Galois representation attached an elliptic curve over $\mathbb Q$ (reviewed)

If $E$ is an elliptic curve defined over $\Q$ and $m\ge2$ then the mod-$m$ Galois representation attached to $E$ is the map \[ \overline\rho_{E,m}: \Gal(\overline{\Q}/\Q) \to \Aut(E[m]), \] giving the action of the absolute Galois group of $\mathbb{Q}$ on the $m$-torsion subgroup $E[m]$.

By identifying the finite abelian group $E[m]$ with $(\Z/m\Z)^2$ we may identify $\Aut(E[m])$ with $\GL(2,\Z/m\Z)$ and hence view the representation as a map \[ \overline\rho_{E,m}: \Gal(\overline{\Q}/\Q) \to \GL(2,\Z/m\Z), \] defined up to conjugation. In the case where $m=p$ is prime we have \[ \overline\rho_{E,p}: \Gal(\overline{\Q}/\Q) \to \GL(2,\F_p). \]

Furthermore, one can combine the mod-$p^n$ representations attached to $E$ for $n\in \mathbb N$ to define the $p$-adic Galois representation attached to $E$, \[ \rho_{E,p}: \Gal(\overline{\Q}/\Q) \to \Aut(T_p(E)) \cong \GL(2,\Z_p), \] which describes the action of the absolute Galois group of $\mathbb{Q}$ on $T_p(E)$, the $p$-adic Tate module of $E$.