CHAPTER 4 Lecture 4

1. The Absolute Galois Group of a Local Field

1.1. Finite Galois Extensions. We continue to let K be a local field, in the sense defined previously, with ring of integers O_{K} and residue field k = k_{A} = O_{K}/𝔪_{A} of characteristic p. For this section we will feel free to think of Galois groups as belonging to extensions of rings of integers; this will simplify notation, and no confusion should result.

Let 𝔤 = Gal(k/𝔽_{p}). Using our construction of Witt vectors, we get a diagram

[figure: A; k; W (k); 𝔤; A^{ur}; 𝔤; k; 𝔤; W (𝔽_{p}); ℤ_{p}; 𝔽_{p}]

In fact, using our study of ramification groups we know that we can further decompose the extension as

A_{p-group} A^{tr}_{⊆k^{∗}} A^{ur}_{𝔤} A

where the notation above is supposed to indicate that we know that Gal(A^{tr}/A^{ur}) is isomorphic to the ramification quotient G_{0}/G_{1}, which canonically embeds in k^{∗}. (See Lecture 3, Section 1.1.)

1.2. Passage to the Limit. We now generalize the above construction. Let k_{n} be the unique extension of k of degree n, and let 𝔤_{n} = Gal(k_{n}/k). Let q_{n} be the order of k_{n} and let w_{n} = q_{n} − 1 be the order of k_{n}^{∗}. Then Hensel’s lemma (see [Ca, Appendix C]) shows that the elements of k_{n}^{∗} lift to W (k_{n}) as roots of unity; precisely, there is an isomorphism

0Last modified February 23, 1998

ω : k_{n}^{∗} ⟶^{≅} μ(W (k_{n})^{∗}),

called the Teichmuller character, such that

ω(a) ≡ a (mod 𝔪)

for all a ∈ k_{n}^{∗}, where 𝔪 is the maximal ideal of W (k_{n}). Somewhat more explicitly, one can define ω(a) by

ω(a) = lim_{ν→∞} ã^{q_{n}^{ν}}

where ã is any choice of lifting of a to W (k_{n}). (The basic idea here is that exponentiating by powers of q_{n} will not affect the root of unity part of ã, while it will “push” the non-root of unity part off to infinity. It is still congruent to a since exponentiating by q_{n} is the identity on the residue field.)

Now let us construct the maximal totally tamely ramified extension of A_{n} = W (k_{n}). This can be done using Kummer theory; simply set

A_{n}^{tr} = W (k_{n})[x]/(x^{w_{n}} − p).

As before, Kummer theory shows that the Galois group of A_{n}^{tr} over W (k_{n}) is canonically isomorphic to k_{n}^{∗}. (See Lecture 1, Section 3.) This further implies that A_{n}^{tr} really is the maximal tamely ramified extension of A_{n}, since we have shown that the Galois group of any tamely ramified extension of A_{n} will inject into k_{n}^{∗}. (See Lecture 3, Proposition 1.1. One must also use the fact that the residue field does not change under totally ramified extensions.) Choose a uniformizer π_{n} of A_{n}^{tr}. Letting T_{n} be the Galois group of A_{n}^{tr} over W (𝔽_{p}) = ℤ_{p}, we have a diagram

[figure: A_{n}^{tr}; 𝔤_{n}; k_{n}^{∗}; T; ℤ_{p}[π_{n}]; A_{n}; 𝔤_{n}; ℤ_{p}]

where the unlabeled extension isn’t even Galois, since ℤ_{p} doesn’t contain the w_{n}^{th}- roots of unity. Nevertheless, we do get a semi-direct product decomposition

T_{n} = k_{n}^{∗} ⋊ 𝔤_{n}

where the action of 𝔤_{n} on k_{n}^{∗} is simply its natural action. (This is immediate, using the fact that the isomorphism of Kummer theory is Galois equivariant.)

Next, we take the limit as n goes to infinity of the above construction. This yields a diagram

[figure: A_{∞}^{tr}; 𝔤; Δ_{𝔽p}; T; ·; A_{∞}; 𝔤; ℤ_{p}]

where A_{∞} = W (\overline{𝔽}_{p}), 𝔤 = Gal(\overline{𝔽}_{p}/𝔽_{p}) ≅ ℤ̂, A_{∞}^{tr} is the tamely ramified extension of A_{∞} generated by all non p-power roots of p, T is the Galois group of the fraction field of A_{∞}^{tr} over ℚ_{p}, and the unnamed entry is some non-canonical non-Galois extension of ℤ_{p}. In particular, as above T = Δ_{𝔽_{p}} ⋊ 𝔤, where the action of 𝔤 on Δ_{𝔽_{p}} is its natural action. We can also write this as an exact sequence

1 → Δ_{𝔽_{p}} → T → 𝔤 → 1

or, via our earlier canonical isomorphisms,

1 → ∏_{l≠p} ℤ_{l}(1) → T → ℤ̂ → 1.

Both of the flanking terms in this exact sequence are topologically cyclic in the sense of Lecture 3, Section 3. Let φ be any lifting of the canonical Frobenius generator of 𝔤; we will refer to it as a Frobenius element of T. Let τ be any topological generator of Δ_{𝔽_{p}}. No matter the choice of φ and τ, they are related by

φτφ^{−1} = τ^{p}

since φ acts as Frobenius on Δ_{𝔽_{p}}, and the action in this setting is given by conjugation. In fact, this is a presentation of T, in a sense that we make precise in the following proposition.

Proposition 1.1. The group T is the profinite completion of the group on two generators τ and φ satisfying the single relation φτφ^{−1} = τ^{p}.

Proof. We sketch the proof. Let F be the group generated by τ and φ with the above relation. One first shows that the profinite completion of the subgroup generated by φ is ℤ̂; this is because combining the above relation with φ^{n} = 1 and τ^{n} = 1 doesn’t introduce any additional conditions solely on φ. On the other hand, the same construction with the subgroup generated by τ does result in conditions on τ of the form

τ^{p^{m}−1} = 1.

It follows that the profinite completion of the subgroup generated by τ is isomorphic to

∏_{l≠p} ℤ_{l}.

Also, the relation shows that this subgroup is normal. Thus we get an exact sequence

1 → ∏_{l≠p} ℤ_{l} → F̂ → ℤ̂ → 1,

where F̂ is the profinite completion of F. Further, we have a lifting of a generator of ℤ̂ to F̂: namely, φ. We also know how φ acts on the other factor, by the defining relation. Thus

F̂ ≅ ∏_{l≠p} ℤ_{l} ⋊ ℤ̂,

with the same action of ℤ̂ on ∏_{l≠p} ℤ_{l} as T. Thus T = F̂.

It follows easily from this characterization that we also have a homeomorphism (although definitely not a homomorphism)

∏_{l≠p} ℤ_{l} × ℤ̂ → T

given by sending α × β to τ^{α}φ^{β}.

Let us now consider a subgroup T′ of T of finite index. T′ must contain some power of some lifting of the canonical generator of ℤ̂, since otherwise all of the powers of φ would lie in different cosets. Similarly, it must contain some power of τ, and we can take this power to be prime to p since p is “invertible” when acting on τ. In other words, there is some integer e prime to p, some integer f and some lifting φ of 1 ∈ ℤ̂ such that

τ^{e}, φ^{f} ⊆ T′.

We will denote by T_{e,f} the closed subgroup of T generated by τ^{e} and φ^{f} for e and f as above; the above argument shows that these are all of the subgroups of T of finite index, where we allow φ to be different liftings of 1 ∈ ℤ̂. (In particular they are open.) They are not all normal, however; one can check easily (by conjugating τ^{e} by φ and φ^{f} by τ) that this occurs if and only if e divides p^{f} − 1.

1.3. Absolute Galois Groups. Since T is the maximal tamely ramified quotient of the absolute Galois group G_{ℚ_{p}} = Gal(\overline{ℚ}_{p}/ℚ_{p}), we have an exact sequence

1 → P → G_{ℚ_{p}} → T → 1

where P is the wild ramification subgroup of G_{ℚ_{p}}; it is pro-p. Letting I be the inertia subgroup of G_{ℚ_{p}}, we have an exact sequence

1 → I → G_{ℚ_{p}} → 𝔤 → 1.

Restricting our attention to I alone, we have an exact sequence

1 → P → I → Δ_{𝔽_{p}} → 1.

This is all sumarized in the diagram

[figure: \overline{ℚ}_{p}; I; G_{ℚp}; ℚ_{p}^{tr}; P; wildly ramified; T; ℚ_{p}^{ur}; Δ_{𝔽p}; tamely ramified; ℚ_{p}; 𝔤; unramified]

Now let K be any finite extension of ℚ_{p}. Let G_{K} be its absolute Galois group, so that G_{K} ⊆ G_{ℚ_{p}}. We set I_{K} = I ∩ G_{K} and P_{K} = P ∩ G_{K}; I_{K} is simply the inertia group of K̄/K. We also let T_{K} be the image of G_{K} under the map G_{ℚ_{p}} → T; this is not the same as the tame inertia group of K in the usual sense if K has wild ramification over ℚ_{p}. One easily checks that T_{K} = T_{e,f} where e is the prime to p part of the inertial degree of K and f is the residue degree of K. We now have an exact sequence

1 → P_{K} → G_{K} → T_{K} → 1.

2. Galois Representations

2.1. Definitions and Reductions. Let K be a local field of residue characteristic p. Let F be a field of scalars; for our purposes we will take F to also be a local field, of residue characteristic l. We will usually, but not always, want l ≠ p. Let V be a vector space over F of finite dimension d. We consider G_{K}- representations on V; of course, we assume them to be continuous and F -linear. So, by a Galois representation we mean a continuous homomorphism

ρ : G_{K} → Aut_{F} (V) ≅ GL_{d}(F),

the isomorphism being given by any choice of an F -basis of V.

We begin by showing that one may actually assume that the image of G_{K} lands in GL_{d}(O_{F}), where O_{F} is the ring of integers in F. More generally, let Γ be any profinite group which acts continuously and F -linearly on V:

Γ → Aut_{F} (V).

Recall that a lattice Ω ⊆ V is a free O_{F} -submodule of V of rank d, such that

Ω ⊗_{O_{F}} F ⟶^{≅} V.

We claim that there is some lattice Ω which is stabilized by Γ. Let Ω_{0} be any lattice of V. Then if we choose the basis for V to also be an O_{F} -basis of Ω_{0}, we have that GL_{d}(O_{F}) is the compact open subgroup of GL_{d}(V) preserving Ω_{0}. The cosets GL_{d}(F)/ GL_{d}(O_{F}) correspond to the set of all lattices of V, the correspondence given by sending g ∈ GL_{d}(F) to gΩ_{0}. Now, for our purposes we may replace Γ by its image in GL_{d}(F), so that Γ is now a compact subgroup of GL_{d}(F). Consider the set

Γ_{0} = Γ ∩ GL_{d}(O_{F}).

This is open in Γ, and thus has finite index since Γ is compact. Let γ_{1}, …, γ_{ν} be a set of coset representatives for Γ/Γ_{0}. Take

Ω = (γ_{1} + ⋯ + γ_{ν})Ω_{0}.

One can check that Ω is a lattice, and it is clearly stabilized by Γ since Ω_{0} is stabilized by Γ_{0}.

Now, as above, the fact that Γ stabilizes the lattice Ω implies that, by choosing the basis of F to be an O_{F} -basis of Ω, the action of Γ on GL_{d}(F) factors through GL_{d}(O_{F}). Thus it will be enough for us to consider representations

Γ → GL_{d}(O_{F}).

We state the Galois case as a proposition.

Proposition 2.1. Let V be an F -vector space of dimension d, and let

ρ : G_{K} → Aut_{F} (V)

be a Galois representation. Then there is a basis of V such that with respect to this basis the image of ρ is in GL_{d}(O_{F}).

2.2. Potentially Semistable Representations. By our presentation of the Galois group T in Section 1.2, we know that a representation of T on V is simply given by specifying two invertible F -linear operators τ and φ satisfying the single relation

φτφ^{−1} = τ^{p}.

By our above reductions we can further assume that τ and φ preserve O_{F}.

Let v ∈ V be an eigenvector of τ with eigenvalue λ. Set v_{n} = φ^{−n}v for all n ≥ 0. Then using the above relation one easily checks that v_{n} is an eigenvector of τ with eigenvalue λ^{p^{n}}. (Each time you “commute” a φ^{−1} with τ the τ is replaced by a τ^{p}.) But V is finite dimensional, so τ has only finitely many eigenvalues. Thus for some m < n,

λ^{p^{m}} = λ^{p^{n}}.

In particular, λ is a (p^{n−m} − 1)p^{m}-root of unity. In fact, we can take the above m to be zero, for we are free to replace λ by λ^{p^{m}}, in which case the above equality becomes

λ = λ^{p^{n−m}}.

Thus λ is a (p^{ν} − 1)^{th} root of unity for some ν. Since all of the eigenvalues of τ^{p^{ν}−1} are thus 1, we see that τ^{p^{ν}−1} is unipotent; that is, it is the sum of the identity matrix and a nilpotent matrix.

We are now in a position to prove a fundamental result of Grothendieck. We first need a definition.

Definition 1. A Galois representation

ρ : G_{K} → GL_{d}(F)

is said to be semistable if the image of I_{K} is unipotent. ρ is said to be potentially semistable if there is some finite extension L/K such that ρ|_{G_{L}} is semistable.

Proposition 2.2 (Grothendieck). Let ρ : G_{K} → GL_{d}(F) be a Galois representation. Suppose that F and K have different residue characteristics l and p respectively. Then ρ is potentially semistable.

Proof. As before, we can conjugate ρ so as to assume that its image is in GL_{d}(O_{F}); this is permitted because conjugation does not affect the property of being unipotent. Now consider the diagram

\xymatrix{ & 0 \ar[d] \\  & ker \ar[d] \\ G_{K} \ar[r] \ar[dr] & GL_{d}(O_{F}) \ar[d] \\  & GL_{d}(k_{F})}

where k_{F} is the residue field of O_{F} and ker is the kernel of the above map. Since GL_{d}(k_{F}) is finite, ker has finite index in GL_{d}(O_{F}). Thus, since we are allowing finite extensions, we can replace K by the fixed field of ker and thus assume that

G_{K} → GL_{d}(k_{F})

is the trivial representation.

It follows that ρ actually maps G_{K} into the above kernel. But we know that this kernel is pro-l (see Lecture 3, Section 2). In particular, since P_{K} ⊆ G_{K} is pro-p, it must vanish under ρ. Thus we have shown that ρ factors through G_{K}/P_{K} = T_{K}.

Now, the image of I_{K} in T_{K} is just Δ_{k}, where k = k_{K} is the residue field of O_{K}. Δ_{k} is generated by τ^{e_{K}}, where τ is a generator of Δ_{𝔽_{p}} and e_{K} is the tame inertial degree of K/ℚ_{p}. We have seen above that τ^{n} is unipotent for some n which is prime to p. It follows that for all δ ∈ Δ_{k}, δ^{n} is unipotent. One now sees easily that taking L to be the compositum of K with the fixed field of T_{ne_{K},f_{K}}, where f_{K} is the residue degree of K, Δ_{ℓ} ⊆ {δ^{n} | δ ∈ Δ_{k}}, where ℓ is the residue field of L. Thus ρ|_{G_{L}} is semistable, which completes the proof.

The above proof rested very heavily on the fact that l was distinct from p; the case where l = p is very far from settled.

2.3. Further Reductions. Suppose for now that l ≠ p. We will now use the methods of the proof of Proposition 2.1 to get additional reductions on our representations.

Let K/ℚ_{p} be a finite extension and let k be the residue field of K. We have a commutative diagram with exact rows relating the groups T and T_{K}.

\xymatrix{1 \ar[r] & Δ_{k} \ar[r] \ar@{^{(}->}[d]^{e} & T_{K} \ar[r] \ar@{^{(}->}[d] & 𝔤_{k} \ar[r] \ar@{^{(}->}[d]^{f} & 1 \\ 1 \ar[r] & Δ_{𝔽_{p}} \ar[r] & T \ar[r] & 𝔤 \ar[r] & 1}

Here 𝔤_{k} is the absolute Galois group of k, e is the tame inertial degree of K and f is the residue degree of K.

We want to consider only the pro-l part of Δ_{k}. More precisely, let T_{l,K} be the quotient of T_{K} by the maximal non-pro-l subgroup of Δ_{k}. This reduces the exact sequence

1 → ∏_{r≠p} ℤ_{r}(1) → T_{K} → ℤ̂ → 1

to an exact sequence

1 → ℤ_{l}(1) → T_{l,K} → ℤ̂ → 1.

The methods of Proposition 2.1 now give the following result.

Proposition 2.3. Let ρ : G_{K} → GL_{d}(F) be a Galois representation with p ≠ l. Then there exists a finite extension L/K such that ρ|_{G_{L}} factors through T_{l,L} and the image of the inertia group of L acts unipotently.

Proof. The proof is virtually the same as that of Proposition 2.1.

We make one final comment on reducing the domain of Galois representations. Let ρ : G_{K} → GL_{d}(O_{F}) be a Galois representation, and consider the residual representation \overline{ρ} : G_{K} → GL_{d}(k_{F}). Let G_{K}^{0} be the kernel of \overline{ρ}; G_{K}^{0} is also the set of elements of G_{K} such that ρ(g) is in the kernel of the natural map

GL_{d}(O_{F}) → GL_{d}(k_{F}).

Then G_{K}^{0} is an open, normal subgroup of G_{K} of finite index. We let G_{K}^{0,l} be its pro-l completion; that is, G_{K}^{0,l} is the projective limit of all finite quotients of G_{K}^{0} of order a power of l. Since G_{K}^{0} is already profinite, there is a natural surjection G_{K}^{0} ↠ G_{K}^{0,l}; let H be the kernel.

1 → H → G_{K}^{0} → G_{K}^{0,l} → 1.

Now, ρ must vanish on H, since H is completely non-pro-l and ρ maps it into the pro-l kernel of GL_{d}(O_{F}) → GL_{d}(k_{F}). Thus ρ factors as a map

G_{K}/H → GL_{d}(O_{F}).

But G_{K}/H is simply an extension of the pro-l group GL^{0,l} by the finite group G_{K}/G_{K}^{0}. In this way one can often gain more control over the Galois representation ρ.