I already posted this question in MO but I'm going to repost it here as well.
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)$.
For context, consider a 2-dimensional Galois representation
$$\rho:\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}\to \mathrm{GL}(V),$$ where $V$ is a 2-dimensional vector space over $\overline{\mathbb{F}}_p$.
Denote $G:=\mathrm{Gal}(K/\mathbb Q)\cong\mathrm{Im}(\rho)$. We define the conductor of $\rho$ as
$$ N(\rho)=\prod_{\ell\neq p\text{ prime}}\ell^{n(\ell,\rho)},\quad \text{ where }\quad n(\ell,\rho)=\sum_{i=0}^\infty \frac{1}{[G_0:G_i]}\dim(V/V_i), $$
and $G_i$ denotes the $i$-th ramification group of $G$, and
$$V_i=\{v\in V:\rho(\sigma)v=v\text{ for all }\sigma\in G_i\}. $$
We can also consider the homomorphism $$ \det\rho:\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}\to \overline{\mathbb{F}}_p^\times$$ as a 1-dimensional representation and associate its conductor.
I want to prove that $N(\det\rho)$ divides $N(\rho)$, which happens if and only if $n(\ell,\det\rho)\leq n(\ell,\rho)$. Since $\dim V=2$ and $\dim\overline{\mathbb{F}}_p=1$, the only case in which this is not clear is when $\dim{V_i}=2$; i.e., $V_i=V$.
Let $U=\overline{\mathbb{F}}_p$, and define $$ U_i = \{u\in U:\det\rho(\sigma)u=u\text{ for all }\sigma\in H_i\},$$ where $H_i$ is the $i$-th ramification group of $H\cong\mathrm{Im}(\det\rho)$.
We need to see that $U_i = U$ as well. But I have trouble proving the statement because I don't understand how the groups $G_i$ and $H_i$ are related. I'm not even sure if Serre is considering these ramification groups defined locally or globally.
I'm aware this is answered here and I'm aware that we can use the Kronecker-Webber theorem and CFT to prove it. But I'm trying to do it just as Serre suggest in his paper, comparing formulas of $N(\rho)$ and $N(\det\rho)$. If someone can give me a hint I would appreciate it a lot. Thank you so much.