fundamental character of level $n$ for the inertia group at $p$

Let $p>2$ be a prime, fix algebraic closures $\overline{\mathbb{Q}}$ and $\overline{{\mathbb{Q}}_p}$, and fix an embedding of $\overline{\mathbb{Q}}\hookrightarrow \overline{{\mathbb{Q}}_p}$. This embedding corresponds with an inclusion of the absolute Galois groups:

$$\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/{\mathbb{Q}}_p)\hookrightarrow \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),\sigma \mapsto {\sigma |}_{\overline{\mathbb{Q}}}.$$

Let $I_p$ be the inertia subgroup of $\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/{\mathbb{Q}}_p)$ which we regard as a subgroup of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ via the injection above (for more information on the inertia subgroup at $p$, $I_p$, see the entry on Galois representations). Let ${𝔽}_{p^n}$ be the finite field of $p^n$ elements. The purpose of this entry is to define ${𝔽}_{p^n}$-valued characters ${\mathrm{\Psi }}_n$, for every $n\ge 1$:

$${\mathrm{\Psi }}_n:I_p⟶{𝔽}_{p^n}^{\times }\cong \mathbb{Z}/(p^n-1)\mathbb{Z}$$

which we will refer to as the fundamental character of level $n$ of $I_p$.

Definition 1.

Let ${\chi }_p\mathrm{:}\mathrm{Gal}\mathit{}\mathrm{(}\overline{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{\to }{\mathrm{Z}}_p^{\mathrm{\times }}$ be the $p$-adic cyclotomic character and let $\overline{{\chi }_p}$ be the reduction of ${\chi }_p$ modulo $p$. The fundamental character of level $\mathrm{1}$ is ${\mathrm{\Psi }}_{\mathrm{1}}\mathrm{=}{\overline{{\chi }_p}\mathrm{|}}_{I_p}$, i.e. ${\mathrm{\Psi }}_{\mathrm{1}}$ is the restriction of the $p$-adic cyclotomic character ${\chi }_p$ to $I_p$, composed with reduction modulo $p$.

Next, we define the fundamental characters in more generality. Let $K_n/{\mathbb{Q}}_p$ be the unique unramified field extension of degree $n$ (it is unique by local field theory). The residue field of $K_n$ is the field $k_n={𝔽}_{p^n}$ (because $k$ must be an extension of degree $n$ of ${𝔽}_p$).

Lemma 1.

The field $K_n$ contains all $\mathrm{(}{p}^n\mathrm{-}\mathrm{1}\mathrm{)}$th roots of unity.

Proof.

Clearly, the polynomial $x^{p^n-1}-1=0$ has $p^n-1$ distinct roots in $k_n={𝔽}_{p^n}$. Using Hensel’s lemma, one can check that each root in $k_n$ lifts to an element of $K_n$.∎

Let $K_n^{\prime }=K_n({(-p)}^{\frac{1}{p^n-1}})$. By the lemma, the $(p^n-1)$th roots of unity are contained in $K_n$. Therefore, the extension $K_n^{\prime }/K_n$ is Galois. Moreover, by Kummer theory one has:

$$\mathrm{Gal}(K_n^{\prime }/K_n)=k_n^{\times }={𝔽}_{p^n}^{\times }.$$

Notice that the fact that $K_n/{\mathbb{Q}}_p$ is unramified implies that the inertia group $I_p$ injects into $\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/K_n)\hookrightarrow \mathrm{Gal}(\overline{{\mathbb{Q}}_p}/{\mathbb{Q}}_p)$. Therefore there is a map:

$I_p\hookrightarrow \mathrm{Gal}(\overline{{\mathbb{Q}}_p}/K_n)\to \mathrm{Gal}(K_n^{\prime }/K_n)\to {𝔽}_{p^n}^{\times }$(1)

where the second map is simply given by restriction to $K_n^{\prime }$.

Definition 2.

The fundamental character of level $n\mathrm{\ge }\mathrm{1}$ is the map ${\mathrm{\Psi }}_n\mathrm{:}{I}_p\mathrm{\to }{\mathrm{F}}_{p^n}^{\mathrm{\times }}$ given by Eq. (1).

Note from the author: I would like to thank Eknath Ghate for explaining this to me.