Hilbert's 9th problem

Let $K$ be an algebraic number field with adele ring $\mathbf{A}_K$, and let $L/K$ be a finite Galois extension with Galois group $G=\mathrm{Gal}(L/K)$. Let

$$\rho : G \longrightarrow GL(V), \qquad V \cong \mathbb{C}^n,$$

be an irreducible complex representation of $G$ of degree $n$.

For a prime $\mathfrak{p}$ of $K$ choose a prime $\mathfrak{P}$ of $L$ above $\mathfrak{p}$, let $I_{\mathfrak{P}}\subseteq G$ be its inertia group, let $\sigma_{\mathfrak{P}}$ be a Frobenius element of the decomposition group at $\mathfrak{P}$ (so that $\sigma_{\mathfrak{P}}$ is determined modulo $I_{\mathfrak{P}}$), let $V^{I_{\mathfrak{P}}}$ be the subspace of $I_{\mathfrak{P}}$-invariant vectors, and let $N\mathfrak{p}$ denote the absolute norm of $\mathfrak{p}$. The Artin $L$-function of $\rho$ is

$$L(s,\rho,L/K)=\prod_{\mathfrak{p}}\det\Bigl(1-\rho(\sigma_{\mathfrak{P}})\big|_{V^{I_{\mathfrak{P}}}}\,N\mathfrak{p}^{-s}\Bigr)^{-1},$$

the product being over all finite primes $\mathfrak{p}$ of $K$ and independent of the choices of $\mathfrak{P}$; it converges for $\operatorname{Re}(s)>1$.

For a cuspidal automorphic representation $\pi \cong \bigotimes_v' \pi_v$ of $GL(n,\mathbf{A}_K)$, unitarily normalized, let

$$L(s,\pi)=\prod_{v \nmid \infty} L(s,\pi_v)$$

be its automorphic $L$-function, the Euler product over the finite places $v$ of $K$ of the standard local $L$-factors attached to the local components $\pi_v$; it converges in a right half-plane.

Then: for every such $K$, $L$, and every irreducible representation $\rho$ of $\mathrm{Gal}(L/K)$ of degree $n$, there exists a cuspidal automorphic representation $\pi$ of $GL(n,\mathbf{A}_K)$ such that

$$L(s,\pi)=L(s,\rho,L/K)$$

as Euler products, i.e. the local factors agree at every finite place of $K$.