Khare–Wintenberger: odd two-dimensional Artin representations of $\mathbb Q$

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 $\rho:\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)\to\mathrm{GL}_2(\mathbb C)$ be a continuous irreducible representation which is odd, i.e. $\det\rho(c)=-1$ for every complex conjugation $c$. Then $\rho$ is automorphic; classically, $\rho$ is attached to a weight-one cuspidal newform.

This is a consequence of Serre's modularity conjecture, proved by Khare and Wintenberger, combined with the Deligne–Serre theorem. It settles the reciprocity conjecture for odd two-dimensional representations of $\mathbb Q$.

Formalization Note By the Artin–Schreier theorem, the elements of order $2$ in $\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)$ are exactly the complex conjugations, i.e. the restrictions of complex conjugation along embeddings $\bar{\mathbb Q}\hookrightarrow\mathbb C$. Oddness is therefore stated as: $\det\rho(c)=-1$ for every $c\neq 1$ with $c^2=1$.