# Artin conjecture for 1-dimensional representations

The Artin conjecture for 1-dimensional representations is the following.

Let $$E/K$$ be a Galois extension of global fields and let $$(V,\rho)$$ be a 1-dimensional non-trivial representation of $$\textrm{Gal}(E/K)$$ with character $$\chi$$. Then the Artin L-function $$L(E/K,\chi,s)=\prod_{\mathfrak{p}}\frac{1}{\textrm{det}(I-N(\mathfrak{p})^{-s}\rho(\sigma_{\mathfrak{P}});V^{I_{\mathfrak{P}}})}$$ admits an analytic continuation holomorphic on $$\mathbb{C}$$.

By inductive invariance, this implies that the Artin conjecture holds for all monomial representations.

I have looked at several proofs of this statement including Neukirch and Artin's original, but they all seem more complicated and unwieldy than they have to be.

If anyone happens to know a resource where a particularly elegant proof of this statement is given, I would appreciate your sharing it with us.

Edit: By proving the holomorphicity of the Artin $$L$$-series for 1-dimensional, I mean proving that they coincide with Hecke $$L$$-series, which are known to be entire functions for non-trivial conductors.

Yes class field theory is complicated, there is no simpler proof : abelian representations of $$Gal(E/K)$$ correspond through the Artin map to finite order Hecke characters of $$K$$, those are defined in term of ideal classes of $$\Bbb{Z}+d O_K$$ and real embeddings letting us construct something analogous to the theta function $$\sum_n \chi(n) e^{-\pi n^2 x}$$ whose Mellin transform is $$\pi^{-s}\Gamma(s)L(2s,\chi)$$, finally the analytic continuation follows from the functional equation
• I don't mind assuming the holomorphicity of the Hecke $L$-series, and that they satisfy the functional equation. What I am looking for is a proof that reduces 1-dimensional Artin $L$-series to Hecke $L$-series. I have edited the post to reflect this. Commented Dec 31, 2019 at 7:27