Reciprocity for $\mathrm{GL}(1)$: Artin reciprocity (class field theory)

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 $K$ be a number field. Every continuous character $\rho:G_K\to\mathrm{GL}_1(\mathbb C)$ is automorphic. That is, there is a unitary idele class character $\chi:K^\times\backslash\mathbb A_K^\times\to\mathbb C^\times$ with $\chi(\varpi_{\mathfrak p})=\rho(\mathrm{Frob}_{\mathfrak p})$ for almost all $\mathfrak p$.

This is the case $n=1$ of the reciprocity conjecture; the article states that it follows from, and is essentially equivalent to, class field theory.

Formalization Note For $n=1$ there are no proper parabolic subgroups, so the cuspidality condition is empty. The only Hecke operator is translation by a uniformizer at $\mathfrak p$, with normalising factor $N\mathfrak p^{0}=1$.