Reciprocity for Artin representations with nilpotent image (Arthur–Clozel)

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 $\rho:G_K\to\mathrm{GL}_n(\mathbb C)$ be a continuous irreducible representation whose image $\rho(G_K)$ is a nilpotent group. Then $\rho$ is automorphic.

This is the strong Artin conjecture for nilpotent Galois extensions, a consequence of the theory of automorphic induction for cyclic extensions of prime degree. It is the main known case in arbitrary dimension $n$.