On the determinant of an odd, continuous Galois representation

Question

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Frob{Frob}$In his paper, Duke paper, Serre consider continuous, odd Galois representation $\rho: G_{\mathbb{Q}}\longrightarrow \GL_{n}(\overline{\mathbb{F}}_{p})$ where $p$ is a rational prime. Roughly, (I don't understand much French except for the help from Google translation) Serre claims (section 1.3) that

$\det(\rho(\Frob_{l})) = \varepsilon(\Frob_{l})\omega^{k}(\Frob_{l})$ for all prime $l\nmid pN$ where $N$ is defined as the level of the representation (with an explicit formula given in the paper) and $\epsilon$ is a Dirichlet character and $k$ is some positive integer.

This seems to be standard since other papers cited it without reproving and I could not find any reference for the proof. In particular, my questions are:

1. Where can I find a proof for this.

2. What exactly is $\varepsilon$, in some paper, there is the claim that $\varepsilon$ is the unique quadratic character mod $p$ ramified only at $p$, and I do not understand where this comes from?

3. How can one find $k$.

For motivation, I think $\det(\rho(\Frob_{l}))$ is an important invariant to compute since, for example, it appears in the attachment equation that associates these representations with modular forms.

Thanks in advance for any insight.

Answer 1

$\det (\rho)$ is a one dimensional rep of the absolute Galois group of the rationals, i.e., it is a character. All such characters can be described by class field theory or, more simply, by the Kronecker-Weber theorem. So is a Dirichlet character and, by the hypotheses, its conductor divides $pN$. Factor it as a character of conductor $p$ (that will be $\epsilon$) times a character of conductor $N$. The latter is a power of the cyclotomic character and $k$ is defined to be that power. The bit about quadratic character must be under additional hypotheses.

Edit: I got $p$ and $N$ switched above. The character of conductor $N$ is $\epsilon$. The character of conductor $p$ is a power of the cyclotomic character because $(\mathbb{Z}/p)^*$ is cyclic.