Langlands–Tunnell: two-dimensional Artin representations with solvable image

Throughout, $K$ is a number field, $G_K=\mathrm{Gal}(\bar K/K)$ its absolute Galois group with the Krull topology, and $\mathbb A_K$ its adele ring. An Artin representation is a continuous homomorphism $\rho: G_K\to\mathrm{GL}_n(\mathbb C)$. We say $\rho$ is automorphic if there are a nonzero bounded cusp form $\varphi$ on $\mathrm{GL}_n(\mathbb A_K)$ with unitary central character and a finite set $S$ of primes of $K$ such that for every prime $\mathfrak p\notin S$: $\rho$ is unramified at $\mathfrak p$, $\varphi$ is right invariant under $\mathrm{GL}_n(\mathcal O_{\mathfrak p})$, and

where $T_{\mathfrak p,i}$ is the Hecke operator of the double coset of $\mathrm{diag}(\varpi,\dots,\varpi,1,\dots,1)$ ($i$ entries $\varpi$) and $e_i(\rho(\mathrm{Frob}_{\mathfrak p}))$ is the $i$-th elementary symmetric function of the eigenvalues of an arithmetic Frobenius. Equivalently, the cuspidal automorphic representation generated by $\varphi$ has the same local $L$-factors as $\rho$ at all $\mathfrak p\notin S$.

Theorem. Let $\rho:G_K\to\mathrm{GL}_2(\mathbb C)$ be a continuous irreducible representation whose image is a solvable group. Then $\rho$ is automorphic: there is a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb A_K)$ with the same local $L$-factors as $\rho$ at almost all primes.

The projective image of such a $\rho$ is dihedral, tetrahedral ($A_4$) or octahedral ($S_4$). The tetrahedral case is due to Langlands (base change for $\mathrm{GL}(2)$), the octahedral case to Tunnell, and the dihedral case goes back to Hecke and Maass. The theorem is the starting point of Wiles' proof of the modularity of semistable elliptic curves.