# Modularity switching for primes $p>7$

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 $$p$$.

• @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 $p=5$.) – Avi Oct 6 at 6:51