# On $L$-function of permutation representation

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.

• What do you mean? You're not clear why you have convergence for Re(s) > 1? Or you don't understand how to prove the Euler product given convergence? In any case, my personal view is that math.stackexchange is a better fit for this as, if I understand correctly, I think you're asking about something that's treated in many standard (graduate) number theory textbooks. May 18, 2020 at 23:09
• Apologies for the misunderstanding, I have made the question clear now. May 18, 2020 at 23:35
• 1. What have you tried so far? 2. Yes, the permutation representation is the standard one. 3. Because $\chi$ is a character of the Galois group of $K$ over $\mathbb Q$, the product is over prime ideals of $\mathbb Q$, not $k$. May 18, 2020 at 23:40
• Thank you for confirming. I have added the attempts I've made so far. May 19, 2020 at 0:14
• You seem to have crossposted this at MSE now: math.stackexchange.com/q/3683157/11323 To be clear, this is generally discouraged (at least here) and I was suggesting that you should've just asked this question there and not here in the first place (the line is kind of blurry, I know). But then if you don't get an answer after a few days, you might try asking here, referencing your MSE post. May 20, 2020 at 13:20

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.

• Thank you for your answer, I'll take some time to read and understand it. Can you give me a reference where I can find an accessible proof of it (preferably a proof where I'll need the least prerequisites)? May 19, 2020 at 0:30
• Sorry I didn't understand why the formula $$\det(I-\rho(\sigma)p^{-s}) = \prod_{C \text{ orbits of } \langle \sigma \rangle \text{ on } G/H} (1-p^{-s|C|})$$ holds. Is it a standard result? Also by "frobenius of O_K/g(Q)" do you mean the Forbenius element of the prime $Q$ lying over $P_g$ (=$O_k \cap g(Q)$)? May 19, 2020 at 0:53
• On each $C$, $\rho(\sigma)$ is permuting cyclically the $|C|$ elements of $C$ so on this subspace of dimension $|C|$ the matrix has order $|C|$. This is enough to say that its minimal/characteristic polynomial is $X^{|C|}-1$. May 19, 2020 at 1:03
• Sorry for bringing this up after a long time, but could you please confirm the following: by "frobenius of $O_K/g(Q)$" do you mean the Frobenius automorphism of the residue field? Thanks. May 21, 2020 at 6:39
• In the residue field there is a unique Frobenius $O_K/g(Q)$ which is $a\to a^p$, for $K/Q$ Galois then it lifts to a unique $g\in Gal(K/Q)$ iff $p$ is unramified, when $p$ is ramified with $g$ a lift then the others are $I_{g(Q)} g$ (the inertia subgroup). This is what I meant with en.wikipedia.org/wiki/… is the main prerequisite. If the concepts are unclear to you then try with a few $p$ in $K=Q(i,\sqrt{2}),k=Q(i)$ @asrxiiviii May 21, 2020 at 17:55