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? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:20:50Z http://mathoverflow.net/feeds/question/24984 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24984/what-is-the-first-galois-cohomology-group-of-the-galois-module-endt-la-for-so 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? Heer 2010-05-17T09:57:02Z 2010-05-17T13:47:09Z <p>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?</p> <p>waht are the Galois cohomology groups of the Tate module of some abelian variety over a finite field or a number field? </p> http://mathoverflow.net/questions/24984/what-is-the-first-galois-cohomology-group-of-the-galois-module-endt-la-for-so/24993#24993 Answer by Xandi Tuni for 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? Xandi Tuni 2010-05-17T12:03:45Z 2010-05-17T13:47:09Z <p>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.</p> <p>Tate modules are <em>not</em> discrete Galois modules, and their cohomology will usually not be torsion. The same goes for $\mathrm{End}(T_\ell A)$.</p> <p>Over finite or local fields the cohomology of $T_\ell A$ is more or less well understood. Not so over global fields.</p>