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.

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2021-07-17 12:55:12

**Referred to by:**

**History:**(expand/hide all)

- 2021-07-17 12:55:12 by Andrew Sutherland (Reviewed)
- 2020-10-19 09:59:07 by John Voight (Reviewed)
- 2020-10-19 09:58:30 by John Voight
- 2020-10-19 09:56:07 by John Voight
- 2020-10-19 09:55:11 by John Voight
- 2020-10-19 09:53:13 by John Voight
- 2020-10-19 09:52:15 by John Voight
- 2020-10-19 07:44:37 by Andrew Sutherland
- 2020-10-18 22:26:21 by Andrew Sutherland
- 2020-10-18 21:12:11 by Andrew Sutherland
- 2020-10-17 14:09:51 by John Voight (Reviewed)
- 2020-10-17 14:09:25 by John Voight
- 2020-10-14 22:24:15 by Andrew Sutherland
- 2020-10-14 12:08:38 by John Voight
- 2020-10-14 11:31:50 by Andrew Sutherland
- 2020-10-14 11:24:47 by Andrew Sutherland
- 2020-10-14 10:44:58 by John Voight
- 2020-09-26 16:49:31 by John Voight
- 2020-09-26 16:48:32 by John Voight
- 2020-09-26 16:47:50 by John Voight
- 2020-09-26 16:46:03 by John Voight
- 2020-09-26 16:45:22 by John Voight
- 2019-10-29 16:44:41 by John Cremona
- 2019-04-05 14:36:52 by Andrew Sutherland
- 2018-06-19 20:11:23 by John Jones (Reviewed)

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