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  "body": "\n## Definition\n\nLet $G$ be an [[abelian group|082GHNPJ53]], written additively, and let $\\ell$ be a prime number.\nFor $n \\ge 1$, let $G[\\ell^n] = \\set{x \\in G : \\ell^n x = 0}$, a subgroup of the [[torsion subgroup|0DBWXQRXQJ]].\nMultiplication by $\\ell$ maps $G[\\ell^{n+1}]$ to $G[\\ell^n]$.\nThe *$\\ell$-adic Tate module* of $G$ is the inverse limit\n$$T_\\ell(G) = \\varprojlim_n G[\\ell^n]$$\nalong these maps.\nIt is a module over the [[$\\ell$-adic integers|V6TJAFZ4MD]] $\\Z_\\ell$.\n\nLet $A$ be an [[abelian variety|R7W03TXRS9]] over a field $k$, and let $k^{s}$ be a separable closure of $k$.\nThe Tate module of $A$ is $T_\\ell(A) = T_\\ell(A(k^{s}))$.\n\n## Remarks\n\nIf $A$ has dimension $g$ and $\\ell$ is not the characteristic of $k$, then $T_\\ell(A)$ is a free $\\Z_\\ell$-module of rank $2g$.\nThe [[absolute Galois group|PGKWBJDTKX]] $\\Gal(k^{s}/k)$ acts on each $A[\\ell^n]$, and so continuously on $T_\\ell(A)$.\n\nExample: let $G = \\mu_{\\ell^\\infty}$ be the group of roots of unity of $\\ell$-power order in $\\C^{*}$.\nThen $G[\\ell^n]$ is cyclic of order $\\ell^n$, and the maps $\\zeta \\mapsto \\zeta^\\ell$ are onto.\nSo $T_\\ell(G) \\cong \\Z_\\ell$, and this module is written $\\Z_\\ell(1)$.\n\nDegenerate example: $G = \\Z$ has no nonzero torsion, so every $G[\\ell^n]$ is $0$ and $T_\\ell(\\Z) = 0$.\nThe Tate module only sees the $\\ell$-power torsion of $G$.\n",
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