Let $l/k$ be a finite galois extension let $A$ be an abelian variety over $k$. Then $A(l)$ is a $Gal(l/k)$-module. Hence it makes sense to study $H^1(Gal(l/k),A(l))$. I know that for $A=\mathbb{G}_m$, we have $H^1(Gal(l/k),\mathbb{G}_m)=0$ (okey, this is not an abelian variety). I'm wondering if there is a good set of conditions on $A$ such that this vanishes?



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