My question is surely a classical one in the algebraic number theory, but I'm not working in it and I ignore the results and references. Please excuse me.

Let $k = \mathbf{F}_q$ be a finite field and let $\ell$ be a prime number invertible in $k$. Let $\mathbf Z_\ell(1) = \varprojlim_{r\to\infty} \mu_{\ell^r}$ be the Tate module, which is a Galois module of $k$, and let $\mathbf{Z}_\ell(m) = \mathbf{Z}_\ell(1)^{\otimes m}$ be the $m$-th tensor power over $\mathbf{Z}_\ell$ for $m\in \mathbf{Z}$.

Question: What is $H^n(k, \mathbf{Z}_\ell(m))$? When is it an $\ell$-torsion abelian group?

My question is surely a classical one in the algebraic number theory, but I'm not working in it and I do not know the results and references. Please excuse me.

Let $k = \mathbf{F}_q$ be a finite field and let $\ell$ be a prime number invertible in $k$. Let $\mathbf Z_\ell(1) = \varprojlim_{r\to\infty} \mu_{\ell^r}$ be the Tate module, which is a Galois module of $k$, and let $\mathbf{Z}_\ell(m) = \mathbf{Z}_\ell(1)^{\otimes m}$ be the $m$-th tensor power over $\mathbf{Z}_\ell$ for $m\in \mathbf{Z}$.

Question: What is $H^n(k, \mathbf{Z}_\ell(m))$? When is it an $\ell$-torsion abelian group?