According to Serre's book 'Galois cohomology', Galois chomology group are always torsion, but it seems to me that H^1(k, End_{Z_l}(T_l(A)))=coker(Frob-1) on End_{Z_l}(T_l(A)), which has the same Z_l rank as End_{k}(T_l(A)) So maybe End_{Z_l}(T_l(A)) is not a discrete galois module. And why is the Tate module a discrete galois module?

waht are the Galois cohomology groups of the Tate module of some abelian variety over a finite field or a number field?