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? 



what is the first Galois cohomology group of the Galois module End(T_l(A)) for some abelian variety A over a finite field k and l some prime number different from the characteristic of the base field?
In general, if $G$ is a profinite group and $M$ a continuous discrete $G$--module, then $H^i(G,M)$ is torsion for $i>0$. This applies in particular to Galois cohomology, i.e. when $G$ is a Galois group.


Tate modules are *not* discrete Galois modules, and their cohomology will usually not be torsion. The same goes for $\mathrm{End}(T\_\ell A)$.


Over finite or local fields the cohomology of $T\_\ell A$ is more or less well understood. Not so over global fields.


