# Artin conjecture for degree 1 representations

The Artin $$L$$-series for Abelian extensions are known to coincide with Hecke $$L$$-series, which in particular implies that if $$E/K$$ is a Abelian extensions, and $$\chi$$ is a non-trivial simple character of $$\textrm{Gal}(E/K)$$, then $$L(E/K,\chi,s)$$ admits an analytic continuation holomorphic on $$\mathbb{C}$$. This proves the Artin conjecture for Abelian extensions.

It is said that this settles the Artin conjecture for all degree 1 representations. But how?

Say we have $$\textrm{Gal}(E/K) \cong S_3$$. How does the above fact imply that $$L(E/K,\chi,s)$$ is entire for any non-trivial simple degree 1 character $$\chi$$?

All help or input would be highly appreciated.

• If $\rho$ is a representation of $Gal(E/K)$ let $H=\ker(\rho)=\{ g,\rho(g)=I\}$, it is normal in $Gal(E/K)$ and $\tilde{\rho}(gH) = \rho(g)$ is a representation of $Gal(E/K)/H=Gal(E^H/K)$ and $L(E/K,\rho,s)= L(E^H/K,\tilde{\rho},s)$. If $\rho(x)\rho(y)=\rho(y)\rho(x)$ then $Gal(E/K)/H$ is abelian and $L(E/K,\rho,s)$ is a product of Hecke L-functions of $K$. Commented Jan 3, 2020 at 18:47

Let $$G=\textrm{Gal}(E/K)$$
By the universal property of the abelianization, $$\chi:G \to \Bbb C^\times$$ factors over $$G^{ab}$$, so we obtain $$\overline{\chi}:\textrm{Gal}(L/K) \to \Bbb C^\times$$ where $$L=E^{[G,G]}$$. Now $$L/K$$ is abelian and $$\chi$$ is the inflation of $$\overline{\chi}$$, so by functoriality of Artin L-functions with respect to inflations we have $$L(E/K,\chi,s)=L(L/K,\overline{\chi},s)$$
• By "$\chi: G \to \mathbb{C}^{\times}$ factors over $G^{ab}$", do you mean that every simple degree 1 character of $G$ can be expressed as $\chi \circ \pi$, where $\chi$ is a simple character of $G_{\textrm{Ab}}$ and $\pi: G \to G_{\textrm{Ab}}$ is the projection? Commented Jan 3, 2020 at 11:59