# 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:

 $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})\hookrightarrow% \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),\quad\sigma\mapsto\sigma|% _{\overline{\mathbb{Q}}}.$

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

 $\Psi_{n}:I_{p}\longrightarrow\mathbb{F}_{p^{n}}^{\times}\cong\mathbb{Z}/(p^{n}% -1)\mathbb{Z}$

which we will refer to as the fundamental character of level $n$ of $I_{p}$.

###### Definition 1.

Let $\chi_{p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathbb{Z}_{p}% ^{\times}$ be the $p$-adic cyclotomic character and let $\overline{\chi_{p}}$ be the reduction of $\chi_{p}$ modulo $p$. The fundamental character of level $1$ is $\Psi_{1}=\overline{\chi_{p}}|_{I_{p}}$, i.e. $\Psi_{1}$ 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 $k_{n}=\mathbb{F}_{p^{n}}$ (because $k$ must be an extension of degree $n$ of $\mathbb{F}_{p}$).

###### Lemma 1.

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

###### Proof.

Clearly, the polynomial $x^{p^{n}-1}-1=0$ has $p^{n}-1$ distinct roots in $k_{n}=\mathbb{F}_{p^{n}}$. 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:

 $\operatorname{Gal}(K_{n}^{\prime}/K_{n})=k_{n}^{\times}=\mathbb{F}_{p^{n}}^{% \times}.$

Notice that the fact that $K_{n}/\mathbb{Q}_{p}$ is unramified implies that the inertia group $I_{p}$ injects into $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/K_{n})\hookrightarrow% \operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$. Therefore there is a map:

 $\displaystyle I_{p}\hookrightarrow\operatorname{Gal}(\overline{\mathbb{Q}_{p}}% /K_{n})\to\operatorname{Gal}(K_{n}^{\prime}/K_{n})\to\mathbb{F}_{p^{n}}^{\times}$ (1)

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

###### Definition 2.

The fundamental character of level $n\geq 1$ is the map $\Psi_{n}:I_{p}\to\mathbb{F}_{p^{n}}^{\times}$ given by Eq. (1).

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

Title fundamental character of level $n$ for the inertia group at $p$ FundamentalCharacterOfLevelNForTheInertiaGroupAtP 2013-03-22 15:36:26 2013-03-22 15:36:26 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 11R04 msc 11R32 msc 11R34