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## 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?

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?

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 Tate modules (with their natural topology) aren't discrete except in trivial cases. – David Loeffler May 17 2010 at 11:58 Thank you very much, David – Heer May 17 2010 at 13:02

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.

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thank you very much. – Heer May 17 2010 at 13:02
williamstein.org/edu/2010/582e/lectures/all/… In Stein's lecture note section 17, it seems that he formulates galois cohomology without mensioning discrete galois module, and says that Tate modules are Galois modules. This makes me very confusing – Heer May 17 2010 at 13:05
@Heer: look more closely at the statement of 17.2 and especially the sentence preceding it; his example of Tate module certainly violates his conventions there. You may also wish to look at the appendix of Rubin's book on Euler systems (where he discusses the general formalism of Galois cohomology with $\ell$-adic coefficients" for arithmetically interesting fields in a concrete way). – BCnrd May 17 2010 at 13:36
@BCnrd: Thank you very much, Prof. Conrad – Heer May 17 2010 at 14:55