Maximal Galois representation (reviewed)

Let $E$ be an elliptic curve over a number field $K$, let $\ell$ be prime, and let \[ \bar\rho_{E,\ell}\colon \Gal(\overline{K}/K)\to \Aut(E[\ell]) \simeq \GL_2(\F_\ell) \] be the mod-$\ell$ Galois representation associated to $E$.

If E does not have potential complex multiplication, then $\bar\rho_{E,\ell}$ is maximal if its image contains $\SL_2(\F_\ell)$.

In general, let $\mathcal{O}$ be the geometric endomorphism ring. Then $E[\ell]$ is an $\mathcal{O}$-module and we view $\Aut_\mathcal{O}(E[\ell]) \leq \Aut(E[\ell]) \simeq \GL_2(\F_\ell)$. We say that $\bar\rho_{E,\ell}$ is maximal if its image contains $\SL_2(\F_\ell) \cap \Aut_{\mathcal{O}}(E[\ell])$.

For $K=\Q$, the image of a maximal $\bar\rho_{E,\ell}$ is $\GL_2(\F_\ell)$, a Borel subgroup, the normalizer of a split Cartan subgroup, or the normalizer of a non-split Cartan subgroup, depending on whether $\mathcal{O}=\Z$ or $\mathcal{O}\ne \Z$ and $\ell$ is ramified, split, or inert in $\mathcal{O}$, respectively.