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Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\mathbb{Q}_l$).

I know that Serre proved that for $p\geq 11$, $\bar{\rho}_{E,p}(G_p)$ cannot be an exceptional group (i.e. isomorphic to one of $A_4,S_4$ or $A_5$) by showing that inertia is "too big". Is something like this known for $\bar{\rho}_{E,p}(G_l)$ when $l \neq p$? I know that the image of inertia $\bar{\rho}_{E,p}(I_l)$ can only be cyclic of order $2,3,4,6$ for $l>3$, however, this is not very helpful in excluding the exceptional groups. My impression is that this $\bar{\rho}_{E,p}(G_l)$ should not be exceptional as a subgroup of $\text{GL}_2(\mathbb{F}_p)$ in general. Are there any cases known when this statement is false or any reference where it is proven?

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  • $\begingroup$ $A_5$ is never the Galois group of an extension of $\mathbb{Q}_{\ell}$. The others can only occur for $\ell=2$. $\endgroup$
    – Chris Wuthrich
    Commented Feb 1 at 15:41
  • $\begingroup$ @ChrisWuthrich, thanks for your answer! Indeed, $A_5$ is not solvable. For the other two cases do you know when it happens that $\bar{\rho}_{E,p}(G_l)$ is isomorphic to $A_4/S_4$, for example do you know if inertia looks in a certain way or any condition on $p$? Thank you! $\endgroup$
    – did
    Commented Feb 1 at 16:16
  • $\begingroup$ You can read off the ramification from the finitely many extension of $\mathbb{Q}_2$ that are left as option for the group $A_4$ and for the group $S_4$. It shouldn't be hard to check if any of them can occur as torsion extensions for some elliptic curve; I haven't done that. $\endgroup$
    – Chris Wuthrich
    Commented Feb 1 at 20:23

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