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If $E$ is an elliptic curve defined over a field $K$ and $m$ is a positive integer, then the mod-$m$ Galois representation attached to $E$ is the continuous homomorphism \[ \overline\rho_{E,m}: \Gal(\overline{K}/K) \to \Aut(E[m]) \] describing the action of the absolute Galois group of $K$ on the $m$-torsion subgroup $E[m]$.

When the characteristic of $K$ does not divide $m\gt 1$, we may identify the finite abelian group $E[m]$ with $(\Z/m\Z)^2$ and hence view the representation as a map \[ \overline\rho_{E,m}: \Gal(\overline{K}/K) \to \GL(2,\Z/m\Z) \] defined up to conjugation. In particular, when $m=\ell$ is a prime different from the characteristic of $K$, we have the mod-$\ell$ Galois representation \[ \overline\rho_{E,\ell}: \Gal(\overline{K}/K) \to \GL(2,\Z/\ell\Z). \] Taking the inverse limit over prime powers $m=\ell^n$ yields the $\ell$-adic Galois representation attached to $E$, \[ \rho_{E,\ell}: \Gal(\overline{K}/K) \to \Aut(T_\ell(E)) \cong \GL(2,\Z_\ell), \] which describes the action of the absolute Galois group of $K$ on $T_\ell(E)$, the $\ell$-adic Tate module of $E$.

When $K$ has characteristic zero one can take the inverse limit over all positive integers $m$ (ordered by divisibility) to obtain the adelic Galois representation \[ \rho_{E}: \Gal(\overline{K}/K) \to \GL(2,\hat \Z). \]

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  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2022-02-09 19:48:52
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