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In Freitas, Le Hung, and Siksek's 2014 paper proving that elliptic curves over totally real quadratic fields are modular they prove (and use) the following result (Theorems 3 and 4): let $\bar \rho_{E,p}: G_K \to \text{GL}_2(\mathbb F_p)$ be the Galois representation given by the action of $G_K$ on the $p$-torsion $E[p]$, where $G_K$ is the absolute Galois group of $K$. Then if $\bar \rho_{E,p}(G_{K(\zeta_p)}$ is absolutely irreducible then $E$ is modular for $p \in \{3,5,7\}$. This is proved by modularity switching a la Wiles (and many others) together with Langlands-Tunnell and a modularity criterion due to Breuil and Diamond (and many others).

Is this, or a similar result, known for any other values of $p$? Is it expected to hold for all or most $p>2$?

Thanks!

Edit: by Theorem 2 in Freitas, Le Hung, and Siksek it suffices to show that $\bar \rho_{E,p}$ is modular. Serre's modularity conjecture (for totally real fields) implies this whenever $\bar \rho_{E,p}$ is totally odd, and as Wojowu pointed out to me this is always the case for these representations by the Galois-equivariance of the Weil pairing so we should expect the result to hold for all odd primes $p$.

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  • $\begingroup$ @JoeT I'm far from an expert, but I think this is only for the $p=3$ case--in that case it's important that $\text{PGL}_2(\mathbb F_3)$ is solvable, but the $p=5,7$ cases are deduced by modularity switching from the $3$ case, not directly. (The analogous statement already fails for $p=5$.) $\endgroup$ – Avi Oct 6 at 6:51

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