Galois representations attached to an elliptic curve (reviewed)

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 \[ \rho_{E}: \Gal(\overline{K}/K) \to \GL(2,\hat \Z). \] If $E$ is an elliptic curve without complex multiplication that is defined over a number field, then the image of $\rho_E$ is an open subgroup of $\GL(2,\hat\Z)$ that has an associated level, index, and genus.