LOCAL CLASS FIELD THEORY

1) Suppose G is a profinite group and M is a discrete G-module. Suppose further that M = M_{0} ⊃ M_{1} ⊃ M_{2}... are G-submodules with ⋂_{i} M_{i} = (0) and that M is complete with respect to the topology induced by the {M_{i}} (i.e. if m_{i} ∈ M satisfy m_{i} − m_{j} ∈ M_{i} for all j ≥ i then there exists m ∈ M such that m − m_{i} ∈ M_{i} for all i). If H^{i}(G, M_{j}/M_{j+1}) = (0) for all j ≥ 0, show that H^{i}(G, M) = (0). [Hint: Let ϕ ∈ Z^{i}(G, M). Show by induction on r that one can find ψ_{r} ∈ C^{i−1}(G, M) with ψ_{r} − ψ_{r−1} ∈ C^{i−1}(G, M_{r−1}) and dψ_{r} ≡ ϕ mod M_{r}.]

2) Suppose that G is a profinite group, H is a closed subgroup and I is a normal closed subgroup. Suppose also that M is a discrete G/I-module.

(a) Show that

H^{i}(G/I, M) ⟶^{Inf} H^{i}(G, M)
↓  ↓
H^{i}(H/I ∩ H, M) ⟶^{Inf} H^{i}(H, M)

is commutative (where the vertical maps are restriction).

(b) Suppose also that H is open in G and acts trivially on M. Show that

H^{i}(G/I, M) ⟶^{Inf} H^{i}(G, M)
↑  ↑
H^{i}(H/I ∩ H, M) ⟶^{Inf} H^{i}(H, M)

is commutative, where the left vertical map is corestriction and the right vertical map is #H/H ∩ I times corestriction.

3) Suppose that G is a profinite group and that H is an open subgroup of G. Suppose also that t : H\G → G is a section to the natural projection.

(a) Suppose that M is a discrete G-module and that ϕ ∈ Z^{1}(H, M). Define Φ : G → M by

Φ(g) = ∑_{x∈H\G} t(x)^{−1}ϕ(t(x)gt(xg)^{−1}).

Show that Φ ∈ Z^{1}(G, M) and that [Φ] = cor_{H}^{G}[ϕ] in H^{1}(G, M).

(b) Show that

V (g) = ∏_{x∈H\G} t(x)gt(xg)^{−1}

defines a homomorphism (the ‘transfer’) V : G^{ab} → H^{ab} such that

ϕ(V (g)) = (cor_{H}^{G}ϕ)(g)

for all g ∈ G and ϕ ∈ H^{1}(H, ℚ/ℤ) = Hom (H, ℚ/ℤ).

4) (a) Suppose that D is a division algebra with centre K and let L be a maximal subfield of D which is Galois over K. If N_{D^{×}}(L) denotes the normaliser of L in D show that N_{D^{×}}(L)/L^{×} →^{∼} Gal (L/K) (via the map that sends n to the automorphism x ↦ nxn^{−1} of L). For τ ∈ Gal (L/K) choose a lifting n(τ) of τ to N_{D^{×}}(L). Show that the n(τ) form a basis of D as an L-vector space (via right multiplication).

(b) Now suppose that Gal (L/K) is cyclic with generator σ. Choose n ∈ N_{D^{×}}(L) lifting σ and set n(σ^{i}) = n^{i} for i = 0, ..., [L : K] − 1. If we identify D ≅ (⨁_{τ∈Gal (L/K)} n(τ)K) ⊗_{K} L then show that conjugation by n(τ) (on D) corresponds to 1 ⊗ τ. This choice of basis also allows us to identify End_{L}(D) with M_{[L:K]}(L). Under the identifications

D ⊗_{K} L ≅ End_{L}(D) ≅ M_{[L:K]}(L)

show that the action of τ on M_{[L:K]}(L) corresponds to the automorphism of D ⊗_{K} L given by

d ⊗ b ⟼ n(τ)dn(τ)^{−1} ⊗ τ(b).

(c) Show further that

ϕ : (τ, τ′) ⟼ n(τ)n(τ′)n(ττ′)^{−1}

is an element of Z^{2}(Gal (L/K), L^{×}) and that [ϕ] ∈ H^{2}(Gal (L/K), L^{×}) inflates to the element of H^{2}(Gal (\overline{K}/K), \overline{K}^{×}) which represents D. [You might be interested to note that [ϕ] is the class classifying the extension

(0) ⟶ L^{×} ⟶ N_{D^{×}}(L) ⟶ Gal (L/K) ⟶ (0).]

5) (a) Let K/ℚ_{p} be a finite extension and let D be a division algebra with centre K and dimension n^{2} over K. Show that the valuation v_{K} extends to a unique homomorphism v_{K} : D^{×} → → \frac{1}{e}ℤ such that v_{K}(a + b) ≥ min\{v_{K}(a), v_{K}(b)\} for all a, b ∈ D (with the convention that v_{K}(0) = ∞). Let O_{D} (resp. P_{D}) denote the set of elements a ∈ D with v_{K}(a) ≥ 0 (resp. > 0). Show that O_{D} is a (non-commutative) ring, that O_{D} is a finite O_{K}-module, that O_{D}[1/p] = D and that P_{D} is a two sided ideal of O_{D}. Show further that O_{D}/O_{P} is a finite field extension of k(K) of degree f say.

(b) Let L/K denote the unramified extension of degree f. Show that L embeds into D over K. [Hint: Consider K(a) where a ∈ O_{D} lifts a primitive element of O_{D}/P_{D}.] Let Π ∈ D have valuation 1/e. Show that O_{D} = O_{L}[Π] and that

dim_{K} D ≤ ef.

Show also that e^{2}| dim_{K} D and f^{2}| dim_{K} D. Deduce that e = f = n and that K(Π) is a totally ramified field extension of K of degree n.

(c) Show that N_{D^{×}}(L)/L^{×} ≅ Gal (L/K) and that Inv_{K}(D) is the image of Frob_{K}^{−1} under the composite

Gal (L/K) ≅ N_{D^{×}}(L)/L^{×} ⟶^{v_{K}} \frac{1}{n}ℤ/ℤ.

6) Suppose that for each finite extension K/ℚ_{p} we are given an isomorphism

s_{K} : K^{×} ⟶^{∼} W_{K}^{ab}

with the following properties:

1. If a ∈ K^{×} and v_{K}(a) = m then s_{K}(a) maps to Frob_{K}^{m} ∈ Gal (\overline{k(K)}/k(K)).

2. If L/K is a finite extension then for b ∈ L^{×} we have

s_{K}(N_{L/K}b) = s_{L}(b)|_{K^{ab}}.

3. If L/K is a finite, abelian, Galois extension then s_{K} induces an isomorphism

s_{K} : K^{×}/N_{L/K}L^{×} ⟶^{∼} Gal (L/K).

The purpose of this question is to prove that s_{K} = r_{K} (the Artin map) for all K.

(a) Show that it suffices to prove that for every finite, totally ramified, abelian extension L/K the two maps

r_{K}, s_{K} : O_{K}^{×} ⟶ Gal (L/K)

are equal. [Hint: Gal (L^{nr}/K) ≅ Gal (L/K) × Gal (K^{nr}/K).]

(b) Let L/K be a finite, totally ramified, abelian extension. If σ ∈ Gal (L/K) let L_{σ} ⊂ L^{nr} denote the fixed field of

(σ, Frob_{K}) ∈ Gal (L/K) × Gal (K^{nr}/K) ≅ Gal (L^{nr}/K).

If L_{σ} = L_{τ} show that σ = τ. Also show that [L_{σ} : K] = [L : K].

(c) Keep the notation of the previous part and let ϖ denote a uniformiser in O_{L}. If a ∈ O_{K}^{×}, show that

N_{L_{s_{K} (a)}/K}L_{s_{K}(a)}^{×} = ⟨N_{L/K}O_{L}^{×}, aN_{L/K}ϖ⟩.

Deduce that s_{K}(a) = r_{K}(a).

7) Suppose that K/ℚ_{p} is a finite extension.

(a) If L/K is a finite Galois extension, show that the cokernel of the natural map W_{L}^{ab} → W_{K}^{ab} is naturally identified with Gal (L/K)^{ab}. Deduce that r_{K} induces an isomorphism

r_{K} : K^{×}/N_{L/K}L^{×} ⟶^{∼} Gal (L/K)^{ab}.

Let L ⊃ M ⊃ K be the maximal subextension with M/K Galois and abelian. Show that N_{L/K}L^{×} = N_{M/K}M^{×}.

(b) Now suppose that L/K is any finite extension and let L ⊃ M ⊃ K be the maximal subextension with M/K Galois and abelian. Show that N_{L/K}L^{×} = N_{M/K}M^{×}. [Hint: Choose a finite Galois extension N/K containing L. Let G = Gal (N/K) and let H = Gal (N/L). Show that M is the fixed field of H[G, G]. Show further that if a ∈ N_{M/K}M^{×} then r_{K}(a) ∈ G^{ab} is in the image of H^{ab} → G^{ab}. Deduce that we can find b ∈ L^{×} such that r_{K}(a) = r_{K}(N_{L/K}b) in G^{ab}.]

8)(Serre) Let ϖ = \sqrt{−1} − 1 and

A = ⟨2, U_{ℚ_{2}(ϖ)}^{4}⟩ ⊂ ℚ_{2}(ϖ)^{×}.

(a) Show that ℚ_{2}(ϖ)^{×}/A is cyclic of order 8 generated by ϖ.

(b) Let L/ℚ_{2}(ϖ) be the abelian extension corresponding to A. Show that L/ℚ_{2} is Galois. Show also that the transfer map V : Gal (L/ℚ_{2})^{ab} → Gal (L/ℚ_{2}(ϖ)) factors as

Gal (L/ℚ_{2}) → → Gal (ℚ_{2}(ϖ)/ℚ_{2}) ↪ Gal (L/ℚ_{2}(ϖ)).

(c) Show that the maximal unramified subextension of L/ℚ_{2} is K = ℚ_{2}(\sqrt{5}). Also show that Gal (L/K) is isomorphic to the quaternion group {±1, ±i, ±j, ±k}. Write Z = {±1} for the centre of Gal (L/K).

(d) Calculate the upper numbering filtration on Gal (L/ℚ_{2}(ϖ)) and hence also calculate the lower numbering filtration. Show that Gal (L/K(ϖ))_{i} has order 4 for i = 1, 2; has order 2 for i = 3, 4; and has order 1 for i > 4.

(e) Deduce that Gal (L/K)_{i} is Gal (L/K) for i = 1, 2; is Z for i = 3, 4; and = {1} for i > 4.