Tate's theorem about abelian variteies in case of abelian scheme

2012-05-09T13:14:53Z

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 !

Tate's theorem about abelian variteies in case of abelian scheme - MathOverflow

Tate's theorem about abelian variteies in case of abelian scheme