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The Artin conjecture states that for any Galois extension $E/K$ of global fields, and for any non-trivial simple character $\chi$ of $\textrm{Gal}(E/K)$. The Artin $L$-series $L(E/K,\chi,s)$ admits an analytic continuation holomorphic on $\mathbb{C}$.

A trivial consequence of inductive invariance is that: $$ \zeta_E(s) = \zeta_K(s) \prod_{\chi \neq \chi_0}L(E/K,\chi,s) $$ The product being taken over all non-trivial simple characters of $\textrm{Gal}(E/K)$.

A corollary of this is that for a Galois extension $E/K$, the ratio of the corresponding $\zeta$-functions, $\frac{\zeta_E(s)}{\zeta_K(s)}$ is holomorphic on $\mathbb{C}$. This has in fact been proved by other means - without assuming the Artin conjecture - and is known as the Aramata-Brauer theorem.

On p. 83 of $\textit{An Introduction to the Langlands Programme}$ Ehud de Shalit says that:

The corresponding theorem for non-Galois $K/k$ is open, although it would follow from the [Artin] conjecture.

That is to say, he claims that the holomorphicity of the ratio of zeta functions follows from the Artin conjecture.

I have been trying to prove this, but have had some difficulties circumventing the fact that we cannot make sense of the $L$-series related to non-Galois extensions $E/K$. One could take the normal closure (say $E \subset N$), but then one would be dealing with $N/K$, which would show that the ratio of the $\zeta$-functions corresponding to this extension is entire, but how could we relate this to $E/K$?

So why does the Artin conjecture imply the holomorphicity of the ratio of the zeta function?

Thank you for your attention.

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If $L/K$ is a finite extension let $N/K$ be the Galois closure or any Galois extension containing $L$, $G=Gal(N/K),H=Gal(N/L),d=|G|/|H|=[N:L]$ then $$\zeta_L(s)=L(N/K,Reg_{G/H},s),\qquad \zeta_K(s)=L(N/K,1,s)$$

where $Reg_{ G/H}$ is the representation sending $g\in G$ to the matrix $\in GL_d(\Bbb{Z})$ of the permutation $xH\mapsto gxH$ of $G/H$,

and $1$ is the trivial representation $G\to \Bbb{C}^*$.

Claim : from the splitting of prime ideals in towers of extensions this is easy to prove for the unramified primes and it is much more delicate for the ramified primes (it suffices to look at the complication in the definition of the Artin L-series in the case of ramified primes).

Thus $$\frac{\zeta_L(s)}{\zeta_K(s)}=L(N/K,\rho,s)$$ where $\rho$ is the representation such that $Reg_{G/H}\cong 1 \oplus \rho$, the Artin conjecture implies that the RHS is entire.

The Dedekind conjecture is that the LHS is entire.

The inflation property follows exactly from the same discussion.

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  • $\begingroup$ Wonderful. And $\rho$ does not contain the trivial rep as $\textrm{Reg}_{G/H}$ contains only one copy of it. The only trouble I am having is understanding why $\zeta_L(s) = L(N/K, \textrm{Reg}_{G/H},s)$, but I am sure I will see it eventually. $\endgroup$ Jan 3, 2020 at 20:51

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