On $L$-function of permutation representation

Question

I came across the statement in a book:

Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\operatorname{Gal}(K/k)$. Let $\chi$ denote the character of the permutation representation of $G$ in $G/G_k$. Then the Artin $L$-function $L(s, \chi)$ is the Dedekind-Zeta function $\zeta_k$ of the extension $k / \mathbb Q$.

Now first of all, I wanted to confirm whether the "permutation representation" here is the 'standard' one given by $\rho: G \longrightarrow \operatorname{Sym}(G/G_k) : \rho(\sigma) (\tau G_k) := \sigma\tau G_k$ for every $\sigma, \tau \in G$ (i.e. the group action $\sigma \cdot (\tau G_k) = \sigma\tau G_k$).

Second, I could see that on account of the ring $\mathcal{O}_k$ of integers of $k$ being a Dedekind Domain, unique prime factorization of ideals holds and we may write the Dedekind Zeta function in the analogous Euler-Product representation: $$\zeta_k(s) \triangleq \sum_{\mathfrak a \lhd \mathcal O_k} \frac{1}{N(\mathfrak a)^s} = \prod_{\mathfrak{p} \in Spec(\mathcal{O}_k)} \frac{1}{1-N(\mathfrak{p})^{-s}}\text{ for }\Re(s)>1 \hspace{3mm} \cdots \hspace{2mm} (1)$$

But it is not clear to me from the definition of the Artin $L$-function why the above equality should hold. The definition of the Artin $L$-function that I am familiar with of a general character $\eta$ of a representation $\rho: G \longrightarrow GL(V)$ (for some complex vector space $V$) is the following one on Wikipedia: $$L(s, \chi) = \prod_{\mathfrak{p}} \frac{1}{\det[I - N(\mathfrak{p})^{-s}\rho(\sigma_{\mathfrak{p}})|V_{\mathfrak{p}, \rho}]} \hspace{3mm} \cdots \hspace{2mm} (2)$$ where the product on the left is over all prime ideals $\mathfrak p$ of $k$.

What am I missing?

Edit: I am sorry it wasn't clear about the question I was asking. First, I just wanted to confirm whether the "permutation representation" referred to in the problem statement is the one I wrote above or not. Second, the only part that is not clear to me is why $L(s, \chi) = \zeta_k(s)$, from the definition of the Artin $L$-function that I know (which is (2)). I am okay with (1) and (2), the only thing I don't understand is why $L(s, \chi) = \zeta_k(s)$.

Edit: We have in this case (as Will Sawin has pointed out) $$L(s, \chi) = \prod_p \frac{1}{\det[I-p^{-s}\rho(\sigma_p)]}$$ where the product on the left is over all integer primes $p$. I tried to show that this is equal to the product occurring on the right hand side of (1) for $\Re(s)>1$. We would therefore be done if could show, for all integer primes $p$ and for all such $s$, the identity $$\det[I-p^{-s}\rho(\sigma_p)|V_{p, \rho}] = \prod_{\mathfrak{p}|p} (1-N(\mathfrak{p})^{-s})$$ To show the last equality, I tried expanding the determinant on the left into a product of eigenvalues, but I'm not sure how to proceed from there.

Answer 1

It is not easy to give all the details so I'll give a sketch in the case of unramified prime

• For $p$ an unramified prime number, $Q\subset O_K$ a prime ideal above $p$, those of $O_k$ are of the form $P_g =g(Q)\cap O_k$ for $g\in G$ with norm $N(P_g)=p^{f_g}$ for some integers $f_g$

  $\sigma$ a Frobenius such that $\forall a\in O_K,\sigma(a)\equiv a^p\bmod Q$

  The Frobenius of $O_K/g(Q)$ is $g^{-1}\sigma g$ and $f_g$ is the least integer such that $(g^{-1}\sigma g)^{f_g}\in H$ ie. such that $\sigma^{f_g} gH = gH$.

  (this is because if $f_g$ was smaller then $P_g$ would appear with multiplicity $>1$ in the factorization of $pO_k$)

• With $\rho$ the representation of $G$ permuting $G/H$ then $\det(I-\rho(\sigma)p^{-s}) = \prod_{C \text{ orbits of } \langle \sigma \rangle \text{ on } G/H} (1-p^{-s|C|})$

  With $C= \langle \sigma \rangle g H$ then $|C|=f_g$

  Thus the unramified Euler factors of $\zeta_k(s)$ and $L(s,\rho,K/\Bbb{Q})$ are equal.

For the ramified primes it works similarly after taking the subfield fixed by the inertia subgroup.