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 map \[ \overline\rho_{E,m}: \Gal(\overline{K}/K) \to \Aut(E[m]) \] 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 $m=p$ is prime different form the characteristic of $K$, we have \[ \overline\rho_{E,p}: \Gal(\overline{K}/K) \to \GL(2,\Z/p\Z). \] Taking the inverse limit over prime powers $m=p^n$ yields the $p$-adic Galois representation attached to $E$, \[ \rho_{E,p}: \Gal(\overline{K}/K) \to \Aut(T_p(E)) \cong \GL(2,\Z_p), \] which describes the action of the absolute Galois group of $K$ on $T_p(E)$, the $p$-adic Tate module of $E$.

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

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- Last edited by John Jones on 2018-06-19 22:22:36

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- 2018-06-19 22:22:36 by John Jones (Reviewed)