Abstract: Wiles' work on modularity of elliptic curves over the rationals, used as a starting point that odd, irreducible represenations $G_Q \rightarrow GL_2 (F_3)$ arise from cohomological cusp forms (i.e. new forms of weight $K \geq 2$).
In ongoing work with Patrick Allen and Jack Thorne we address the question of showing that representations $G_K \rightarrow GL_2 (F_3)$ arise from cohomological cusp forms on $GL_2(A_K)$ for $K$ a CM field like $Q(i)$. I will describe some of the main ideas of this work, in which instead of invoking the Langlands-Tunnell theorem like Wiles (which does not seem directly useful in this setting), we isntead rely on (restricted) 2-adic automorphy lifting theorems (extending results in the 10-author paper Patrick Allen will talk about at the conference) in the residually dihedral case.
A starting point is a Diophantine argument ("2-3" switch) that gives a criterion for mod 6 represenations to arise from elliptic curves over $K$.