Definition of the Weil group: Question about exact sequence with Inertia Group and absolute Galois group over a local field

Question

Let $K$ be a local field, $k$ be its residue field, $G_K, G_k$ be the absolute Galois groups of $K, k$ and $I_K$ be the inertia group of $K$. In several books and papers, I found the following exact sequence:

$$ 1 \longrightarrow I_K \longrightarrow \ G_K \longrightarrow G_k \longrightarrow 1. $$

Now I would like to know why because my motivation is it to understand the definition of the Weil group $W_K$. If I recall correctly, this is defined by the exact sequence

$$ 1 \longrightarrow I_K \longrightarrow W_K \longrightarrow \mathbb{Z} \longrightarrow 0,$$

but I did not understand every detail of it.

First, let me assemble what I understood (or not understood entirely):

• I heard that $G_k$ is isomorphic to $\hat{\mathbb{Z}} = \varprojlim\limits_n \mathbb{Z}/n\mathbb{Z}$. I think this follows from the fact that $k$ is a finite field and the absolute Galois group is a profinite group but I was not yet able to put these facts together.
• Anyway, I think that $\mathbb{Z}$ is a subgroup of $\hat{\mathbb{Z}}$ but I do not know how the embedding would look like.
• One defines $W_K$ to be the inverse image of $\mathbb{Z}$ under the map $\phi: G_K \to \hat{\mathbb{Z}} $.
• I know that $I_K = \{ \sigma \in G_K : \sigma(x) \equiv x \mod{\mathcal{P}_K} \: \forall x \in \mathcal{O}_K \} $ where $\mathcal{P}_K = \{ x \in K : |x|<1 \}$ denotes the unique maximal ideal over $K$ and $\mathcal{O}_K$ denotes the ring of integers over $K$. As the first sequence is exact, this must mean that $I_K$ is the kernel of some map $G_K \to G_k$ resp. $G_K \to \hat{\mathbb{Z}}$. But I do now know how this map would look like.
• How does the Frobenius automorphism comes into play? The Frobenius automorphism is the map $\operatorname{Frob}_k \in G_k$ with $\operatorname{Frob}_k : x \mapsto x^q $ where $q$ is the size of the finite field $k$.

Could you please explain this to me? Thank you!

Answer 1

Let $K$ be your non-archimedean local field, $\mathcal O_K$ be the ring of integers, $\mathfrak p$ be the maximal ideal of the ring, and $k := \mathcal O_K / \mathfrak p$ be the residue field.

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If $k$ is a finite field and $|k| = q$, then we assemble $\overline k$ this way: by the classification of finite fields, for every $n$ there is a unique field of order $q^n$ up to isomorphism, and the field of order $q^m$ is a subfield of the field of order $q^n$ iff $m \mid n$, so we take the "union" (really a direct limit) of the fields of order $q^n$, and that is $\overline k$.

And as to the Galois groups, if we denote by $\Bbb F_{q^n}$ the field of order $q^n$, then $\operatorname{Gal}(\Bbb F_{q^n} / \Bbb F_q) \cong \Bbb Z/n\Bbb Z$, with the canonical generator being the automorphism $x \mapsto x^q$. The Galois group of a direct limit is the inverse limit of the Galois groups, i.e.: $$\operatorname{Gal}(\overline{\Bbb F_q} / \Bbb F_q) \cong \operatorname{Gal}(\varinjlim \Bbb F_{q^n} / \Bbb F_q) \cong \varprojlim \operatorname{Gal}(\Bbb F_{q^n} / \Bbb F_q) \cong \varprojlim \Bbb Z/n\Bbb Z = \widehat{\Bbb Z}$$ And the topological generator is still the automorphism $x \mapsto x^q$. "Topological generator" means that this automorphism generates the dense cyclic subgroup $\Bbb Z$ of $\widehat{\Bbb Z}$.

So this clears the $\Bbb Z$ part of your query.

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Now we discuss the map $G_K \to G_k$

The following three things are equivalent:

1. Finite unramified extension $L/K$
2. Finite unramified ring extension $\mathcal O_L / \mathcal O_K$
3. Finite extension $l/k$

Moreover, from the classification of finite fields, $\Bbb F_{q^n}$ can be described as $\Bbb F_q(\mu_{q^n-1})$, where $\mu_{q^n-1}$ is the set of the $(q^n-1)$^st roots of unity.

From the classification of unramified extensions, for each $n$ there is a unique unramified extension $L/K$ with $[L:K] = n$, and $L = K(\mu_{q^n-1})$. Again we have $\operatorname{Gal}(L/K) \cong \Bbb Z/n\Bbb Z$, the generator being the automorphism that sends a primitive $(q^n-1)$^th root of unity to its $q$-th power (no, it doesn't send everything else to their $q$-th powers). We call this map Frobenius.

Using the same machinery, we can establish that $\operatorname{Gal}(K^{ur}/K) \cong \widehat{\Bbb Z}$.

Note that $K^{ur}$, being a union of normal extensions, is itself a normal extension. Therefore, we have a map $G_K \to \operatorname{Gal}(K^{ur}/K) \cong \operatorname{Gal}(\overline k/k) = G_k$ as required.

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A caveat is that the topology on the Weil group is not the subspace topology, but finer than the subspace topology. We require that (the image of) the inertia group be open.

Answer 2

Let me try and answer each of your questions. Let's stick to the case where $K$ is a finite extension of $\mathbb Q_p$, and $k = \mathbb F_q$, with $q= p^r$ for some $r$.

• What is $G_k$? By definition, it is $$\mathrm{Gal}(\overline k/k)=\varprojlim_{\ell/k\text{ finite}}\mathrm{Gal}(\ell/k).$$ But $k = \mathbb F_q$, and the only finite extensions of $\mathbb F_q$ are $\mathbb F_{q^n}$ for various $n$. Moreover, each $\mathrm{Gal}(\mathbb F_{q^n}/\mathbb F_q)$ is canonically isomorphic to $\mathbb Z/n\mathbb Z$, with canonical generator $\mathrm{Frob}_k$. It follows that $$G_k=\varprojlim_n(\mathbb Z/n\mathbb Z).$$

• By construction, $G_k=\widehat{\mathbb Z}$ is topologically generated by $\mathrm{Frob}_k$. The subgroup $\langle \mathrm{Frob}_k\rangle$ is isomorphic to $\mathbb Z$.

• Your definition of the intertia group is incorrect: the group you've defined is the relative inertia group of the extension $K/\mathbb Q_p$. The group you need is the absolute inertia group.

• If $\sigma\in G_K$, then $\sigma$ acts on $\overline{\mathbb Z}_p$ and fixes the prime ideal $\mathfrak p$ of $\overline{\mathbb Z}_p$ lying over $p$. Hence, it acts on $\overline k=\overline{\mathbb Z}_p/\mathfrak p$, and fixes $k$. This process defines a map $$G_K\to G_k.$$ The absolute inertia group, $I_K$, is defined to be the kernel of this map. As the other answer shows, $I_K=\mathrm{Gal}(\overline K/K^{ur})$.

• This construction should be somewhat familiar. We can define a map $$\mathrm{Gal}(K/\mathbb Q_p)\to\mathrm{Gal}(k/\mathbb F_p),$$ and the inertia group that you described is the kernel of this map. In the case of the Weil group, we need to carry this out for the infinite extension $\overline K/K$.

• We call a $\sigma\in G_K$ a Frobenius element if it is a lift of $\mathrm{Frob}_k\in G_k$. But in reality, the Frobenius element is only well-defined in $G_K/I_K$.

Answer 3

Allow a late comer to add a few complements to the answers given by @Kenny Lau and @Mathmo 123. Actually, the Weil group $W_K$ of a local or global field $K$ is not merely a group, but a triple $(W_K , \phi, {r_L})$ , where $L$ runs through the finite extensions of $F$, which must satisfy a certain number of properties coming from class field theory. More precisely, $W_K$ is a topological group, $\phi: W_K \to G_K$ (= absolute Galois group of $K$ with the profinite topology) is a continuous homomorphism with dense image. This means that $\phi^{-1}(G_L)$ is open for every $L$, and $\phi$ induces a bijection between the homogeneous spaces $W_K/W_L$ and $G_L/G_K$, which is an isomorphism if $L/K$ is Galois. The arithmetic lies in the topological isomorphisms $r_L: A_L \to {W_L}^{ab}$ $:= W_L/{W_L}^c$, where ${W_L}^c$ denotes the closure of the commutator subgroup, and $A_L$ is the multiplicative group $L^*$ if $L$ is local, the idèle class group $C_L$ if $L$ is global. Extra conditions are required, in particular that the composite maps $A_L \to {W_L}^{ab} \to {G_L}^{ab}$ coincide with the reciprocity homomorphisms of CFT, and the natural map $W_K \to pro.lim W_{L/K}$ is a topological isomorphism, where $W_{L/K}:= W_K /{W_L}^c (\neq W_K/W_L)$.

The main work consists in proving the existence and uniqueness (up to suitable isomorphisms) of the Weil group. This is done in detail in Artin-Tate, CFT, chap.14. One key point is thm.5, p.245, which gives canonical isomorphisms $A_K/Norm_{L/K} A_L \cong W_K/W_L{W_K}^c \cong G(L/K)^{ab}$ if $L/K$ is Galois, and justifies a first motivation of the construction : "[This] shows that the entire theory of the reciprocity law is contained in the theory of Weil groups (...) The reciprocity relationship between levels $A_L$ and Galois groups $G(L/K)$ becomes easy to vizualize when one identifies $A_L$ with $W_L/{W_L}^c$ (by means of $r_L$) and $G(L/K)$ with $W_L /W_K$ (by means of $\phi$). In this way, all the facts are wrapped up in one neat non - abelian bundle, namely a suitable Weil group" (p.246). Unfortunately for the layman, chap.14 of Artin-Tate is written in the language of "class formations", which is itself introduced in chap.13 as an axiomatization of the cohomological properties of CFT. So it is perhaps not superfluous to outline the construction of $W_F$ in two particular cases.

1) $F$ is a local $p$-adic field : This case is a relatively simple application of local CFT. The residue field $k$ of $K$ is a finite field of order $q$= a power of $p$ and its absolute Galois group $G_k$ is topologically isomorphic to the profinite completion $\hat {\mathbf Z}$ of $\mathbf Z$ , topologically generated by the Frobenius automorphism $Frob_k:x \to x^q$. It is classically known (see e.g. Cassels-Fröhlich, ANT, chap.1, coroll.2 of thm.2 , p.28) that the maximal unramified extension $K_{nr}$ of $K$ has Galois group isomorphic to $G_k$, hence an inertia exact sequence $1 \to I_K \to G_K \to G_k \cong \hat {\mathbf Z} \to 1$, where $I_K$ is the absolute inertia subgroup. Take $W_K$ = the inverse image in $G_K$ of $<Frob_k>$, so that $W_K/I_K \cong \mathbf Z$, and $\phi$ = inclusion. The topology in $W_K$ will be that for which $I_K$ gets the profinite topology induced by $G_K$, and is open in $G_K$. Consider the local reciprocity map $\theta_K:K^* \to {G_K}^{ab}$. By prop.2 of Cassels-Fröhlich, chap.6, p.141, the composition $K^* \to {G_K}^{ab} \to G_{K_{nr}} \cong \hat {\mathbf Z}$ induces the valuation map $v:K^* \to \mathbf Z$, whose kernel is the group of units $U_K$, hence $\theta_K$ provides a commutative diagram linking the valuation exact sequence to the abelian inertia exact sequence $1 \to J_K \to {G_K}^{ab} \to \hat {\mathbf Z}\to 1$ [I wish I could type the diagram !]. The so called "existence theorem" of local CFT states that $\theta_K$ induces an isomorphism $U_K \cong J_K$ (Cassels-Fröhlich,thm.3a, p.144). From this and the reciprocity isomorphism $A_K \cong {G_K}^{ab}$, the desired property for the maps $r_L$ follows easily.

2) $F$ is a number field : This case is much more involved. At the present time, there exists no direct approach as in 1), the only known construction being somewhat roundabout. Let $L/K$ be Galois with group $G$, and consider the maximal abelian extension $L^{ab}/L$ with Galois group $A\cong C_L/D_L$, where $D_L$ is the connected component of $1$ in $C_K$. By maximality, $L^{ab}/K$ is also Galois, and the extension of groups $1\to A \to G(L^{ab}/K) \to G \to 1$ is described by a 2-cohomology class $\epsilon \in H^2(G, A)$. Besides, define $W_{L/K}$ by the extension of groups $1\to C_L \to W_{L/K} \to G\to 1$ corresponding to the CFT fundamental class $u\in H^2(G, C_L)$, and the Safarevic-Weil theorem asserts that, under the natural map $H^2(G, C_L) \to H^2(G, A)$ induced by reciprocity, the class $u$ maps to the class $\epsilon$ . This gives an obvious commutative diagram linking the two exact sequences of group extensions, hence in particular a surjective map $\phi_L: W_{L/K} \to G(L^{ab}/K)$, with kernel $D_L$ (NB: this also works word for word in the local case, but then $D_L$ is trivial). One finally defines $W_K$ as being $pro.lim W_{L/K}$, and $\phi: W_K \to G_K$ is surjective, with a huge kernel $D_K$ isomorphic to the direct product of $\mathbf R$, $r_2$ circles and $r_1+r_2-1$ solenoids. But, as stressed by Weil (1951), what is actually sought is a Galois-like interpretation of $W_K$ (Weil, 1951).

Thus a second motivation is the study of (continuous) representations of the Weil group $W_K$. In support of the idea that the Weil group behaves like a Galois group, the L-functions attached by Weil (1951) to representations of $W_K$ include as special cases the abelian L-series of Hecke, made with "Grössencharakteren" (=continuous homomorphisms $C_K \to \mathbf C^*$) and the non-abelian L-functions of Artin, made with representations of $G_K$. And beyond lies the Langlands program, but this is outside our scope ./.