Tate module

Remarks

If $A$ has dimension $g$ and $\ell$ is not the characteristic of $k$, then $T_\ell(A)$ is a free $\Z_\ell$-module of rank $2g$. The absolute Galois group $\Gal(k^{s}/k)$ acts on each $A[\ell^n]$, and so continuously on $T_\ell(A)$.

Example: let $G = \mu_{\ell^\infty}$ be the group of roots of unity of $\ell$-power order in $\C^{*}$. Then $G[\ell^n]$ is cyclic of order $\ell^n$, and the maps $\zeta \mapsto \zeta^\ell$ are onto. So $T_\ell(G) \cong \Z_\ell$, and this module is written $\Z_\ell(1)$.

Degenerate example: $G = \Z$ has no nonzero torsion, so every $G[\ell^n]$ is $0$ and $T_\ell(\Z) = 0$. The Tate module only sees the $\ell$-power torsion of $G$.