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  "body": "\n## Summary\n\nFix a prime $\\ell$.\nThe $\\ell$-adic Tate module of an abelian group collects its elements of $\\ell$-power order into one module over the [[$\\ell$-adic integers|V6TJAFZ4MD]].\nLet $A$ be an [[abelian variety|R7W03TXRS9]] of dimension $g$ over a field $k$ whose characteristic is not $\\ell$.\nThen the Tate module of $A$ is free of rank $2g$.\nThe [[absolute Galois group|PGKWBJDTKX]] of $k$ acts on it, which gives an $\\ell$-adic [[Galois representation|JQEYEQJY2G]].\nThese representations carry much of the arithmetic of $A$.\n\n## Explanations\n\n### L1 human\n\nOn an elliptic curve over $\\C$ there are $\\ell^{2n}$ points that give zero when added to themselves $\\ell^n$ times.\nThese torsion points form a tower as $n$ grows, each level mapping onto the one below by multiplication by $\\ell$.\nThe Tate module is the limit of this tower.\nIt behaves like a lattice $\\Z_\\ell^2$, an $\\ell$-adic version of the lattice that defines the curve as a quotient of $\\C$.\nOver a number field, the Galois group permutes the torsion points, and so it acts on the Tate module.\n",
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