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I'm trying to understand the concept (definition) of the fundamental characters of level 1,2, etc. as Serre defined those in his Inventiones'72 paper and how they are related to Serre's conjecture. The paper is in French and hard to read, I would greatly appreciate clarifications and more detailed explanations. My background is algebraic number theory at the level of Marcus and Milnes's note and some class field theory from Milne's notes.

Edit: link to the paper.

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$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\Q}{\mathbf Q} \newcommand{\F}{\mathbf F} \newcommand{\s}{\mathrm{s}} \newcommand{\nr}{\mathrm{nr}} \newcommand{\t}{\mathrm{t}}$

Let me explain Serre's definition in the case of a finite extension $K$ of $\Q_p$ ($p$ prime). In his notation, $K_\s$ is an algebraic closure of $K$, $K_\t$ is the maximal tamely ramified extension of $K$ in $K_\s$, $K_\nr$ is the maximal unramified extension of $K$ in $K_\t$, and $I_\t=\Gal(K_\t|K_\nr)$. The common residue field $k_\s$ of $K_\s$, $K_\t$, and $K_\nr$ is an algebraic closure of the finite field $\F_p$.

So inside $k_\s$ you have all the finite extensions $\F_{p^n}$ of $\F_p$, with norm maps $\F_{p^m}^\times\to\F_{p^n}^\times$ whenever $n|m$. The first thing to observe is that there is a natural identification $\theta$ of $I_\t$ with the inverse limit of this system of norm maps (prop. 2)

As such, for each power $q=p^n$ of $p$, there is a natural projection $\theta_{q-1}:I_\t\to\F_q^\times$, which you might call the fundamental character of level $n$. When you compose it with some power $\varphi^i$ of the automorphism $\varphi:x\mapsto x^p$ of $\F_q^\times$, you get all the other fundamental characters of level $n$. There are thus $n$ fundamental characters of level $n$, namely $\varphi^i\circ\theta_{q-1}=\theta_{q-1}^{p^i}$ for $i\in[0,n[$.

Here is how to think of $\theta_{q-1}$ concretely. Take a uniformiser $\pi$ of $K_\nr$ and adjoin its $(q-1)$-th root $\root q-1\of\pi$ to get the extension $L_{q-1}$ of $K_\nr$ in $K_\t$; this extension does not depend on the choice of $\pi$. On the one hand, the group $\Gal(L_{q-1}|K_\nr)$ is a quotient of $I_\t$. On the other hand, $\Gal(L_{q-1}|K_\nr)$ is naturally isomorphic to the group $\mu_{q-1}$ of $(q-1)$-th roots of $1$ (under $\zeta\mapsto(\root q-1\of\pi\mapsto\zeta.\root q-1\of\pi$)), which is naturally isomorphic (under reduction modulo $\pi$) to the group $\F_q^\times$. The character $\theta_{q-1}$ is just the composite $$ I_\t\to\Gal(L_{q-1}|K_\nr)\to\mu_{q-1} \to\F_q^\times. $$

I'll leave it to the experts to elucidate the role of fundamental characters in Serre's modularity conjecture (now a theorem). I believe they serve to define the optimal weight of a cuspidal eigenform from which a given odd irreducible representation $\Gal(\bar{\Q}|\Q)\to\GL_2(\bar{\F}_p)$ arises.

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