian) Mat. Sb. (N.S.) 95(137) (1974), 461–470, 472. – Felipe Voloch May 9 '12 at 15:50

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