Artin conductor of a character and factorisation through $(\mathbb{Z} / N\mathbb{Z})^{*}$

Question

This is from Serre's paper on modular representations of degree $2$ of $Gal(\bar{\mathbb{Q}} : \mathbb{Q} ) $.

We consider a representation $\rho : Gal(\bar{\mathbb{Q}}:\mathbb{Q}) \rightarrow GL(2,\bar{\mathbb{F}}_p)$ and its determinant $det \rho$. We define the (global) Artin conductor $N(\rho)$ of $\rho$ as

$$N(\rho) = \prod_{l \neq p} l^{n(l,\rho)} $$

Where $n(l,\rho)$ is the (local) Artin conductor of $\rho$ at $l$ (which is $0$ if and only if $\rho$ is unramified).

Comparing the formulae for $N(\rho)$ and $N(det \rho)$, we have that $N(det \rho) | pN(\rho)$.

Therefore, we can factor $det \rho$ through $(\mathbb{Z} / pN(\rho)\mathbb{Z})^{*}$.

I am looking for a proof of the last statement. I have been told that it is a fairly basic fact of class field theory but I can't seem to find it anywhere.

Thanks in advance!

Answer 1

The determinant $\det \rho$ even factors through $(\mathbb{Z}/pN(\det\rho)\mathbb{Z})^\times$. This follows from a result about the local class field theoretic conductor at $l$: Say your representation factors over the finite abelian Galois group of some extension $L|\mathbb{Q}$. For a prime $l\neq p$ choose a prime $\lambda$ lying over $l$ in $L$ and denote by $G$ the decomposition group $G:=D_{\lambda|l} \subseteq Gal(L|\mathbb{Q})$, that we identify with $Gal(L_\lambda|\mathbb{Q}_l)$ . Then global class field theory gives you a surjective map from the ideles of $\mathbb{Q}$

$$\mathbb{I}_\mathbb{Q}/\mathbb{Q}^\times \to Gal(L|\mathbb{Q})$$

that factors over $Gal(\mathbb{Q}^{ab}|\mathbb{Q})$. Local class field theory tells you that you get a commutative diagram $$\require{AMScd} \begin{CD} \mathbb{I}_\mathbb{Q}/\mathbb{Q}^\times @>>> Gal(L|\mathbb{Q}) @>>> \overline{\mathbb{F}}_p^\times\\ @AAA @AAA \\ \mathbb{Q}^\times_l @>\phi_l>> Gal(L_\lambda|\mathbb{Q}_l) \end{CD}$$ (If you follow the notes by J. Milne, this is the diagram on page 11). Now if $L|\mathbb{Q}$ is unramified at $l$, then $\mathbb{Z}^\times_l$ is already in the kernel of $\phi_l$. In general, the local conductor of class field theory is the smallest $n$ such that the subgroup $1+l^n\mathbb{Z}_l\subseteq \mathbb{Z}_l^\times$ is contained in the kernel of the composition $\mathbb{Z}^\times_l \hookrightarrow \mathbb{Q}^\times_l \to Gal(L_\lambda|\mathbb{Q_l})$, that is the smallest $n$ such that the map factors as

$$\require{AMScd} \begin{CD} \mathbb{Q}^\times_l @>\phi_l>> G\\ @AAA @AAA \\ \mathbb{Z}^\times_l @>>> (\mathbb{Z}/l^n\mathbb{Z})^\times \end{CD}$$

The connection of the two types of conductors is then provided by the following pleasant result relating the local conductor of class field theory to the ramification groups of $G$ in upper numbering:

We have $1+l^n\mathbb{Z}_l \subseteq \ker \phi_l$ if and only if $G^n= 1$

(If you like a reference, this is a consequence of XV §2 in Serre's local fields).

You will be able to figure it out from there using a bit of algebraic number theory input about how $\mathbb{I}_\mathbb{Q}$ and $Gal(\mathbb{Q}^{ab}|\mathbb{Q})$ look like.