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

If E does not obtain complex multiplication (CM) over an extension, then $\bar\rho_{E,p}$ is maximal if its image contains $\SL_2(\F_p)$.

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

For $K=\Q$, the image of a maximal $\bar\rho_{E,p}$ is $\GL_2(\F_p)$, 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 $p$ is ramified, split, or inert in $\mathcal{O}$, respectively.

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  • Last edited by John Voight on 2020-10-19 09:59:07
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