Algebraic Number Theory

Chapter 9
Local class field theory

9.1. The Brauer group of a local field

Fix a local field $K$ and a separable closure $K^{sep}$ of $K$ . All separable extensions of $K$ will be supposed to lie in $K^{sep}$ . In this section, we construct invariant maps ${inv}_{L} : Br (L) \to ℚ ∕ ℤ$ for finite separable extensions $L ∕ K$ such that $({(K^{sep})}^{\times}, inv)$ forms a class formation for $K$ .

Notation 9.1.1. §

For a Galois extension $E ∕ F$ of fields, we set

Br (E ∕ F) = H^{2} (Gal (E ∕ F), E^{\times}) .

We construct ${inv}_{K}$ by first defining it on the subgroup $Br (K^{ur} ∕ K)$ of $Br (K)$ corresponding to the maximal unramified extension $K^{ur}$ of $K$ . For this, note that the unique extension $w_{K} : {(K^{ur})}^{\times} \to ℤ$ of the additive valuation on $K$ to $K^{ur}$ is a map of $Gal (K^{ur} ∕ K)$ -modules, hence induces a map

w_{K}^{*} : Br (K^{ur} ∕ K) \to H^{2} (Gal (K^{ur} ∕ K), ℤ),

where we identify $Gal (K^{ur} ∕ K)$ with $\hat{ℤ}$ via the isomorphism taking the Frobenius element to $1$ .

Definition 9.1.2. §

The invariant map ${inv}_{K^{ur} ∕ K} : Br (K^{ur} ∕ K) \to ℚ ∕ ℤ$ is the composition

Br (K^{ur} ∕ K) \to_{}^{w_{K}^{*}} H^{2} (Gal (K^{ur} ∕ K), ℤ) \to_{}^{δ^{- 1}} H^{1} (Gal (K^{ur} ∕ K), ℚ ∕ ℤ) \to_{}^{{ev}_{φ_{K}}} ℚ ∕ ℤ,

where $δ$ is the connecting homomorphism arising from $0 \to ℤ \to ℚ \to ℚ ∕ ℤ \to 0$ , and ${ev}_{φ_{K}}$ is evaluation at the Frobenius $φ_{K}$ in $Gal (K^{ur} ∕ K)$ .

We will show that ${inv}_{K^{ur} ∕ K}$ is an isomorphism, which amounts to showing that $w_{K}^{*}$ is an isomorphism. We require a preliminary lemma.

Lemma 9.1.3. §

Let $G$ be a finite group, and let $M$ be a $G$ -module such that there exists a decreasing sequence ${(M_{n})}_{n \geq 0}$ with $M_{0} = M$ of $G$ -submodules of $M$ for which

M = \underset{n}{lim_{\leftarrow}} M ∕ M_{n} .

Let $i \geq 0$ , and suppose that $H^{i} (G, M_{n} ∕ M_{n + 1}) = 0$ for all $n \geq 0$ . Then $H^{i} (G, M) = 0$ as well.

Proof.

Let $f \in Z^{i} (G, M)$ . Suppose that we have inductively defined $f_{n} \in Z^{i} (G, M_{n})$ and $h_{j} \in C^{i - 1} (G, M_{j})$ for $0 \leq j \leq n - 1$ such that

f = f_{n} + \sum_{j = 0}^{n - 1} d^{i - 1} (h_{j}) .

(Note that we take $C^{- 1} (G, M_{j}) = 0$ .) The image of $f_{n}$ in $Z^{i} (G, M_{n} ∕ M_{n + 1})$ is a coboundary by assumption, say of ${\bar{h}}_{n} \in C^{i - 1} (G, M_{n} ∕ M_{n + 1})$ . Lifting ${\bar{h}}_{n}$ to any $h_{n} \in C^{i - 1} (G, M_{n})$ , we then set $f_{n + 1} = f_{n} - d^{i - 1} (h_{i})$ . Since $M = {lim_{\leftarrow}}_{n} M ∕ M_{n}$ , the sequence of partial sums $\sum_{j = 1}^{n} h_{j}$ converges to an element of $C^{i - 1} (G, M)$ with coboundary $f$ . □

Proposition 9.1.4. §

Let $L$ be a finite Galois extension of $K$ . Then there exists an open $Gal (L ∕ K)$ -submodule $V$ of $𝒪_{L}^{\times}$ that is cohomologically trivial.

Proof.

Let $G = Gal (L ∕ K)$ . By the normal basis theorem, there exists $α \in L$ such that ${σ (α) ∣ σ \in G}$ forms a $K$ -basis of $L$ . By multiplying by an element $K^{\times}$ , we may suppose that $α \in 𝒪_{L}$ . Let $Λ^{'}$ be the $𝒪_{K}$ -lattice in $L$ spanned by the $σ (α)$ . Let $π$ be a uniformizer in $𝒪_{K}$ . Then $[𝒪_{L} : Λ^{'}]$ is finite, so $π^{n} 𝒪_{L} \subseteq Λ^{'}$ for $n$ sufficiently large. Set $Λ = π^{n + 1} Λ^{'}$ . Then

Λ \cdot Λ = π^{2 (n + 1)} Λ^{'} \subseteq π^{2 (n + 1)} 𝒪_{L} \subseteq π^{n + 2} 𝒪_{L} \subseteq π Λ .

Then $V = 1 + Λ$ is a $G$ -submodule of $𝒪_{L}^{\times}$ , and $V$ in turn has a decreasing filtration $V_{i} = 1 + π^{i} Λ$ for $i \geq 0$ of $G$ -submodules. We have isomorphisms

V_{i} ∕ V_{i + 1} \to_{}^{\sim} Λ ∕ π Λ, (1 + π^{i} λ) V_{i + 1} \mapsto λ + π Λ .

Since $Λ ∕ π Λ$ is a free $𝔽_{p} [G]$ -module, it is induced, so cohomologically trivial. Lemma 9.1.3 then tells us, in particular, that $H^{i} (G_{p}, V) = 0$ for all $i \geq 1$ for each Sylow subgroup $G_{p}$ of $G$ , and the cohomological triviality then follows from Theorem 1.11.11. □

We use Proposition 9.1.4 first to study the case of cyclic, and then more specifically, unramified extensions.

Corollary 9.1.5. §

Let $L$ be a finite cyclic extension of $K$ . Then the Herbrand quotient of $𝒪_{L}^{\times}$ with respect to group $Gal (L ∕ K)$ is $1$ .

Proof.

Let $V$ be as in Proposition 9.1.4. The exact sequence

1 \to V \to 𝒪_{L}^{\times} \to 𝒪_{L}^{\times} ∕ V \to 1

gives rise to the identity of Herbrand quotients

h (𝒪_{L}^{\times}) = h (V) h (𝒪_{L}^{\times} ∕ V) = 1

since $𝒪_{L}^{\times} ∕ V$ is finite as $V$ is open. □

Corollary 9.1.6. §

Let $L ∕ K$ be a finite unramified extension. Then $𝒪_{L}^{\times}$ is a cohomologically trivial $Gal (L ∕ K)$ -module.

Proof.

It clearly suffices to show that ${\hat{H}}^{i} (G, 𝒪_{L}^{\times}) = 0$ for $G = Gal (L ∕ K)$ and all $i$ , since any subgroup of $G$ is the Galois group of an unramified extension of local fields. The additive valuation $v_{L}$ on $𝒪_{L}$ restricts to the valuation $v_{K}$ on $𝒪_{L}$ since $L ∕ K$ is unramified. The short exact sequence

1 \to 𝒪_{L}^{\times} \to L^{\times} \to_{}^{v_{L}} ℤ \to 0

then gives rise to a long exact sequence starting

1 \to 𝒪_{K}^{\times} \to K^{\times} \to_{}^{v_{K}} ℤ \to H^{1} (G, 𝒪_{L}^{\times}) \to H^{1} (G, L^{\times}),

with the last group zero by Hilbert’s Theorem 90 and the map $v_{K}$ surjective. Thus $H^{1} (G, 𝒪_{L}^{\times}) = 0$ , and since $L ∕ K$ is cyclic, the result follows from the triviality of the Herbrand quotient and the periodicity of Tate cohomology. □

Proposition 9.1.7. §

The invariant map ${inv}_{K^{ur} ∕ K} : Br (K^{ur} ∕ K) \to ℚ ∕ ℤ$ is an isomorphism.

Proof.

For any $i \geq 1$ , Proposition 9.1.6 implies that

H^{i} (Gal (K^{ur} ∕ K), 𝒪_{K^{ur}}^{\times}) ≅ \underset{n}{lim_{\to}} H^{i} (Gal (K_{n} ∕ K), 𝒪_{K_{n}}^{\times}) = 0

where $K_{n}$ is the unique unramified extension of $K$ (in $K^{sep}$ ) of degree $n$ . The valuation map yields an exact sequence

1 \to 𝒪_{K^{ur}}^{\times} \to {(K^{ur})}^{\times} \to_{}^{w_{K}} ℤ \to 0,

we therefore have that

w_{K}^{*} : Br (K^{ur} ∕ K) \to H^{2} (Gal (K^{ur} ∕ K), ℤ)

is an isomorphism. The other maps in the definition of ${inv}_{K^{ur} ∕ K}$ are clearly isomorphisms, so the result holds. □

Notation 9.1.8. §

For a finite separable extension $L$ of $K$ , we use ${Res}_{L ∕ K}$ to denote the map

{Res}_{L ∕ K} : Br (K^{ur} ∕ K) \to Br (L^{ur} ∕ L)

defined by the compatible pair consisting of restriction $Gal (L^{ur} ∕ L) \to Gal (K^{ur} ∕ K)$ and the inclusion ${(K^{ur})}^{\times} \to {(L^{ur})}^{\times}$ .

Remark 9.1.9. §

For $L ∕ K$ finite separable, the map ${Res}_{L ∕ K}$ fits into a commutative diagram

Restriction on unramified and absolute Brauer groups. A full diagram description follows.

Diagram description: Restriction on unramified and absolute Brauer groups

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Br (K superscript (ur) / K); column 2: Br (L superscript (ur) / L).
Row 2, from left to right: column 1: Br (K); column 2: Br (L).

Arrows and lines:

An arrow from Br (K superscript (ur) / K) to Br (L superscript (ur) / L), labelled Res subscript (L / K).
An arrow from Br (K superscript (ur) / K) to Br (K), labelled Inf.
An arrow from Br (L superscript (ur) / L) to Br (L), labelled Inf.
An arrow from Br (K) to Br (L), labelled Res subscript (L / K).

The following describes how our invariant map behaves after finite extension of the base field.

Proposition 9.1.10. §

Let $L$ be a finite separable extension of $K$ . Then

{inv}_{L^{ur} ∕ L} \circ {Res}_{L ∕ K} = [L : K] {inv}_{K^{ur} ∕ K} .

Proof.

We claim that the diagram

Restriction and the local invariant map. A full diagram description follows.

Diagram description: Restriction and the local invariant map

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Br (K superscript (ur) / K); column 2: H superscript (2)( Gal (K superscript (ur) / K), blackboard Z ); column 3: H superscript (1)( Gal (K superscript (ur) / K), blackboard Q / blackboard Z ); column 4: blackboard Q / blackboard Z.
Row 2, from left to right: column 1: Br (L superscript (ur) / L); column 2: H superscript (2)( Gal (L superscript (ur) / L), blackboard Z ); column 3: H superscript (1)( Gal (L superscript (ur) / K), blackboard Q / blackboard Z ); column 4: blackboard Q / blackboard Z.

Arrows and lines:

An arrow from Br (K superscript (ur) / K) to Br (L superscript (ur) / L), labelled Res subscript (L / K).
An arrow from Br (K superscript (ur) / K) to H superscript (2)( Gal (K superscript (ur) / K), blackboard Z ), labelled w subscript (K) superscript (star).
An arrow from H superscript (2)( Gal (K superscript (ur) / K), blackboard Z ) to H superscript (2)( Gal (L superscript (ur) / L), blackboard Z ), labelled e subscript (L / K) Res subscript (L / K).
An arrow from H superscript (2)( Gal (K superscript (ur) / K), blackboard Z ) to H superscript (1)( Gal (K superscript (ur) / K), blackboard Q / blackboard Z ), labelled delta superscript (minus 1).
An arrow from H superscript (1)( Gal (K superscript (ur) / K), blackboard Q / blackboard Z ) to H superscript (1)( Gal (L superscript (ur) / K), blackboard Q / blackboard Z ), labelled e subscript (L / K) Res subscript (L / K).
An arrow from H superscript (1)( Gal (K superscript (ur) / K), blackboard Q / blackboard Z ) to blackboard Q / blackboard Z (row 1, column 4), labelled ev subscript (varphi subscript (K)).
An arrow from blackboard Q / blackboard Z (row 1, column 4) to blackboard Q / blackboard Z (row 2, column 4), labelled [L:K].
An arrow from Br (L superscript (ur) / L) to H superscript (2)( Gal (L superscript (ur) / L), blackboard Z ), labelled w subscript (L) superscript (star).
An arrow from H superscript (2)( Gal (L superscript (ur) / L), blackboard Z ) to H superscript (1)( Gal (L superscript (ur) / K), blackboard Q / blackboard Z ), labelled delta superscript (minus 1).
An arrow from H superscript (1)( Gal (L superscript (ur) / K), blackboard Q / blackboard Z ) to blackboard Q / blackboard Z (row 2, column 4), labelled ev subscript (varphi subscript (L)).

commutes (where ${Res}_{L ∕ K}$ in the middle two arrows denotes the corresponding composition of restriction and inflation), from which the result follows. The commutativity of the middle square is straightforward. Since the restriction of $w_{L}$ to ${(K^{ur})}^{\times}$ is $e_{L ∕ K} w_{K}$ , the leftmost square commutes. Since the restriction of $φ_{L}$ to $K^{ur}$ is the $f_{L ∕ K}$ -power of $φ_{K}$ , the rightmost square commutes. □

Having defined the invariant map on $Br (K^{ur} ∕ K)$ and shown that it satisfied the desired property with respect to change of base field, our next goal is to show that the inflation map

Inf : Br (K^{ur} ∕ K) \to Br (K)

is an isomorphism. At the finite level, we note the following.

Corollary 9.1.11. §

Let $L ∕ K$ be a finite Galois extension of local fields, and set

Br {(L ∕ K)}^{ur} = Br (K^{ur} ∕ K) \cap Br (L ∕ K) \leq Br (K) .

Then $Br {(L ∕ K)}^{ur}$ is cyclic of order $[L : K]$ .

Proof.

By the inflation-restriction sequence for Brauer groups, we have

Br {(L ∕ K)}^{ur} = {α \in Br (K^{ur} ∕ K) ∣ {Res}_{L ∕ K} (α) = 0} .

By Proposition 9.1.10, this coincides with the kernel of $[L : K]$ on $Br (K^{ur} ∕ K)$ , which is cyclic of order $n$ . □

We require a special case of the following cohomological lemma.

Lemma 9.1.12. §

Let $G$ be a finite group, let $A$ be a $G$ -module, and let $i, r \geq 0$ . Suppose that for all subgroups $H$ of $G$ , we have that ${\hat{H}}^{j} (H, A) = 0$ for all $1 \leq j \leq i - 1$ and that the order ${\hat{H}}^{i} (H ∕ K, A^{K})$ divides ${[H : K]}^{r}$ for all normal subgroups $K$ of $H$ of prime index. Then the order of ${\hat{H}}^{i} (G, A)$ divides $| G |^{r}$ .

Proof.

If we replace $G$ by a Sylow $p$ -subgroup for a prime $p$ , then the conditions of the lemma are still satisfied. By Corollary 1.8.24, we see that $| H^{i} (G, A) |$ divides $\prod_{p} | H^{i} (G_{p}, A) |$ , which if we prove the lemma for each $G_{p}$ will divide $\prod_{p} | G_{p} |^{r} = | G |^{r}$ .

Thus, we can and do assume that $G$ is a $p$ -group. Let $H$ be a normal subgroup of $G$ of index $p$ . By hypothesis, we have that $| {\hat{H}}^{i} (G ∕ H, A^{H}) |$ divides $p^{r}$ , and we may suppose by induction on the order of $G$ that $| {\hat{H}}^{i} (H, A) |$ divides $| H |^{r}$ . If $i \geq 1$ , then by the triviality of $H^{j} (H, A)$ for $1 \leq j \leq i - 1$ , we have an exact inflation-restriction sequence

0 \to H^{i} (G ∕ H, A^{H}) \to H^{i} (G, A) \to H^{i} (H, A),

so the order of $H^{i} (G, A)$ divides $| G |^{r} = {(p | H |)}^{r}$ . For $i = 0$ , we merely replace the inflation-restriction sequence with the exact sequence

{\hat{H}}^{0} (H, A) \to_{}^{Cor} {\hat{H}}^{0} (G, A) \to {\hat{H}}^{0} (G ∕ H, A^{H}),

where we recall that corestriction in degree $0$ coincides with the sum over left coset representatives of $G ∕ H$ . □

Theorem 9.1.13. §

The inflation map $Inf : Br (K^{ur} ∕ K) \to Br (K)$ is an isomorphism.

Proof.

It suffices to see that $Br {(L ∕ K)}^{ur} = Br (L ∕ K)$ for every finite Galois extension $L ∕ K$ , since the union under (injective) inflation maps of the groups $Br {(L ∕ K)}^{ur}$ is $Br (K^{ur} ∕ K)$ and the union under inflation maps of the groups $Br (L ∕ K)$ is $Br (K)$ . For this, it suffices by Corollary 9.1.11 to show that $Br (L ∕ K)$ has order dividing $n = [L : K]$ .

First, suppose that $L ∕ K$ is cyclic. We consider Herbrand quotients for $G = Gal (L ∕ K)$ . The exact sequence defined by the valuation on $L$ yields

h (L^{\times}) = h (𝒪_{L}^{\times}) h (ℤ) .

We have $h (𝒪_{L}^{\times}) = 1$ by Lemma 9.1.5, while $h (ℤ) = n$ since ${\hat{H}}^{0} (G, ℤ) ≅ℤ ∕ 𝑛ℤ$ and ${\hat{H}}^{- 1} (G, ℤ) = 0$ . Since $h_{1} (L^{\times}) = 1$ by Hilbert’s Theorem 90, we have $| Br (L ∕ K) | = h_{0} (L^{\times}) = n$ .

Now take $L ∕ K$ to be any finite Galois extension. With $G = Gal (L ∕ K)$ and $A = L^{\times}$ , the hypotheses of Lemma 9.1.12 are satisfied with $i = 2$ and $r = 1$ by Hilbert’s Theorem 90 and the case of cyclic extensions. Consequently, $Br (L ∕ K)$ has order dividing $n$ , as we aimed to show. □

By Theorem 9.1.13, we may make the following definition of the invariant map for $K$ (and hence for any local field).

Definition 9.1.14. §

The invariant map ${inv}_{K} : Br (K) \to ℚ ∕ ℤ$ for a local field $K$ is the composition

{inv}_{K} : Br (K) \to_{}^{{Inf}^{- 1}} Br (K^{ur} ∕ K) \to_{}^{{inv}_{K^{ur} ∕ K}} ℚ ∕ ℤ .

Theorem 9.1.15. §

The pair $({(K^{sep})}^{\times}, inv)$ is a class formation for $K$ .

Proof.

For $L ∕ K$ finite separable, we have $H^{1} (G_{L}, {(K^{sep})}^{\times}) = 0$ by Hilbert’s Theorem 90. The invariant map ${inv}_{L}$ is an isomorphism by Theorem 9.1.13 and Proposition 9.1.7. Moreover, we have

{inv}_{L} \circ {Res}_{L ∕ K} = [L : K] {inv}_{K}

as a consequence of Proposition 9.1.10, noting Remark 9.1.9. Thus, the axioms of a class formation are satisfied. □

9.2. Local reciprocity

We continue to let $K$ denote a local field and $K^{sep}$ a separable closure of $K$ .

Definition 9.2.1. §

The (local) reciprocity map for $K$ is the reciprocity map $ρ_{K} : K^{\times} \to G_{K}^{ab}$ attached to the class formation $({(K^{sep})}^{\times}, inv)$ of Theorem 9.1.15.

Let us proceed directly to the statement of the main theorem.

Theorem 9.2.2 (Local reciprocity). §

Let $K$ be a nonarchimedean local field. Then the local reciprocity map

ρ_{K} : K^{\times} \to G_{K}^{ab}

satisfies

i.

for each uniformizer $π$ of $K$ , the element $ρ_{K} (π)$ is a Frobenius element in $G_{K}^{ab}$ , and

ii.

for any finite abelian extension $L$ of $K$ , the map

ρ_{L ∕ K} : K^{\times} \to Gal (L ∕ K)

defined by $ρ_{L ∕ K} (a) = ρ_{K} (a) |_{L}$ for all $a \in K^{\times}$ is surjective with kernel $N_{L ∕ K} L^{\times}$ .

Proof.

By Theorem 8.1.13, the reciprocity map $ρ_{K}$ satisfies (ii). We show that it satisfies (i). For this, take any finite unramified extension $L ∕ K$ , and let $G = Gal (L ∕ K)$ . Let $φ$ denote the Frobenius element in $G$ . Let $χ : G \to ℚ ∕ ℤ$ be an injective homomorphism. It suffices to show that $χ (ρ_{L ∕ K} (π)) = χ (φ)$ . By Proposition 8.1.10, we have

χ (ρ_{L ∕ K} (π)) = {inv}_{L ∕ K} (π \cup δ (χ)),

where $δ$ is the connnecting homomorphisms for $0 \to ℤ \to ℚ \to ℚ ∕ ℤ \to 0$ . For the valuation $v_{L}$ on $L$ , we have

v_{L}^{*} (π \cup δ (χ)) = v_{L} (π) \cup δ (χ) = δ (χ) .

Since ${inv}_{L ∕ K} = {ev}_{φ} \circ δ^{- 1} \circ v_{L}^{*}$ by definition, we have

{inv}_{L ∕ K} (π \cup δ (χ)) = {ev}_{φ} (χ) = χ (φ) .

□

Remark 9.2.3. §

Theorem 9.2.2 is also referred to as the local reciprocity law.

We can quickly see a connection with class field theory over a finite field.

Proposition 9.2.4. §

Let $K$ be a nonarchimedean local field and $π$ a uniformizer of $K$ . Let

ι_{π} : ℤ \to ⟨ π ⟩

denote the isomorphism that sends $1$ to $π$ . Let

Res : G_{K}^{ab} \to G_{κ (K)}

be induced by the restriction map to $Gal (K^{ur} ∕ K)$ and its natural isomorphism with $G_{κ (K)}$ , as in Proposition 6.4.10. Then

Res \circ ρ_{K} \circ ι_{π} : ℤ \to G_{κ (K)}

is the reciprocity map $ρ_{κ (K)}$ for the finite field $κ (K)$ .

Remark 9.2.5. §

Given a nonarchimedean local field $K$ and a finite abelian extension $L$ of $K$ , we at times also denote by $ρ_{L ∕ K}$ the induced isomorphism

ρ_{L ∕ K} : K^{\times} ∕ N_{L ∕ K} L^{\times} \to_{}^{\sim} Gal (L ∕ K)

and refer to it also as the local reciprocity map for $L ∕ K$ .

Remark 9.2.6. §

In the case $K$ is $ℝ$ or $ℂ$ , we can also define a reciprocity map. In the case of $ℂ$ , the group $G_{ℂ}$ is trivial, so the reciprocity map is trivial $ρ_{ℂ} : ℂ^{\times} \to 1$ . In the case of $ℝ$ , it is the unique homomorphism

ρ_{ℝ} : ℝ^{\times} \to Gal (ℂ ∕ ℝ)

with kernel the positive reals $ℝ_{> 0}$ .

We remark that the following compatibilities among local reciprocity maps follow immediately from Proposition 8.1.18.

Proposition 9.2.7. §

Let $K$ be a local field, and let $L ∕ K$ be a finite separable extension. Then we have commutative diagrams

\begin{array}{rcl}  \end{array}

Local reciprocity: norm and restriction. A full diagram description follows.

Local reciprocity: inclusion and transfer. A full diagram description follows.

Local reciprocity: conjugation. A full diagram description follows.

Diagram description: Local reciprocity: norm and restriction

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: L superscript (times); column 2: G subscript (L) superscript (ab).
Row 2, from left to right: column 1: K superscript (times); column 2: G subscript (K) superscript (ab).

Arrows and lines:

An arrow from L superscript (times) to G subscript (L) superscript (ab), labelled rho subscript (L).
An arrow from L superscript (times) to K superscript (times), labelled N subscript (L / K).
An arrow from G subscript (L) superscript (ab) to G subscript (K) superscript (ab), labelled R subscript (L / K).
An arrow from K superscript (times) to G subscript (K) superscript (ab), labelled rho subscript (K).

Diagram description: Local reciprocity: inclusion and transfer

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: K superscript (times); column 2: G subscript (K) superscript (ab).
Row 2, from left to right: column 1: L superscript (times); column 2: G subscript (L) superscript (ab).

Arrows and lines:

An arrow from K superscript (times) to G subscript (K) superscript (ab), labelled rho subscript (K).
An arrow from K superscript (times) to L superscript (times), labelled i subscript (L / K).
An arrow from G subscript (K) superscript (ab) to G subscript (L) superscript (ab), labelled V subscript (L / K).
An arrow from L superscript (times) to G subscript (L) superscript (ab), labelled rho subscript (L).

Diagram description: Local reciprocity: conjugation

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: L superscript (times); column 2: G subscript (L) superscript (ab).
Row 2, from left to right: column 1: sigma (L) superscript (times); column 2: G subscript (sigma (L)) superscript (ab).

Arrows and lines:

An arrow from L superscript (times) to G subscript (L) superscript (ab), labelled rho subscript (L).
An arrow from L superscript (times) to sigma (L) superscript (times), labelled sigma.
An arrow from G subscript (L) superscript (ab) to G subscript (sigma (L)) superscript (ab), labelled sigma superscript (star).
An arrow from sigma (L) superscript (times) to G subscript (sigma (L)) superscript (ab), labelled rho subscript (sigma (L)).

where $R_{L ∕ K}$ denotes the restriction map, $i_{L ∕ K}$ is the inclusion map, $V_{L ∕ K}$ is the transfer map, and for any embedding $σ : L ↪ K^{sep}$ , the map $σ^{*}$ is induced by conjugation $τ \mapsto 𝜎𝜏 σ^{- 1}$ for $τ \in G_{L}$ .

It follows immediately from Theorem 9.2.2(ii), or Corollary 8.2.5, that we have

[L : K] = [K^{\times} : N_{L ∕ K} L^{\times}]

(9.2.1)

for any finite abelian extension $L$ of $K$ in $K^{sep}$ . Moreover, in the present context, Proposition 8.2.6 says the following.

Proposition 9.2.8. §

Let $K$ be a local field, and let $L$ and $M$ finite abelian extensions of $K$ . Then we have the following:

a.: $N_{L ∕ K} L^{\times} \cap N_{M ∕ K} M^{\times} = N_{𝐿𝑀 ∕ K} {(𝐿𝑀)}^{\times}$ ,
b.: $N_{L ∕ K} L^{\times} \cdot N_{M ∕ K} M^{\times} = N_{(L \cap M) ∕ K} {(L \cap M)}^{\times}$ ,
c.: $N_{M ∕ K} M^{\times} \subseteq N_{L ∕ K} L^{\times}$ if and only if $L \subseteq M$ ,
d.: for any subgroup $A$ of $K^{\times}$ containing $N_{L ∕ K} L^{\times}$ , there exists an intermediate field $E$ in $L ∕ K$ with $A = N_{E ∕ K} E^{\times}$ .

Remark 9.2.9. §

The equality of degrees and the statements of Proposition 9.2.8 quite obviously hold for archimedean local fields as well. The extension $ℂ ∕ ℝ$ is of course the only nontrivial extension in that setting, with norm group $N_{ℂ ∕ ℝ} ℂ^{\times} = ℝ_{> 0}^{\times}$ of index $2$ in $ℝ^{\times}$ .

9.3. Norm residue symbols

Notation 9.3.1. §

In this section, we fix an integer $n \geq 1$ and suppose that $K$ is a local field of characteristic not dividing $n$ such that $K$ contains $μ_{n}$ , the $n$ th roots of unity in a separable closure $K^{sep}$ of $K$ .

Definition 9.3.2. §

The $n$ th norm residue symbol (or Hilbert symbol, or Hilbert norm residue symbol) for the field $K$ is the pairing

{(,)}_{n, K} : K^{\times} \times K^{\times} \to μ_{n}

defined on $a, b \in K^{\times}$ by

{(a, b)}_{n, K} = \frac{ρ_{K} (b) (a^{1 ∕ n})}{a^{1 ∕ n}},

where $a^{1 ∕ n}$ is an $n$ th root of $α$ in $K^{ab}$ .

Remark 9.3.3. §

Since $K$ contains $μ_{n}$ , the $n$ th roots of $a$ lie in $K^{ab}$ , and every element of $G_{K}^{ab}$ acts trivially on $μ_{n}$ , so the quantity $σ (a^{1 ∕ n}) a^{- 1 ∕ n}$ for $σ \in G_{K}^{ab}$ is independent of the choice of $n$ th root $a^{1 ∕ n}$ of $a$ .

Proposition 9.3.4. §

The $n$ th norm residue symbol for $K$ has the following properties:

a.: it is bimultiplicative: i.e., for all $a, b, a^{'}, b^{'} \in K^{\times}$ , we have
$\begin{array}{rcl} {(a a^{'}, b)}_{n, K} = {(a, b)}_{n, K} {(a^{'}, b)}_{n, K} & and & {(a, b b^{'})}_{n, K} = {(a, b)}_{n, K} {(a, b^{'})}_{n, K}, \end{array}$
b.: ${(a, b)}_{n, K} = 1$ for $a, b \in K^{\times}$ if and only if $b \in N_{K (a^{1 ∕ n}) ∕ K} K {(a^{1 ∕ n})}^{\times}$ ,
c.: ${(a, 1 - a)}_{n, K} = 1$ for all $a \in K - {0, 1}$ ,
d.: ${(a, - a)}_{n, K} = 1$ for all $a \in K^{\times}$ ,
e.: it is skew-symmetric: i.e., ${(a, b)}_{n, K} = {(b, a)}_{n, K}^{- 1}$ for all $a, b \in K^{\times}$ ,
f.: it induces a perfect pairing on $K^{\times} ∕ K^{\times n}$ : i.e., ${(a, b)}_{n, K} = 1$ for a fixed $a \in K^{\times}$ (resp., $b \in K^{\times}$ ) for all $b \in K^{\times}$ (resp., $a \in K^{\times}$ ) if and only if $a \in K^{\times n}$ (resp., $b \in K^{\times n}$ ).

Proof.

a.

Note that we may choose ${(a a^{'})}^{1 ∕ n}$ to equal $a^{1 ∕ n} {(a^{'})}^{1 ∕ n}$ . Since $ρ_{K} (b)$ is a homomorphism, we have

{(a a^{'}, b)}_{n, K} = \frac{ρ_{K} (b) ({(a a^{'})}^{1 ∕ n})}{{(a a^{'})}^{1 ∕ n}} = \frac{ρ_{K} (b) (a^{1 ∕ n})}{a^{1 ∕ n}} \frac{ρ_{K} (b) ({(a^{'})}^{1 ∕ n})}{{(a^{'})}^{1 ∕ n}} = {(a, b)}_{n, K} {(a^{'}, b)}_{n, K},

and since $ρ_{K}$ is a homomorphism and $ρ_{K} (b) \in G_{K}^{ab}$ acts trivially on $μ_{n}$ , we have

{(a, b b^{'})}_{n, K} = \frac{ρ_{K} (b b^{'}) (a^{1 ∕ n})}{a^{1 ∕ n}} = \frac{ρ_{K} (b) (a^{1 ∕ n})}{a^{1 ∕ n}} \cdot ρ_{K} (b) (\frac{ρ_{K} (b^{'}) (a^{1 ∕ n})}{a^{1 ∕ n}}) = {(a, b)}_{n, K} {(a, b^{'})}_{n, K} .

b.

Note that ${(a, b)}_{n, K} = 1$ if and only if $ρ_{K} (b) (a^{1 ∕ n}) = a^{1 ∕ n}$ , so if and only if $ρ_{K} (b)$ fixes $K (a^{1 ∕ n})$ , and so if and only if $ρ_{K (a^{1 ∕ n}) ∕ K} (b) = 1$ . But this occurs if and only if $b$ is a norm from $K (a^{1 ∕ n})$ by local reciprocity.

c.

For any $c \in K^{\times}$ and a primitive $n$ th root $ζ_{n}$ of $1$ in $K$ , we may write

c^{n} - a = \prod_{i = 0}^{n - 1} (c - ζ_{n}^{i} a^{1 ∕ n}) = N_{K (a^{1 ∕ n}) ∕ K} (c - a^{1 ∕ n}) .

We then take $c = 1$ and apply (b).

d.

Take $c = 0$ in the proof of (c) and again apply (b).

e.

By (d) and (a), we have

1 = {(𝑎𝑏, - 𝑎𝑏)}_{n, K} = {(a, - a)}_{n, K} {(a, b)}_{n, K} {(b, a)}_{n, K} {(b, - b)}_{n, K} = {(a, b)}_{n, K} {(b, a)}_{n, K} .

f.

If $a \in K^{\times n}$ , then write $a = c^{n}$ with $c \in K^{\times}$ , and note that ${(a, b)}_{n, K} = {(c, b)}_{n, K}^{n} = 1$ . On the other hand, if ${(a, b)}_{n, K} = 1$ for all $b \in K^{\times}$ , then $ρ_{K (a^{1 ∕ n}) ∕ K}$ is the trivial homomorphism by the argument of (a). Local reciprocity then tells us that $K (a^{1 ∕ n}) = K$ , so $a \in K^{\times n}$ . The analogous statement switching the variables now follows immediately from (e).

□

Remark 9.3.5. §

Part (b) of the Proposition 9.3.4, says that ${(a, b)}_{n, K} = 1$ for $a, b \in K^{\times}$ if and only if $b$ is a norm from $K (a^{1 ∕ n})$ . It is this property for which the norm residue symbol is named. Much less obvious from the definition is the fact then obtained using the skew-symmetry of the symbol in part (e) of said proposition: that is, we also have ${(a, b)}_{n, K} = 1$ if and only if $a$ is a norm from $K (b^{1 ∕ n})$ . So, $a$ is a norm from $K (b^{1 ∕ n})$ if and only if $b$ is a norm from $K (a^{1 ∕ n})$ .

Let us compute the norm residue symbol for $n = 2$ , when $p = 2$ , which is sometimes simply referred to as the Hilbert symbol.

Proposition 9.3.6. §

Let $a, b \in ℤ_{2}^{\times}$ . Then ${(2, 2)}_{2, ℚ_{2}} = 1$ ,

\begin{array}{rcl} {(a, b)}_{2, ℚ_{2}} = {(- 1)}^{(a - 1) (b - 1) ∕ 4} & and & {(2, b)}_{2, ℚ_{2}} = {(- 1)}^{(b^{2} - 1) ∕ 8} . \end{array}

Proof.

Note first that the expression $f (a, b) = (a - 1) (b - 1) mod 8$ for $a, b \in ℤ_{2}^{\times}$ satisfies

f (a a^{'}, b) \equiv (a^{'} (a - 1) + a^{'} - 1) (b - 1) \equiv f (a, b) + f (a^{'}, b) mod 8

and the expression $g (b) = b^{2} - 1 mod 16$ satisfies

g (b b^{'}) \equiv {(b b^{'})}^{2} - 1 \equiv {(b^{'})}^{2} (b^{2} - 1) + ({(b^{'})}^{2} - 1) \equiv g (b) + g (b^{'}) mod 16,

so the right-hand sides of the equations of interest are multiplicative in the variables $a$ and $b$ . Since they are also continuous, it suffices to verify the formulas on a set of topological generators.

Recall that $ℚ_{2}^{\times}$ is topologically generated by $- 1$ , $2$ , and $5$ . First, we claim that $- 1$ is not a norm from $ℚ_{2} (i)$ , which is to say an element of the form $a^{2} + b^{2}$ with $a, b \in ℚ_{2}^{\times}$ . For this, note that it suffices to consider $a, b \in ℤ_{2}^{\times}$ and then congruence modulo $4$ eliminates the possibility. We therefore have ${(- 1, - 1)}_{2, ℚ_{2}} = - 1$ . Since $2$ and $5$ are norms from $ℚ_{2} (i)$ , we have ${(2, - 1)}_{2, ℚ_{2}} = 1$ and ${(5, - 1)}_{2, ℚ_{2}} = 1$ . We then have (noting (d) of Proposition 9.3.4) that

{(2, 2)}_{2, ℚ_{2}} = {(2, - 2)}_{2, ℚ_{2}} {(2, - 1)}_{2, ℚ_{2}} = 1

and similarly ${(5, 5)}_{2, ℚ_{2}} = 1$ . Finally we calculate ${(2, 5)}_{2, ℚ_{2}}$ . The question becomes whether $5 = a^{2} - 2 b^{2}$ for some $a, b \in ℤ_{2}^{\times}$ , but $a^{2}, b^{2}$ lie in ${0, 1, 4}$ modulo $8$ , and a quick check shows that the equality cannot hold modulo $8$ and therefore ${(2, 5)}_{2, ℚ_{2}} = - 1$ . These values all agree with the stated values, as needed. □

In the case that $n$ and the residue characteristic of $K$ are coprime, the norm residue symbol is also not too difficult to compute. Note that in this case, the extensions $K (a^{1 ∕ n})$ with $a \in K^{\times}$ are tamely ramified.

Definition 9.3.7. §

Suppose that $n$ is relatively prime to the residue characteristic of $K$ . Then ${(,)}_{n, K}$ is called a tame symbol.

Theorem 9.3.8. §

Suppose that $n$ is not divisible by the residue characteristic of $K$ . Let $q$ denote the order of the residue field $κ$ of $K$ , and note that $n$ divides $q - 1$ since $K$ is assumed to contain $μ_{n}$ . For $a, b \in K^{\times}$ , let ${[a, b]}_{K} \in κ^{\times}$ be defined by

{[a, b]}_{K} = {(- 1)}^{v_{K} (a) v_{K} (b)} \frac{a^{v_{K} (b)}}{b^{v_{K} (a)}} mod π_{K},

where $π_{K}$ is a uniformizer of $K$ . We then have

{(a, b)}_{n, K} = {[a, b]}_{K}^{(q - 1) ∕ n} \in μ_{n},

identifying elements of $μ_{n} (κ)$ with their unique lifts to $n$ th roots of unity in $K$ .

Proof.

Let us first compute ${(a, π_{K})}_{n, K}$ for a unit $a \in 𝒪_{K}^{\times}$ . Let $L = K (a^{1 ∕ n})$ , and note that $L ∕ K$ is unramified. Therefore, $ρ_{L ∕ K} (π_{K})$ is the unique Frobenius element in $Gal (L ∕ K)$ . In particular, we have

ρ_{K} (π_{K}) (a^{1 ∕ n}) \equiv {(a^{1 ∕ n})}^{q} mod π_{K} 𝒪_{L},

{(a, π_{K})}_{n, K} \equiv a^{(q - 1) ∕ n} mod π_{K},

as desired. Note that if $b \in 𝒪_{K}^{\times}$ as well, then $ρ_{L ∕ K} (b)$ is trivial since $L ∕ K$ is unramified, so ${(a, b)}_{n, K} = 1$ .

In general, take $a, b \in K^{\times}$ , and write $a = π_{K}^{v (a)} α$ and $b = π_{K}^{v (b)} β$ with $α, β \in 𝒪_{K}^{\times}$ . Writing $v = v_{K}$ for short, the properties of the norm residue symbol and the cases already computed yield

\begin{matrix} {(a, b)}_{n, K} = {(π_{K}, π_{K})}_{n, K}^{v (a) v (b)} {(α, π_{K})}^{v (b)} {(β, π_{K})}^{- v (a)} \\ = {(π_{K}, - 1)}^{v (a) v (b)} α^{v (b) (q - 1) ∕ n} β^{- v (a) (q - 1) ∕ n} = {({(- 1)}^{v (a) v (b)} a^{v (b)} b^{- v (a)})}^{(q - 1) ∕ n}, \end{matrix}

as originally asserted. □

Remark 9.3.9. §

We may also speak of the $2$ nd norm residue symbol for $ℝ$ , which is defined in the same manner as for nonarchimedean local fields. It satisfies

{(a, b)}_{2, ℝ} = {\begin{matrix} - 1 & if a, b < 0, \\ 1 & if a, b > 0, \end{matrix}

since $- 1$ is not a norm from $ℂ$ to $ℝ$ . We can also speak of $n$ th norm residue symbols for $ℂ$ for any $n$ , but they are all of course trivial.

We end with a more cohomological description of norm residue symbols which can be useful. From now on, let us identify $ℤ ∕ 𝑛ℤ$ with $\frac{1}{n} ℤ ∕ ℤ$ via the inverse of multiplication by $n$ . We denote the resulting injection $ℤ ∕ 𝑛ℤ ↪ ℚ ∕ ℤ$ by $ι$ .

Lemma 9.3.10. §

The invariant map on $Br (K)$ induces a canonical isomorphism

H^{2} (G_{K}, μ_{n} \otimes_{ℤ} μ_{n}) \to_{}^{\sim} μ_{n} .

Proof.

First, note that $μ_{n}$ is a trivial $G_{K}$ -module since $μ_{n}$ is contained in $K$ . Set $μ_{n}^{\otimes 2} = μ_{n} \otimes_{ℤ} μ_{n}$ .

First, there is a canonical isomorphism

H^{i} (G_{K}, μ_{n}) \otimes_{ℤ} μ_{n} \to H^{i} (G_{K}, μ_{n}^{\otimes 2})

for all $i \in ℤ$ that is induced by the map of complexes

C^{\cdot} (G_{K}, μ_{n}) \otimes_{ℤ} μ_{n} \to C^{\cdot} (G_{K}, μ_{n}^{\otimes 2})

that takes an $f \otimes ζ$ , where $f \in C^{i} (G_{K}, μ_{n})$ and $ζ \in μ_{n}$ to the cochain with value on $x \in G^{i}$ given by $f (x) \otimes ζ$ .

Secondly, recall from Proposition 2.5.4 that $H^{2} (G_{K}, μ_{n}) ≅ Br (K) [n]$ , and the invariant map induces an isomorphism $Br (K) [n] \to_{}^{\sim} \frac{1}{n} ℤ ∕ ℤ \to_{}^{ι^{- 1}} ℤ ∕ 𝑛ℤ$ . In total, we have

H^{i} (G_{K}, μ_{n}^{\otimes 2}) ≅ μ_{n} \otimes ℤ ∕ 𝑛ℤ≅ μ_{n},

hence the result. □

Recall that Kummer theory provides an isomorphism $K^{\times} ∕ K^{\times n} \to_{}^{\sim} H^{1} (G_{K}, μ_{n})$ , hence a canonical surjection from $K^{\times}$ to the latter group.

Proposition 9.3.11. §

The pairing $(,)$ defined by the cup product through the composition

(,) : K^{\times} \times K^{\times} ↠ H^{1} (G_{K}, μ_{n}) \times H^{1} (G_{K}, μ_{n}) \to_{}^{\cup} H^{2} (G_{K}, μ_{n} \otimes_{ℤ} μ_{n}) \to_{}^{\sim} μ_{n}

is equal to the norm residue symbol for $K$ .

Proof.

Let $a, b \in K^{\times}$ . Let $χ_{a} : G_{K} \to μ_{n}$ be the Kummer character attached to $a$ . Fix a primitive $n$ th root of unity $ζ_{n}$ , let $\tilde{χ} : G_{K} \to ℤ ∕ 𝑛ℤ$ be defined by $χ_{a} (σ) = ζ_{n}^{\tilde{χ} (σ)}$ , and let $χ = ι \circ \tilde{χ} : G_{K} \to ℚ ∕ ℤ$ . By definition and Proposition 8.1.10, we have

{(a, b)}_{n, K} = χ_{a} (ρ_{K} (b)) = ζ_{n}^{ι^{- 1} ({inv}_{K} (b \cup δ (χ)))},

where $δ$ is again the connecting homomorphism for $0 \to ℤ \to ℚ \to ℚ ∕ ℤ \to 0$ . On the other hand,

χ_{a} \cup χ_{b} = (\tilde{χ} \cup χ_{b}) \otimes ζ_{n},

and we see that

(a, b) = ζ_{n}^{ι^{- 1} ({inv}_{K} (\tilde{χ} \cup χ_{b}))}

by construction of the isomorphism in Lemma 9.3.10.

It thus suffices to see that $b \cup δ (χ) = \tilde{χ} \cup χ_{b}$ in $Br (K) [n]$ . We compare $2$ -cocycles representing these classes, noting the antisymmetry

\tilde{χ} \cup χ_{b} = - (χ_{b} \cup \tilde{χ}) .

Lift $\tilde{χ}$ to a map $χ : G_{K} \to ℤ$ , and choose an $n$ th root $β$ of $b$ . Let $σ, τ \in G_{K}$ . By construction of $δ$ , we have

δ (χ) (σ, τ) = \frac{1}{n} (ψ (σ) + ψ (τ) - ψ (𝜎𝜏)) \in ℤ,

from which it follows that

(b \cup δ (χ)) (σ, τ) = b^{δ (χ) (σ, τ)} = β^{ψ (σ) + ψ (τ) - ψ (𝜎𝜏)}

while, via the identification $μ_{n} \otimes ℤ ∕ 𝑛ℤ = μ_{n}$ , we have

(χ_{b} \cup \tilde{χ}) (σ, τ) = χ_{b} (σ) \otimes \tilde{χ} (τ) = {(\frac{σ (β)}{β})}^{ψ (τ)} \in μ_{n} .

The product of these $2$ -cocycles in $Z^{2} (G_{K}, {(K^{sep})}^{\times})$ is

β^{ψ (σ)} β^{- ψ (𝜎𝜏)} σ {(β)}^{ψ (τ)},

which is the value on $(σ, τ)$ of the coboundary of the $1$ -cochain

σ \mapsto β^{ψ (σ)} .

□

9.4. The existence theorem

Let $K$ be a local field. We give its separable closure $K^{sep}$ the topology defined by the unique extension of the valuation on $K$ to $K^{sep}$ and then endow ${(K^{sep})}^{\times}$ with the subspace topology. We will show that $({(K^{sep})}^{\times}, inv)$ is a topological class formation. Note that the Galois group $G_{K}$ acts continuously on ${(K^{sep})}^{\times}$ since it its action preserves the valuation of elements.

Proposition 9.4.1. §

For any finite separable extension $L$ of $K$ , the norm map $N_{L ∕ K} : L^{\times} \to K^{\times}$ has closed image and compact kernel.

Proof.

Recall from Proposition 5.5.6 that $𝒪_{L}^{\times}$ is compact. Since $v_{K} \circ N_{L ∕ K} = f_{L ∕ K} v_{L}$ , the kernel of the continuous map $N_{L ∕ K}$ is a closed subgroup of $𝒪_{L}^{\times}$ , hence is compact.

Since $𝒪_{L}^{\times}$ is compact, $N_{L ∕ K} (𝒪_{L}^{\times})$ is a closed subset of $𝒪_{K}^{\times}$ . Note that

N_{L ∕ K} 𝒪_{L}^{\times} = 𝒪_{K}^{\times} \cap N_{L ∕ K} L^{\times},

[𝒪_{K}^{\times} : N_{L ∕ K} 𝒪_{L}^{\times}] \leq [K^{\times} : N_{L ∕ K} L^{\times}] = [L : K]

is finite. Being of finite index in $𝒪_{K}^{\times}$ , the closed subgroup $N_{L ∕ K} (𝒪_{L}^{\times})$ is open in $𝒪_{K}^{\times}$ . Since $𝒪_{K}^{\times}$ is open in $K^{\times}$ , the group $N_{L ∕ K} (𝒪_{L}^{\times})$ is open in $K^{\times}$ as well. Finally, as a union of $N_{L ∕ K} (𝒪_{L}^{\times})$ -cosets, the subgroup $N_{L ∕ K} L^{\times}$ is open in $K^{\times}$ , hence closed. □

Proposition 9.4.2. §

Let $p$ be a prime, and suppose that the characteristic of $K$ is not $p$ . For any finite separable extension $L$ of $K (μ_{p})$ , the $p$ th power map on $L^{\times}$ has finite kernel $μ_{p}$ and image $L^{\times p}$ containing $D_{L} = \ker ρ_{L}$ .

Proof.

The first statement is obvious. If $a \in \ker ρ_{L}$ , then $a \in N_{M ∕ L} M^{\times}$ for every finite abelian extension $M$ of $L$ . In particular, for all $b \in L^{\times}$ , we have that $a \in N_{L (b^{1 ∕ p}) ∕ L} L {(b^{1 ∕ p})}^{\times}$ , so ${(a, b)}_{p, L} = 1$ . Proposition 9.3.4f then implies that $a \in L^{\times p}$ . □

Proposition 9.4.3. §

Every closed subgroup of $K^{\times}$ of finite index that contains $𝒪_{K}^{\times}$ is a norm group.

Proof.

Note that $K^{\times} ∕ 𝒪_{K}^{\times} ≅ℤ$ via the valuation map, which is in fact a homeomorphism if we give the left-hand side the quotient topology and the right-hand side the discrete topology. The closed subgroups of finite index in $ℤ$ are the nontrivial subgroups, so the closed subgroups of finite index in $K^{\times}$ that contain $𝒪_{K}^{\times}$ are those of the form $𝒩_{n} = ⟨ π^{n} ⟩ 𝒪_{K}^{\times}$ for some $n \geq 1$ and a fixed uniformizer $π$ . If $K_{n}$ is the unramified extension of $K$ of degree $n$ , then every element of $K_{n}^{\times}$ has the form for some $i \in ℤ$ and $u \in 𝒪_{K_{n}}^{\times}$ , and we have

N_{K_{n} ∕ K} (π^{i} u) = π^{𝑛𝑖} N_{K_{n} ∕ K} (u) \in 𝒩_{n} .

The indices of the two subgroups $N_{K_{n} ∕ K} K_{n}^{\times} \leq 𝒩_{n}$ of $𝒪_{K}^{\times}$ are both $n$ , the former being the consequence (9.2.1) of local reciprocity, so they must be equal. □

Theorem 9.4.4 (Existence theorem of local CFT). §

The closed subgroups of $K^{\times}$ of finite index are exactly the norm subgroups $N_{L ∕ K} L^{\times}$ with $L$ a finite abelian extension of $K$ .

We omit the proof of Theorem 9.4.4 for Laurent series fields and focus on the characteristic zero setting.

Proof for

p

-adic fields.

The three properties of Definition 8.2.10 are satisfied by Propositions 9.4.1, 9.4.2, 9.4.3, so the result follows from Theorem 8.2.14. □

We also have the following, which is actually immediate from part (c) of Proposition 9.2.8.

Theorem 9.4.5 (Uniqueness theorem of local CFT). §

Let $L$ and $M$ be distinct finite abelian extensions of $K$ . Then $N_{L ∕ K} L^{\times} \neq N_{M ∕ K} M^{\times}$ .

Remark 9.4.6. §

Taken together, the existence and uniqueness theorems that there is a one-to-one correspondence between finite abelian extensions $L$ of $K$ and open subgroups of finite index in $K^{\times}$ given by taking $L$ to $N_{L ∕ K} L^{\times}$ . In part for historical reasons, we have stated them separately. We have seen additional properties of this (inclusion-reversing) correspondence in parts (a) and (b) of Proposition 9.2.8.

Next, we see that local reciprocity and the existence theorem imply the following.

Theorem 9.4.7. §

Let $K$ be a nonarchimedean local field.

a.: The reciprocity map $ρ_{K}$ is continuous and injective with dense image.
b.: The restriction of $ρ_{K}$ to $𝒪_{K}^{\times}$ provides a topological isomorphism between $𝒪_{K}^{\times}$ and the inertia subgroup of $G_{K}^{ab}$ .

Proof.

That $ρ_{K}$ is continuous with dense image follows from Proposition 8.2.15. Recall that, by definition, the groups $U_{i} (K)$ with $i \geq 1$ form a basis of open neighborhoods of $1$ in the topology on $K$ . They are not, however, of finite index in $K^{\times}$ . However, the subgroups $⟨ π^{n} ⟩ \times U_{i} (K)$ with $i, n \geq 1$ are, and are clearly open. Moreover, their intersection is ${1}$ . By the existence theorem, there exists a finite abelian extension $L_{n, i} ∕ K$ such that $N_{L_{n, i} ∕ K} L_{n, i}^{\times} = ⟨ π^{n} ⟩ U_{i} (K)$ . For $a \in K^{\times}$ with $a \neq 1$ , we may then choose $n$ and $i$ large enough so that $a \notin N_{L_{n, i} ∕ K} L_{n, i}^{\times}$ , and therefore local reciprocity tells us that $ρ_{L_{n, i} ∕ K} (a)$ is nontrivial, so $ρ_{K} (a)$ is nontrivial. That is, $ρ_{K}$ is injective. This proves (a).

That $ρ_{K}$ maps $𝒪_{K}^{\times}$ into the inertia group in $G_{K}^{ab}$ is as follows. Every element $u \in 𝒪_{K}^{\times}$ may be written as a quotient of two uniformizers $π$ and $π^{'}$ by taking $π^{'} = 𝑢𝜋$ for any uniformizer $π$ . By property (i) in the local reciprocity law, we have

ρ_{K} (u) |_{K^{ur}} = ρ_{K} (π^{'}) |_{K^{ur}} {(ρ_{K} (π) |_{K^{ur}})}^{- 1} = φ_{K} φ_{K}^{- 1} = 1,

where $φ_{K}$ is the Frobenius element of $Gal (K^{ur} ∕ K)$ . Hence, $ρ_{K} (u)$ lies in the inertia subgroup, and for the same reason, this occurs only when $u$ is a unit. As for surjectivity onto inertia, the element $ρ_{K} (π)$ gives a choice of Frobenius, hence a splitting of the surjection $G_{K}^{ab} \to Gal (K^{ur} ∕ K)$ . Via this splitting, the reciprocity map is the direct product of the continuous maps from $𝒪_{K}^{\times}$ to inertia and the group generated by $π$ to $Gal (K^{ur} ∕ K)$ . Since $ρ_{K}$ has dense image, the image of $𝒪_{K}^{\times}$ in inertia is therefore dense as well, and it suffices to see that $ρ_{K} (𝒪_{K}^{\times})$ is closed. But $ρ_{K}$ is continuous and $𝒪_{K}^{\times}$ is compact Hausdorff, so indeed this is the case, proving (b). □

We leave the following remark to the reader as an exercise.

Lemma 9.4.8. §

Let $K$ be a nonarchimedean local field, and let $π$ be a uniformizer of $K$ . Let $K_{π}$ denote the fixed field of $ρ_{K} (π)$ in $K^{ab}$ . Then $K^{ab} = K_{π} \cdot K^{ur}$ . Moreover, $K_{π}$ is a maximal totally ramified extension of $K$ in $K^{ab}$ .

We next prove the uniqueness of the local reciprocity map to complete the proof of the local reciprocity law.

Theorem 9.4.9 (Uniqueness of the local reciprocity map). §

The reciprocity map $ρ_{K}$ is the unique map satisfying properties (i) and (ii) of Theorem 9.2.2.

Proof.

We prove that if a homomorphism $ϕ : K^{\times} \to G_{K}^{ab}$ satisfies properties (i) and (ii) of Theorem 9.2.2 with $ρ_{K}$ replaced by $ϕ$ , then it is $ρ_{K}$ . Consider the open subgroup $A_{n} = ⟨ π ⟩ U_{n} (K)$ of finite index in $K^{\times}$ . By the existence theorem, there exists a finite abelian extension $L_{n}$ of $K$ with norm group equal to $A_{n}$ . The union of the fields $L_{n}$ is the field $K_{π}$ of Lemma 9.4.8. Being that $π \in A_{n}$ for all $n$ , we have that by property (ii) of the local reciprocity law that $ϕ (π) |_{K_{π}} = 1$ ,. On the other hand, by property (i), we have that $ϕ (π) |_{K^{ur}}$ is the Frobenius element of $Gal (K^{ur} ∕ K)$ . On the other hand, $ρ_{K} (π)$ also has both of these properties and $K^{ab} = K_{π} \cdot K$ , so $ρ_{K} (π) = ϕ (π)$ . Since this holds for every uniformizer of $K$ and any $a \in K^{\times}$ can be written as $a = π^{v_{K} (a) - 1} \cdot π^{'}$ where $π^{'}$ is a uniformizer defined by this equality, the two maps $ρ_{K}$ and $ϕ$ are equal. □

We end with a few remarks on the topology of $K^{\times}$ .

Proposition 9.4.10. §

Let $K$ be a $p$ -adic field. Then every subgroup of $K^{\times}$ of finite index is open.

Proof.

Since $A$ be a subgroup of finite index in $K^{\times}$ , and let $m$ be the exponent of $K^{\times} ∕ A$ . Then $K^{\times m} \subseteq A$ , and Proposition 6.3.9 tells us that $K^{\times} ∕ K^{\times m}$ is a finite abelian group (in fact, isomorphic to a subgroup of ${(ℤ ∕ 𝑚ℤ)}^{[K : ℚ_{p}] + 2}$ ). Thus $K^{\times m}$ has finite index in $K^{\times}$ . As the $m$ th power map is continuous and $𝒪_{K}^{\times}$ is compact, $𝒪_{K}^{\times m}$ is closed in $𝒪_{K}^{\times}$ . Letting $π_{K}$ denote a uniformizer for $K$ , we then have that $K^{\times m} = 𝒪_{K}^{\times m} ⟨ π_{K}^{m} ⟩$ is closed in $K^{\times}$ , therefore open. As $A$ is a union $K^{\times}$ -cosets, it is open as well. □

Corollary 9.4.11. §

Let $K$ be a $p$ -adic field. Then $ρ_{K}$ induces a topological isomoprhism

\hat{K^{\times}} \to_{}^{\sim} G_{K}^{ab},

where $\hat{K^{\times}}$ is the profinite completion of $K^{\times}$ .

Proof.

By definition $G_{K}^{ab}$ is isomorphic to the inverse limit of the system of groups $Gal (L ∕ K)$ for $L ∕ K$ finite abelian with respect to restriction maps. On the other hand, local reciprocity provides a series of isomorphisms

ρ_{L ∕ K} : K^{\times} ∕ N_{L ∕ K} L^{\times} \to_{}^{\sim} Gal (L ∕ K)

that are compatible with the natural quotient maps on the left and restriction maps on the right. In other words, local reciprocity sets up an isomorphism

\underset{L}{lim_{\leftarrow}} K^{\times} ∕ N_{L ∕ K} L^{\times} \to G_{K}^{ab},

(9.4.1)

but as $L$ runs over the finite abelian extensions, Proposition 9.4.10 tells us that the groups $N_{L ∕ K} L^{\times}$ run over all subgroups of finite index in $K^{\times}$ . Therefore, the inverse limit in (9.4.1) is just the profinite completion of $K^{\times}$ . □

Remarks 9.4.12. §

a.: The converse to Proposition 9.4.10 is false: for instance, $𝒪_{K}^{\times}$ is open in $K^{\times}$ but not of finite index.
b.: If $K$ is a Laurent series field, then its multiplicative group has subgroups of finite index that are not closed. To see this, recall that $K$ is isomorphic to $𝔽_{q} ((t))$ for some $q$ . Recall from Proposition 6.3.10 that $U_{1} (K)$ and $\prod_{i = 1}^{\infty} ℤ_{p}$ are topologically isomorphic. Note that $\oplus_{i = 1}^{\infty} ℤ_{p}$ is dense in $\prod_{i = 1}^{\infty} ℤ_{p}$ but not closed. Any subgroup of finite index in the latter group containing the former group will therefore not be closed. Choose such a group $U$ , and consider $⟨ t ⟩ U$ . (We leave it as an exercise to apply Zorn’s lemma to see that $U$ exists.) This is a subgroup of finite index in $K^{\times}$ that is not closed.
c.: For a Laurent series field $K$ , the isomorphism (9.4.1) still holds, but the inverse limit of the multiplicative group modulo norm groups, while a profinite group, is no longer isomorphic to the profinite completion of $K^{\times}$ .

9.5. Class field theory over $ℚ_{p}$

In this section, we will determine the abelian extensions of $ℚ_{p}$ and make explicit the reciprocity law for $ℚ_{p}$ . We shall not assume the results of the previous section.

Lemma 9.5.1. §

Let $p$ be a prime, and let $K$ be a field of characteristic not equal to $p$ . Let $a \in K {(μ_{p})}^{\times}$ . For a generator $δ$ of $Gal (K (μ_{p}) ∕ K)$ , let $c \in ℤ$ be such that $δ (ζ_{p}) = ζ_{p}^{c}$ for any generator $ζ_{p}$ of $μ_{p}$ . Then $M = K (μ_{p}, a^{1 ∕ p})$ is abelian over $K$ if and only if

δ (a) a^{- c} \in K {(μ_{p})}^{\times p} .

Proof.

We may suppose without loss of generality that $a \notin K {(μ_{p})}^{\times p}$ . Let $τ \in Gal (M ∕ K (μ_{p}))$ be a generator such that $τ (a^{1 ∕ p}) = ζ_{p} a^{1 ∕ p}$ .

Suppose first that $M ∕ K$ is abelian. Lift $δ$ to a generator of $Gal (M ∕ L)$ , where $L$ is the unique abelian subextension in $M ∕ K$ of degree $p$ over $K$ , and denote this also by $δ$ . We have

τ (δ (a^{1 ∕ p})) = δ (τ (a^{1 ∕ p})) = δ (ζ) δ (a^{1 ∕ p}) = ζ_{p}^{c} δ (a^{1 ∕ p}) .

In terms of Kummer duality, this says that the Kummer pairing of $τ$ and $δ (a)$ is $ζ_{p}^{c}$ . Since $M ∕ K (μ_{p})$ is generated by a $p$ th root of $a$ and $τ$ pairs with $a^{c}$ to $ζ_{p}^{c}$ as well, we have by the nondegeneracy of the Kummer pairing that $δ (a) a^{- c} \in K {(μ_{p})}^{\times p}$ .

Now, suppose that $δ (a) a^{- c} = x^{p}$ for some $x \in K {(μ_{p})}^{\times}$ . Extend $δ$ to an embedding of $M$ in $\bar{K}$ . Note that $δ (a^{1 ∕ p})$ is a $p$ th root of $a^{c} x^{p}$ , hence of the form $ζ_{p}^{j} {(a^{1 ∕ p})}^{c} x$ for some $j \in ℤ$ , and this is an element of $M$ . It follows that $M ∕ K$ is Galois. Moreover, we have

τ (δ (a^{1 ∕ p})) = τ (ζ_{p}^{j} a^{c ∕ p} x) = ζ_{p}^{j + c} a^{c ∕ p} x = ζ_{p}^{c} δ (a^{1 ∕ p}) = δ (ζ a^{1 ∕ p}) = δ (τ (a^{1 ∕ p}))

and

τ (δ (ζ_{p})) = ζ_{p}^{c} = δ (τ (ζ_{p}))

since $τ$ fixes $μ_{p}$ . Thus, the generators $δ$ and $τ$ of $Gal (M ∕ K)$ commute, and so $M ∕ K$ is abelian. □

The following is a straightforward exercise using Lemma 6.3.7.

Lemma 9.5.2. §

For any prime $p$ , we have

U_{1} (ℚ_{p} (μ_{p})) \cap ℚ_{p} {(μ_{p})}^{\times p} = U_{p + 1} (ℚ_{p} (μ_{p})) .

We also have the following.

Lemma 9.5.3. §

Let $p$ be an odd prime. Let $δ$ be a generator of $Gal (ℚ_{p} (μ_{p}) ∕ ℚ_{p})$ , and let $c \in ℤ$ be such that $δ (ζ_{p}) = ζ_{p}^{c}$ for $ζ_{p}$ generating $μ_{p}$ . For any positive integer $i \leq p$ and $a \in U_{i} (ℚ_{p} (μ_{p})) - U_{i + 1} (ℚ_{p} (μ_{p}))$ , one has

δ (a) \equiv a^{c^{i}} mod {(1 - ζ_{p})}^{i + 1} .

Proof.

Set $λ = 1 - ζ_{p}$ . Note first that for any $k \geq 1$ , one has

1 - ζ_{p}^{k} = λ \sum_{j = 0}^{k - 1} ζ_{p}^{j} \equiv 𝑘𝜆 mod λ^{2} .

In particular, we have $δ (λ) \equiv 𝑐𝜆 mod λ^{2}$ . It follows from the binomial theorem that

δ {(λ)}^{i} \equiv {(𝑐𝜆 + (δ (λ) - 𝑐𝜆))}^{i} \equiv {(𝑐𝜆)}^{i} mod λ^{i + 1} .

Write $a = 1 + u λ^{i}$ for some $u \in ℤ {[μ_{p}]}^{\times}$ . One then has

δ (a) \equiv δ (1 + u λ^{i}) \equiv 1 + 𝑢𝛿 {(λ)}^{i} \equiv 1 + u c^{i} λ^{i} \equiv {(1 + u λ^{i})}^{c^{i}} \equiv a^{c^{i}} mod λ^{i + 1} .

□

Proposition 9.5.4. §

a.: Let $p$ be an odd prime. The maximal abelian extension of $ℚ_{p}$ of exponent $p$ has Galois group isomorphic to ${(ℤ ∕ 𝑝ℤ)}^{2}$ .
b.: The maximal abelian extension of $ℚ_{2}$ of exponent $4$ has Galois group isomorphic to ${(ℤ ∕ 4 ℤ)}^{2} \times ℤ ∕ 2 ℤ$ .

Proof.

Let $p$ be a prime and $L$ be the maximal abelian extension of $ℚ_{p}$ of exponent $p$ . The restriction map

Gal (L (μ_{p}) ∕ ℚ_{p} (μ_{p})) \to Gal (L ∕ ℚ_{p})

(9.5.1)

is an isomorphism since $[ℚ_{p} (μ_{p}) : ℚ_{p}]$ and $[L : ℚ_{p}]$ are relatively prime,. By Kummer theory, there exists a unique subgroup $Δ$ of $L {(μ_{p})}^{\times}$ containing $L {(μ_{p})}^{\times p}$ such that $L (μ_{p}) = ℚ_{p} (μ_{p}, \sqrt[p]{Δ})$ . By Lemma 9.5.1, we have

Δ = {a \in ℚ_{p} {(μ_{p})}^{\times} ∣ δ (a) a^{- c} \in ℚ_{p} {(μ_{p})}^{\times p}} .

Now suppose that $p$ is odd. Let us set $U_{i} = U_{i} (ℚ_{p} (μ_{p}))$ for each $i \geq 1$ . Note first that since the valuation of an element is unchanged by application of $δ$ , any element of $Δ$ must lie in $⟨ λ^{p} ⟩ ℤ_{p} {[μ_{p}]}^{\times}$ . Moreover, every element of $μ_{p - 1} (ℚ_{p})$ is a $p$ th power, so

Δ = ℚ_{p} {(μ_{p})}^{\times p} \cdot (U_{1} \cap Δ) .

(9.5.2)

Now, it follows from Lemmas 9.5.1, 9.5.2, and 9.5.3, any non $p$ th power in $U_{1} \cap Δ$ lies either in $U_{1} - U_{2}$ or $U_{p} - U_{p + 1}$ . We know that $μ_{p} \subseteq Δ$ , in that the group $μ_{p^{2}}$ generates an abelian extension of $ℚ_{p}$ . If any other element $x$ of $U_{1} - U_{2}$ were in $Δ$ , then there would exist a $p$ th root of unity $ξ$ such that $x ξ^{- 1} \in U_{2} \cap Δ$ , which would imply $x ξ^{- 1} \in U_{p}$ . Moreover, since $U_{p + 1} \leq ℚ_{p} {(μ_{p})}^{\times p}$ , we have that $U_{p}$ itself is contained in $Δ$ . It follows that $U_{1} \cap Δ = μ_{p} U_{p}$ . Recall that $U_{p} ∕ U_{p + 1} ≅ℤ ∕ 𝑝ℤ$ . Applying (9.5.2), we see that

Δ ∕ ℚ_{p} {(μ_{p})}^{\times p} ≅ {(ℤ ∕ 𝑝ℤ)}^{2} .

Kummer theory tells us that

Gal (L (μ_{p}) ∕ ℚ_{p} (μ_{p})) ≅ Hom (Δ ∕ ℚ_{p} {(μ_{p})}^{\times p}, μ_{p}) ≅ Hom ({(ℤ ∕ 𝑝ℤ)}^{2}, ℤ ∕ 𝑝ℤ) ≅ {(ℤ ∕ 𝑝ℤ)}^{2} .

Recalling (9.5.1), this implies the result.

If $p = 2$ , then we note that $Δ = ℚ_{2}^{\times}$ has a minimal set of topological generators consisting of $- 1$ , $2$ , and $3$ . Otherwise, we omit the proof of part b. □

We now turn to the local Kronecker-Weber theorem.

Theorem 9.5.5 (Local Kronecker-Weber). §

Let $p$ be a prime number. Then every finite abelian extension of $ℚ_{p}$ is contained in $ℚ_{p} (μ_{n})$ for some $n \geq 1$ .

Proof.

Since any finite abelian extension of $ℚ_{p}$ will be a compositum of such a finite abelian extension of $p$ -power and a finite abelian extension of prime-to- $p$ power degree, it suffices to consider such fields separately. We recall that finite abelian extensions of $ℚ_{p}$ of degree prime to $p$ are tamely ramified. The maximal tamely ramified abelian extension of $ℚ_{p}$ is equal to $ℚ_{p}^{ur} ({(- p)}^{1 ∕ (p - 1)})$ , since $- p$ is a uniformizer of $ℚ_{p}$ , and we know that $ℚ_{p} ({(- p)}^{1 ∕ (p - 1)}) = ℚ_{p} (μ_{p})$ while $ℚ_{p}^{ur}$ is the field given by adjoining to $ℚ_{p}$ all prime-to- $p$ roots of unity. Hence, we have the result for such fields.

So, let $L$ be an abelian extension of $ℚ_{p}$ of exponent $p^{r}$ for some $r \geq 1$ , and set $G = Gal (L ∕ ℚ_{p})$ . First consider odd $p$ . By Proposition 9.5.4a, the group $G ∕ G^{p}$ is a quotient of ${(ℤ ∕ 𝑝ℤ)}^{2}$ . By the structure theorem for finite abelian groups, $G$ is then isomorphic to a quotient of ${(ℤ ∕ p^{r} ℤ)}^{2}$ . On the other hand, $ℚ_{p} (μ_{p^{r + 1}})$ is a totally ramified abelian extension of $ℚ_{p}$ with Galois group

Gal (ℚ_{p} (μ_{p^{r + 1}}) ∕ ℚ_{p}) ≅ ℤ ∕ p^{r} ℤ \times {(ℤ ∕ 𝑝ℤ)}^{\times},

and the field $ℚ_{p} (μ_{p^{p^{r}} - 1})$ is an unramified abelian extension of $ℚ_{p}$ with Galois group isomorphic to $ℤ ∕ p^{r} ℤ$ . It follows that $ℚ_{p} (μ_{p^{r + 1} (p^{p^{r}} - 1)})$ has a subfield with Galois group ${(ℤ ∕ p^{r} ℤ)}^{2}$ over $ℚ_{p}$ , and so said field is $L$ . The result follows for odd $p$ .

In the case that $p = 2$ , Proposition 9.5.4b tells us that $G ∕ G^{4} ≅ {(ℤ ∕ 4 ℤ)}^{2} \times ℤ ∕ 2 ℤ$ . It follows that $G$ is isomorphic to a quotient of ${(ℤ ∕ 2^{r} ℤ)}^{2} \times ℤ ∕ 2 ℤ$ . Now, we know that

Gal (ℚ_{2} (μ_{2^{r + 2}}) ∕ ℚ_{2}) ≅ ℤ ∕ 2^{r} ℤ \times ℤ ∕ 2 ℤ

As with $p$ odd, we have an unramified cyclotomic extension of $ℚ_{2}$ , linearly disjoint from the totally ramified $ℚ_{2} (μ_{2^{r + 2}})$ over $ℚ_{2}$ , with Galois group $ℤ ∕ 2^{r} ℤ$ . So, there exists a cyclotomic extension of $ℚ_{2}$ with Galois group ${(ℤ ∕ 2^{r} ℤ)}^{2} \times ℤ ∕ 2 ℤ$ , which must then be $L$ . □

Corollary 9.5.6. §

For any prime $p$ , the maximal abelian extension of $ℚ_{p}$ is given by adjoining all roots of unity in $\bar{ℚ_{p}}$ . That is, we have

ℚ_{p}^{ab} = ℚ_{p} (μ_{\infty}),

where $μ_{\infty}$ is the group of all roots of unity in $\bar{ℚ_{p}}$ .

With the knowledge of the maximal abelian extension of $ℚ_{p}$ in hand, we are now prepared to give an explicit construction of the reciprocity map for $ℚ_{p}$ .

Remark 9.5.7. §

If $ζ$ is a $p^{k}$ th root of unity in $\bar{ℚ_{p}}$ for some prime $p$ and $k \geq 1$ , then $ζ^{a}$ for any $a \in ℤ_{p}$ is the well-defined root of unity equal to $ζ^{b}$ for any $b \in ℤ$ with $b \equiv a mod p^{k}$ .

Proposition 9.5.8. §

For each $n \geq 1$ , let $ζ_{n}$ denote a primitive $n$ th root of unity in $ℚ_{p}^{ab}$ . There exists a unique homomorphism $ρ : ℚ_{p}^{\times} \to G_{ℚ_{p}}^{ab}$ which, for $m \geq 1$ prime to $p$ and $k \geq 1$ , satisfies

i.: $ρ (p) (ζ_{p^{k}}) = ζ_{p^{k}}$ and $ρ (p) (ζ_{m}) = ζ_{m}^{p}$ , and
ii.: $ρ (u) (ζ_{p^{k}}) = ζ_{p^{k}}^{u^{- 1}}$ and $ρ (u) (ζ_{m}) = ζ_{m}$ for every $u \in ℤ_{p}^{\times}$ .

The map $ρ$ takes uniformizers in $ℚ_{p}$ to Frobenius elements, and its restriction to $ℤ_{p}^{\times}$ is an isomorphism onto the inertia subgroup of $G_{ℚ_{p}}^{ab}$ .

Proof.

Recall that $ℚ_{p}^{ur}$ is given by adjoining all prime-to- $p$ roots of unity in $\bar{ℚ_{p}}$ . Corollary 9.5.6 then tells us that

ℚ_{p}^{ab} = ℚ_{p}^{ur} (μ_{p^{\infty}}) = ℚ_{p}^{ur} \cdot ℚ_{p} (μ_{p^{\infty}}),

where $μ_{p^{\infty}}$ is the group of $p$ -power roots of unity in $\bar{ℚ_{p}}$ . Since $ℚ_{p} (μ_{p^{\infty}}) ∕ ℚ_{p}$ is totally ramified, we have

ℚ_{p}^{ur} \cap ℚ_{p} (μ_{p^{\infty}}) = ℚ_{p},

and so

\begin{matrix} G_{ℚ_{p}}^{ab} = Gal (ℚ_{p}^{ab} ∕ ℚ_{p}) ≅ Gal (ℚ_{p}^{ab} ∕ ℚ_{p}^{ur}) \times Gal (ℚ_{p}^{ab} ∕ ℚ_{p} (μ_{p^{\infty}})) \\ ≅ Gal (ℚ_{p} (μ_{p^{\infty}}) ∕ ℚ_{p}) \times Gal (ℚ_{p}^{ur} ∕ ℚ_{p}), & (9.5.3) \end{matrix}

the latter isomorphism being the product of restriction maps.

We claim the automorphisms $ρ (p)$ and $ρ (u)$ for $u \in ℤ_{p}^{\times}$ of $μ_{\infty}$ specified in the statement of the theorem are actually restrictions of elements of $G_{ℚ_{p}}^{ab}$ . Given this, since every root of unity is the product of roots of unity of prime-to- $p$ and $p$ -power order and $ℚ_{p}^{\times} ≅ ⟨ p ⟩ \times ℤ_{p}^{\times}$ , it follows that $ρ$ is indeed a homomorphism to $G_{ℚ_{p}}^{ab}$ , and it is uniquely specified by the given conditions.

For the claim, it suffices by (3) to see that these automorphisms define automorphisms of the prime-to- $p$ and $p$ -power roots of unity that are the restrictions of Galois elements in $Gal (ℚ_{p}^{ur} ∕ ℚ_{p})$ and $Gal (ℚ_{p} (μ_{p^{\infty}}) ∕ ℚ_{p})$ , respectively. First, we note that $ρ (p)$ has the same action as the trivial element on $p$ -power roots of unity and as the Frobenius element on prime-to- $p$ roots of unity. In particular, $ρ (p)$ does extend to a Frobenius element of $G_{ℚ_{p}}^{ab}$ .

On the other hand, $ρ (u)$ acts trivially on $p$ -power roots of unity, so we need only see that its action on $p$ -power roots of unity is the restriction of a Galois element. Note that the cyclotomic character The cyclotomic character

χ : Gal (ℚ_{p} (μ_{p^{\infty}}) ∕ ℚ_{p}) \to \underset{k}{lim_{\leftarrow}} {(ℤ ∕ p^{k} ℤ)}^{\times} ≅ ℤ_{p}^{\times}, χ (σ) (ζ_{p^{k}}) = ζ_{p^{k}}^{χ (σ)}

for $ℚ_{p}$ is an isomorphism in that $[ℚ_{p} (μ_{p^{k}}) : ℚ_{p}] = p^{k - 1} (p - 1)$ for each $k$ . Thus, we have that there exists $σ_{u} \in Gal (ℚ_{p} (μ_{p^{\infty}}) ∕ ℚ_{p})$ with $χ (σ_{u}) = u$ . We then have that $ρ (u)$ as defined is indeed the restriction of $σ_{u}^{- 1}$ on $μ_{p^{\infty}}$ . That is $ρ (u)$ does extend to a well-defined element of $Gal (ℚ_{p}^{ab} ∕ ℚ_{p})$ . with $ρ (u) |_{ℚ_{p} (μ_{p^{\infty}})} = σ_{u}^{- 1}$ . Moreover, as $ρ$ on $ℤ_{p}^{\times}$ followed by restriction to $ℚ_{p} (μ_{p^{\infty}})$ is the inverse map to the map taking $σ$ to $χ {(σ)}^{- 1}$ , we have that $ρ |_{ℤ_{p}^{\times}}$ is an isomorphism to inertia in $Gal (ℚ_{p}^{ab} ∕ ℚ_{p})$ .

Finally note that $ρ (p)$ is by definition a Frobenius element and $ρ (u)$ for $u \in ℤ_{p}^{\times}$ has image in inertia, so $ρ (𝑝𝑢)$ is a Frobenius element as well. Since $u$ was arbitrary, $ρ$ takes uniformizers to Frobenius elements. □

Though we omit the proof, it is possible to show using the uniqueness in Theorem 9.2.2 (after computations of norm groups of abelian extensions of $ℚ_{p}$ ) that the map $ρ$ of Proposition 9.5.8 must indeed be the local reciprocity map for $ℚ_{p}$ .

Theorem 9.5.9. §

The map $ρ$ constructed in Proposition 9.5.8 is the local reciprocity map $ρ_{ℚ_{p}}$ .

9.6. Ramification groups and the unit filtration

Definition 9.6.1. §

Let $L ∕ K$ be a Galois extension of local fields with Galois group $G$ . Then $ψ_{L ∕ K} : [- 1, \infty) \to [- 1, \infty)$ be defined to be the inverse of the function $ϕ_{L ∕ K}$ of Definition 6.5.19.

This allows us to define ramification groups in the upper numbering.

Definition 9.6.2. §

Let $L ∕ K$ be a Galois extension of local fields with Galois group $G$ . For any real number $s \geq - 1$ , we define the $s$ th ramification group $G^{s}$ of $L ∕ K$ in the upper numbering (or upper ramification group) by $G^{s} = G_{ψ_{L ∕ K} (s)}$ .

Remarks 9.6.3. §

Suppose that $L ∕ K$ is a Galois extension of local fields with Galois group $G$ .

a.

Since $ϕ_{L ∕ K} = ψ_{L ∕ K}^{- 1}$ , we have $G_{t} = G^{ϕ_{L ∕ K} (t)}$ for all $t \geq - 1$ .

b.

For the same reason, we have

ψ_{L ∕ K} (s) = \int_{0}^{s} [G^{0} : G^{y}] 𝑑𝑦

for any $s \geq 0$ .

Example 9.6.4. §

Let $F_{n} = ℚ_{p} (μ_{p^{n}})$ for a prime $p$ and $n \geq 1$ . As a consequence of Example 6.5.21, we have

ψ_{F_{n} ∕ ℚ_{p}} (s) = {\begin{matrix} s & if - 1 \leq s \leq 0, \\ p^{k - 1} (1 + (p - 1) (s - k + 1)) - 1 & if k - 1 \leq s \leq k with 1 \leq k \leq n - 1, \\ p^{n - 1} (1 + (p - 1) (s - n + 1)) - 1 & if s \geq n - 1 \end{matrix}

for all $s \geq - 1$ .

The following property of the $ψ$ -function is immediate from Proposition 6.5.26.

Lemma 9.6.5. §

Let $L ∕ K$ be a Galois extension of local fields and $E$ a normal subextension of $K$ in $L$ . Then

ψ_{L ∕ K} = ψ_{L ∕ E} \circ ψ_{E ∕ K} .

We also see that ramification groups in the upper numbering are compatible with quotients.

Proposition 9.6.6. §

Let $L ∕ K$ be a Galois extension of fields with Galois group $G$ , let $E ∕ K$ be a Galois subextension, and set $N = Gal (L ∕ E)$ . For any $s \geq - 1$ , one has

{(G ∕ N)}^{s} = G^{s} N ∕ N .

Proof.

By definition of the upper numbering and the function $ψ_{L ∕ E}$ , Herbrand’s theorem, and Lemma 9.6.5, we have

{(G ∕ N)}^{s} = {(G ∕ N)}_{ψ_{E ∕ K} (s)} = G_{ψ_{L ∕ E} (ψ_{E ∕ K} (s))} N ∕ N = G_{ψ_{L ∕ K} (s)} N ∕ N = G^{s} N ∕ N .

□

We therefore have the following example.

Proposition 9.6.7. §

Let $p$ be a prime and $n \geq 1$ . Then for any $s \geq - 1$ , we have

Gal {(ℚ_{p} (μ_{p^{n}}) ∕ ℚ_{p})}^{s} = {\begin{matrix} Gal (ℚ_{p} (μ_{p^{n}}) ∕ ℚ_{p}) & if - 1 \leq s \leq 0, \\ Gal (ℚ_{p} (μ_{p^{n}}) ∕ ℚ_{p} (μ_{p^{k}})) & if k - 1 < s \leq k with 1 \leq k \leq n - 1 \\ 1 & if s > n - 1 . \end{matrix}

Proof.

This is quickly calculated using Proposition 6.5.12 and Example 9.6.4. □

Definition 9.6.8. §

Let $L ∕ K$ be a Galois extension of local fields with Galois group $G$ . A real number $s \in [- 1, \infty)$ is said to be a jump in the ramification filtration of $L ∕ K$ (in the upper numbering) if $G^{s} \neq G^{s + 𝜖}$ for all $𝜖 > 0$ .

Example 9.6.9. §

The jumps in the ramification filtration of $ℚ_{p} (μ_{p^{n}}) ∕ ℚ_{p}$ are $0, 1, 2, \dots, n - 1$ .

Note that the jumps in the ramification filtration of $ℚ_{p} (μ_{p^{n}}) ∕ ℚ_{p}$ for a prime $p$ and $n \geq 1$ are always integers, though there may seem to be no a priori reason for them to be so. In fact, the jumps in the ramification filtration of an abelian extension of local fields are always integers. The following related result is known as the Hasse-Arf theorem: in the form stated it is actually due to Hasse. We state it without proof.

Theorem 9.6.10 (Hasse). §

Let $K$ be a local field and $L$ be a finite abelian extension of $K$ with Galois group $G$ . Then the jumps in the ramification filtration of $G$ (in the upper numbering) are all integers.

We next state, also without proof, the following remarkable connection between the reciprocity map and ramification groups in the upper numbering.

Theorem 9.6.11. §

Let $K$ be a local field and $L$ be a finite abelian extension of $K$ with Galois group $G$ . Then $ρ_{L ∕ K} (U_{i} (K)) = G^{i}$ for all $i \geq 0$ .

We have the following immediate corollary.

Corollary 9.6.12. §

Let $L ∕ K$ be a finite abelian extension of local fields with Galois group $G$ . Then $G^{i}$ is trivial for some $i \geq 1$ if and only if

U_{i} (K) \subseteq N_{L ∕ K} L^{\times} .

We make the following definition.

Definition 9.6.13. §

Let $L ∕ K$ be a finite abelian extension of local fields. The conductor $𝔣_{L ∕ K}$ of the extension $L ∕ K$ is the ideal $𝔪_{K}^{r}$ , where $𝔪_{K}$ is the maximal ideal of the valuation ring of $K$ and $r$ is the smallest positive integer such that $U_{r} (K) \subseteq N_{L ∕ K} L^{\times}$ .

Remark 9.6.14. §

By Corollary 9.6.12, the conductor of a finite abelian extension $L ∕ K$ of local fields is $𝔪_{K}^{r}$ , where $r$ is the smallest integer such that the upper ramification group $Gal {(L ∕ K)}^{r}$ is trivial. This $r$ is one more than the last jump in the ramification filtration of $Gal (L ∕ K)$ , recalling the integrality of the jumps that is the Hasse-Arf theorem.

We leave as an exercise to the reader the computation of the conductor of an arbitrary finite abelian extension of $ℚ_{p}$ using local Kronecker-Weber and the computation of the upper ramification groups of $ℚ_{p} (μ_{p^{n}}) ∕ ℚ_{p}$ . The result is as follows.

Example 9.6.15. §

The conductor of a finite abelian extension $L$ of $ℚ_{p}$ is $(p^{n})$ , where $n$ is maximal such that $L$ is contained in an unramified extension of $ℚ_{p} (μ_{p^{n}})$ .

Let us consider one nontrivial example.

Proposition 9.6.16. §

Let $p$ be an odd prime and $K = ℚ_{p} (μ_{p})$ . Set $L = K ({(1 - p)}^{1 ∕ p})$ . The conductor of the extension $L ∕ K$ is ${(1 - ζ_{p})}^{2}$ .

Proof.

Note that $L ∕ ℚ_{p}$ is totally ramified of degree $p (p - 1)$ . Let $ζ_{p}$ be a primitive $p$ th root of unity in $K$ . We have that

p = N_{ℚ_{p} ({(1 - p)}^{1 ∕ p}) ∕ ℚ_{p}} (1 - {(1 - p)}^{1 ∕ p}),

so $π = 1 - {(1 - p)}^{1 ∕ p}$ is a uniformizer of $ℚ_{p} ({(1 - p)}^{1 ∕ p})$ . It follows that $v_{L} (π) = p - 1$ and then, since $v_{L} (1 - ζ_{p}) = p$ , that $λ = (1 - ζ_{p}) ∕ π$ is a uniformizer of $L$ .

For $σ \in Gal (L ∕ K)$ with $σ ({(1 - p)}^{1 ∕ p}) = ζ_{p} {(1 - p)}^{1 ∕ p}$ , we have

\frac{σ (λ)}{λ} = \frac{π}{σ (π)} .

and

σ (π) = 1 - ζ_{p} {(1 - p)}^{1 ∕ p} = π + (1 - ζ_{p}) {(1 - p)}^{1 ∕ p} .

Thus, noting that $v_{L} (1 - ζ_{p}) = p$ , we have

v_{L} (\frac{𝜎𝜆}{λ} - 1) = 1 .

It follows that the first (and last) jump in the upper numbering in the ramification filtration of $Gal (L ∕ K)$ is at $1$ , and therefore by Remark 7.4.19, we have $𝔣_{L ∕ K} = {(1 - ζ_{p})}^{2}$ , as asserted. □

9.7. Lubin-Tate formal groups

Let $R$ denote a commutative ring.

Remark 9.7.1. §

Consider the power series ring $A = R ⟦ x_{1}, \dots, x_{n} ⟧$ in $n$ variables over $R$ . The composition $f \circ g$ of $f, g \in A$ is well-defined in $A$ so long as $g$ has zero constant term, i.e., $g \in (x_{1}, \dots, x_{n})$ .

Lemma 9.7.2. §

The following are equivalent for a power series $f \in 𝑥𝑅 ⟦ x ⟧$ :

i.: $f$ has a left inverse under composition,
ii.: $f$ has a right inverse under composition,
iii.: $f \equiv 𝑢𝑥 mod (x^{2})$ with $u \in R^{\times}$ .

Moreover, if $f$ has an inverse, then it is unique.

Proof.

Suppose that $f \equiv 𝑢𝑥 mod (x^{2})$ with $u \in R^{\times}$ . Let $g_{1} = u^{- 1} x$ , and suppose we have found $g_{n} \in R [x]$ of degree at most $n$ such that $f \circ g_{n}$ and $g_{n} \circ f$ are both $x$ in $R [x] ∕ (x^{n + 1})$ . Write $f \circ g_{n} = x + a x^{n + 1} mod (x^{n + 2})$ for some $a \in R$ . We then set $g_{n + 1} = g_{n} - u^{- 1} a x^{n + 1}$ and note that

f \circ g_{n + 1} = f \circ (g_{n} - u^{- 1} a x^{n + 1}) \equiv f \circ g_{n} - a x^{n + 1} \equiv x mod (x^{n + 2})

in that ${(g_{n} - u^{- 1} a x^{n + 1})}^{k} \equiv g_{n}^{k} mod (x^{n + 2})$ for any $k \geq 2$ . Let $g = \lim_{n \to \infty} g_{n} \in R ⟦ X ⟧$ so that $f \circ g = x$ . Now, $g$ also has some right inverse $h$ , and so $x = g \circ h = g \circ f \circ g \circ h = g \circ f$ . Moreover, note that $g_{n}$ specified recursively as above is unique with the property that $f \circ g_{n} \equiv x mod (x^{n + 1})$ .

Finally, suppose that $f, g \in 𝑥𝑅 ⟦ x ⟧$ . If $f \equiv 𝑎𝑥 mod (x^{2})$ and $g \equiv 𝑏𝑥 mod x^{2}$ , then $f \circ g \equiv 𝑎𝑏𝑥 mod x^{2}$ , so if $f \circ g = x$ , then $a$ and $b$ must both be units in $R$ . □

Definition 9.7.3. §

A (commutative) formal group law over $R$ is a polynomial $F \in R ⟦ x, y ⟧$ such that

i.: $F (x, y) \equiv x + y mod {(x, y)}^{2}$ ,
ii.: $F (F (x, y), z) = F (x, F (y, z))$ in $R ⟦ x, y, z ⟧$ , and
iii.: $F (x, y) = F (y, x)$ .

Lemma 9.7.4. §

Let $F \in R ⟦ x, y ⟧$ be a formal group law. Then

a.: $F (x, y) \equiv x + y mod (𝑥𝑦)$ , and
b.: there exists a unique $ι_{F} (x) \in R ⟦ x ⟧$ such that $F (x, ι_{F} (x)) = 0$ .

Proof.

For part (a), set $f = F (x, 0)$ , so $f \equiv x mod (x^{2})$ . We also have $F (F (x, 0), 0) = F (x, 0)$ , so $f \circ f = f$ , which forces $f = x$ . For part (b), we leave it to the reader to check recursively that for any $F \in R ⟦ x, y ⟧$ having the form in part (a), there exists a unique $ι_{F} (x) \equiv - x mod (x^{2})$ with the desired property. □

Examples 9.7.5. §

a.: We have the additive formal group law $F (x, y) = x + y$ . Here, we have $ι_{F} (x) = - x$ .
b.: We have the multiplicative formal group law $G (x, y) = x + y + 𝑥𝑦$ . Note that $ι_{G} = {(x + 1)}^{- 1} - 1$ , as $G (x, y) = (x + 1) (y + 1) - 1$ .

Definition 9.7.6. §

A homomorphism $f : F \to G$ of formal group laws $F$ and $G$ is a power series $f \in 𝑥𝑅 ⟦ x ⟧$ such that $f (F (x, y)) = G (f (x), f (y))$ . We write $f \circ F = G \circ f$ for to denote that $f$ is such a homomorphism.

We can compose homomorphisms of formal group laws by composing the power series which define them, and we can add them as well.

Definition 9.7.7. §

Let $F$ and $G$ be formal groups over $R$ .

a.: The group of homomorphisms from $F$ to $G$ is the set $Hom (F, G)$ of homomorphisms from $F$ to $G$ with the operation of addition of power series.
b.: The ring of endomorphisms of $F$ is the set $End (F)$ of endomorphisms of $F$ with the operations of addition and composition of power series.

Remark 9.7.8. §

An isomorphism of formal group laws $f : F \to_{}^{\sim} G$ is a homomorphism given by a power series with an inverse $f^{- 1}$ under composition.

If $R$ is a complete local ring with maximal ideal $𝔪$ , any power series in $R ⟦ x ⟧$ converges on $𝔪$ . Given the existence of the inverse power series of Lemma 9.7.4(b), a commutative formal group law then defines the structure of an abelian group on $𝔪$ .

Definition 9.7.9. §

For a complete local ring $R$ with maximal ideal $𝔪$ , a formal group is $𝔪$ together with the group law $a +_{F} b = F (a, b)$ for $a, b \in 𝔪$ , where $F \in R ⟦ x, y ⟧$ is a formal group law.

Notation 9.7.10. §

a.: The additive formal group, with formal group law $x + y$ . is denoted $𝔾_{a}$ .
b.: The multiplicative formal group, with formal group law $x + y + 𝑥𝑦$ , is denoted $𝔾_{m}$ .

Our interest is in a class of formal groups particularly useful for studying abelian extensions of local fields. Let $K$ denote a local field with valuation ring $𝒪$ , and maximal ideal $𝔪$ . Let $q$ denote the order of the residue field $κ = 𝒪 ∕ 𝔪$ .

Definition 9.7.11. §

A Lubin-Tate power series for $K$ is a power series $f \in 𝒪 ⟦ x ⟧$ such that $f (x) \equiv x^{q} mod π$ and $f (x) \equiv 𝜋𝑥 mod (x^{2})$ , where $π$ is a uniformizer of $K$ .

Notation 9.7.12. §

For a uniformizer $π$ of $K$ , we let $𝔉_{π}$ denote the set of Lubin-Tate power series over $K$ with $f (x) \equiv 𝜋𝑥 mod (x^{2})$ .

Let us fix a uniformizer $π$ of $K$ . We omit, for now, the proof of the following key result.

Proposition 9.7.13. §

Let $f, g \in F_{π}$ . Let $ℓ = \sum_{i = 1}^{n} a_{i} x_{i}$ with $a_{i} \in 𝒪$ for $1 \leq i \leq n$ , and where the $x_{i}$ are indeterminates. Then there exists a unique $F \in 𝒪 ⟦ x_{1}, \dots, x_{n} ⟧$ such that $F \equiv ℓ mod {(x_{1}, \dots, x_{n})}^{2}$ and $f (F (x_{1}, \dots, x_{n})) = F (g (x_{1}), \dots, g (x_{n}))$ .

Proof.

Let $I = (x_{1}, \dots, x_{n})$ . Set $ℓ_{1} = ℓ$ and $F_{0} = 0$ , and suppose we have constructed $F_{k} = F_{k - 1} + ℓ_{k}$ for some $k$ , where $ℓ_{k} \in 𝒪 [x_{1}, \dots, x_{n}]$ is homogeneous of degree $k$ , such that

f \circ F_{k} \equiv F_{k} \circ g mod π I^{k + 1} .

Let $H \equiv 0 mod π$ be the homogeneous of degree $k + 1$ part of $f \circ F_{k} = F_{k} \circ g$ , and set $ℓ_{k + 1} = - {(π - π^{k + 1})}^{- 1} H \in 𝒪 [x]$ . Set $F_{k + 1} = F_{k} + ℓ_{k + 1}$ . Then

f \circ F_{k + 1} \equiv f \circ F_{k} + π ℓ_{k + 1} mod I^{k + 2},

while

F_{k + 1} \circ g \equiv F_{k} \circ g + π^{k + 1} ℓ_{k + 1} mod I^{k + 2}

so subtracting the two equations, we have

f \circ F_{k + 1} - F_{k + 1} \circ g \equiv H + (π - π^{k + 1}) ℓ_{k + 1} \equiv 0 mod I^{k + 2} .

Since $f, g \equiv x^{q} mod π$ , we also have

f \circ F_{k + 1} - F_{k + 1} \circ g \equiv {(F_{k + 1})}^{q} - F_{k + 1} (x^{q}) \equiv 0 mod π,

where ${(F_{k + 1})}^{q}$ denotes $q$ th power in the power series ring. Thus, the difference lies in $π I^{k + 1}$ , and we may continue the recursion. Setting $F = \sum_{k = 1}^{\infty} ℓ_{k}$ , the uniqueness is clear from the uniqueness of $H$ at each step. □

Definition 9.7.14. §

A Lubin-Tate formal group law associated to $f \in F_{π}$ is a formal group law $F_{f} \in 𝒪 ⟦ x, y ⟧$ , where $f \in 𝒪 [x]$ is a Lubin-Tate power series which is an endomorphism for $F_{f}$ , which is to say $f \circ F_{f} = F_{f} \circ f$ .

Taking the linear form in Proposition 9.7.13 to be $x + y$ , we see that $F_{f}$ is uniquely specified by $f$ .

Corollary 9.7.15. §

Given $f \in F_{π}$ , there exists a unique Lubin-Tate formal group law associated to $f$ .

Proof.

The proposition provides a power series $F_{f} \in 𝒪 ⟦ x, y ⟧$ with $F_{f} \equiv x + y mod {(x, y)}^{2}$ such that $f$ is an endomorphism of $F_{f}$ . That $F_{f} (x, y) = F_{f} (y, x)$ follows by the uniqueness therein, since $x + y = x + y$ . Similarly, that $F_{f} (x, F_{f} (y, z)) = F_{f} (F_{f} (x, y), z)$ follows as both commute with $f$ and have linear terms $x + y + z$ . □

We also have the following.

Corollary 9.7.16. §

Let $f \in F_{π}$ . For any $a \in 𝒪$ , there exists a unique power series ${[a]}_{f} \in 𝒪 ⟦ x ⟧$ with ${[a]}_{f} \equiv 𝑎𝑥 mod (x^{2})$ and which commutes with $f$ under composition. In particular, ${[π]}_{f} = f$ . Moreover, ${[a]}_{f}$ is an endomorphism of $F_{f}$ , and the resulting map ${[]}_{f} : 𝒪 \to End (F_{π})$ is an injective ring homomorphism.

Proof.

We take $n = 1$ , $ℓ (x) = 𝑎𝑥$ , and $g = f$ in Proposition 9.7.13 to define ${[a]}_{f}$ . To see that ${[a]}_{f}$ is an endomorphism of $F_{f}$ , note that $F_{f} \circ {[a]}_{f}$ and ${[a]}_{f} \circ F_{f}$ both have linear terms $𝑎𝑥 + 𝑎𝑦$ and commute with $f$ , so we can again use uniqueness in the proposition. The rest follows similarly by uniqueness of the power series ${[a]}_{f}$ , aside from the injectivity of the ring homomorphism they determine, which follows as ${[a]}_{f} \equiv 𝑎𝑥 mod x^{2}$ , and $𝑎𝑥 \equiv 𝑏𝑥 mod x^{2}$ if and only if $a = b$ . □

Corollary 9.7.17. §

Let $f, g \in F_{π}$ be Lubin-Tate power series for $K$ . Then $F_{f}$ and $F_{g}$ are isomorphic.

Proof.

Suppose $f \in F_{π}$ and $g \in F_{π}$ . Apply Proposition 9.7.13 with $ℓ = x$ to get a power series $h \in 𝒪 ⟦ x ⟧$ with $f \circ h = h \circ g$ . Then $F_{f} \circ h$ and $h \circ F_{g}$ both have linear terms $x + y$ . Since $f \circ (F_{f} \circ h) = F_{f} \circ (f \circ h) = (F_{f} \circ h) \circ g$ and similarly with $h \circ F_{g}$ , uniqueness gives that $F_{f} \circ h = h \circ F_{g}$ . As $h$ is invertible, we have that $h$ provides the isomorphism. □

Example 9.7.18. §

For $K = ℚ_{p}$ , set $f (x) = {(x + 1)}^{p} - 1 \in F_{p}$ . Then the multiplicative formal group law $G = (x + 1) (y + 1) - 1$ satisfies $f \circ F = {(x + 1)}^{p} {(y + 1)}^{p} - 1 = F \circ f$ , so $G = F_{f}$ . That is, the associated Lubin-Tate formal group to $f$ is $𝔾_{m}$ . We have ${[a]}_{f} = {(x + 1)}^{a} - 1 \in ℤ_{p} ⟦ x ⟧$ for $a \in ℤ_{p}$ .

The power series ${[u]}_{f}$ associated to a unit $u \in 𝒪^{\times}$ is an isomorphism of $F_{f}$ , so it can have no zeros in the maximal ideal of the valuation ring of the completion of an algebraic closure of $K$ . On the other hand, ${[π]}_{f}$ certainly can and does.

Definition 9.7.19. §

For $n \geq 0$ , the $π^{n}$ -torsion in the formal group associated to $f$ is the kernel $W_{f, n}$ of ${[π^{n}]}_{f}$ on the maximal ideal in the completion of an algebraic closure of $K$ . We refer to $W_{f, n} - W_{f, n - 1}$ for $n \geq 1$ as the primitive $π^{n}$ -torsion. The torsion in the formal group of $f$ is $W_{f} = ⋃_{n = 1}^{\infty} W_{f, n}$ .

Theorem 9.7.20. §

For $n \geq 1$ and $f \in F_{π}$ , we have the following.

a.: The field extension $K_{π, n} = K (W_{f, n})$ is a totally ramified Galois extension of $K$ , independent of $f$ .
b.: Any primitive $n$ -torsion element $ϖ_{n} \in W_{f}^{n}$ is a uniformizer in $K_{n}$ .
c.: The group $W_{f}^{n}$ is a free $(𝒪 ∕ 𝔪^{n})$ -module of rank $1$ for the action of $𝒪$ via ${[]}_{f}$ .
d.: There is an isomorphism of groups $χ_{f, n} : Gal (K_{n} ∕ K) \to {(𝒪 ∕ 𝔪^{n})}^{\times}$ with inverse taking the image of $a \in 𝒪^{\times}$ to the Galois element $σ$ such that ${[a]}_{f} (ϖ_{n}) = σ (ϖ_{n})$ .

Proof.

Suppose that $f = 𝜋𝑥 + x^{q}$ , which is $x$ times an Eisenstein polynomial. In general, we see that ${[π^{n}]}_{f} = f \circ f \circ \dots \circ f$ is ${[π^{n - 1}]}_{f}$ times an Eisenstein polynomial of degree $q^{n - 1} (q - 1)$ which has as its roots the primitive $π^{n}$ -torsion of $F_{f}$ . Setting $K_{n} = K (W_{f, n})$ , this forces $K_{n} ∕ K$ to be not just algebraic, but Galois and totally ramified of degree $q^{n - 1} (q - 1)$ , having any $ϖ_{n} \in W_{f}^{n} - W_{f}^{n - 1}$ as a uniformizer. As ${[a]}_{f} (ϖ_{n}) \equiv a ϖ_{n} mod ϖ_{n}^{2}$ for $a \in 𝒪^{\times}$ , we see that $W_{f}^{n} - W_{f}^{n - 1}$ is free of rank $1$ over ${(𝒪 ∕ 𝔪^{n})}^{\times}$ under the action of ${[]}_{f}$ . It follows that $σ (ϖ_{n}) \in W_{f}^{n}$ equals ${[a]}_{f} (ϖ_{n})$ for some $a \in 𝒪^{\times}$ , unique modulo $𝔪^{n}$ . It is then clear that $χ_{f, n}$ defines an isomorphism.

In general, let $g \in F_{π}$ , and suppose that $h$ is an isomorphism from $F_{f}$ to $F_{g}$ so that $h \circ f = g \circ h$ . Then ${[π^{n}]}_{g} (h (ϖ_{n})) = 0$ , and it follows $h$ defines an $𝒪$ -module isomorphism between $W_{f, n}$ and $W_{g, n}$ . Since $h \in 𝒪 ⟦ x ⟧$ with $h \equiv x mod x^{2}$ , we have that $h (ϖ_{n}) \in W_{g, n} - W_{g, n - 1}$ converges to a uniformizer in $K (W_{f, n})$ , and therefore $K_{n} = K (W_{g, n})$ . For $σ \in Gal (K_{n} ∕ K)$ , we have $σ (h (ϖ_{n})) = h (σ (ϖ_{n}))$ , and if $χ_{f, n} (σ) = a mod 𝔪^{n}$ , then ${[a]}_{g} (h (ϖ_{n})) = h ({[a]}_{f} (ϖ_{n})) \equiv h (σ (ϖ_{n})) \equiv σ (h (ϖ_{n})) mod 𝔪^{n}$ . Thus, the proposition holds for $g$ as it holds for $f$ . □

Observing that $𝒪^{\times} ≅ {lim_{\leftarrow}}_{n} {(𝒪 ∕ 𝔪^{n})}^{\times}$ , we have the following.

Corollary 9.7.21. §

The field $K_{π, \infty} = K (W_{f, \infty})$ is a totally ramified Galois extension of $K$ with, independent of $f \in F_{π}$ , with Galois group isomorphic to $𝒪^{\times}$ .

We omit the proof of the following lemma.

Lemma 9.7.22. §

Let $h = \sum_{i = 0}^{n} a_{i} x^{i} \in 𝔽_{q} [x]$ be monic of degree $n \geq 1$ with $\gcd (q, n) = 1$ . Then there exists $k \geq 1$ and $r \in 𝔽_{q} [x]$ with $\deg r \leq k$ and $r (0) = 1$ such that $g = x^{k} h + r$ has no multiple zeros.

Proof.

Let $m \geq 1$ be such that $q^{m} > n$ and $𝔽_{q^{m}}$ contains the roots of $h^{'}$ . Then set $k = q^{m + 1}$ and $r = - x^{q} h + 1$ . We then have $g = (x^{q^{m + 1}} - x^{q}) h + 1$ and $g^{'} = (x^{q^{m + 1}} - x^{q}) h^{'}$ . If $α$ is a root of $x^{q^{m + 1}} - x^{q}$ , then $g (α) = 1$ . If $α$ is a root of $h^{'}$ , then it lies in $𝔽_{q^{m}}$ , so $g (α) = (α^{q} - α^{q}) h (α) + 1 = 1$ as well. □

Proposition 9.7.23. §

For $n \geq 1$ , we have $N_{K_{π, n} ∕ K} K_{π, n}^{\times} = ⟨ π ⟩ U_{n} (K)$ .

Proof.

We know that $K_{π, n} = K (W_{f}^{n})$ with $f = 𝜋𝑥 + x^{q}$ . Moreover, ${[π^{n}]}_{f} = {[π^{n - 1}]}_{f} \cdot h_{n}$ , where $h_{n}$ is $π + {[π^{n - 1}]}_{f}^{q - 1}$ . This has leading coefficient $π$ , and its roots are the primitive $π^{n}$ -torsion elements for $F_{f}$ . In particular, $π = N_{K_{π, n} ∕ K} (ϖ_{n})$ if $K_{π, n} ∕ K$ is nontrivial (i.e., other than the case that $q$ even and $n = 1$ ).

It is now enough to show that the norms of units from $K_{π, n}$ are contained in $U_{n} (K)$ since local reciprocity implies that $q^{n - 1} (q - 1) = [K^{\times} : N_{K_{π, n} ∕ K} K_{π, n}^{\times}]$ . The norms of elements of $μ_{q - 1}$ are clearly trivial, so it suffices to consider norms of $1$ -units.

Let $u = 1 + \sum_{i = 1}^{\infty} a_{i} ϖ_{n}^{i} \in U_{1} (K_{π, n})$ with $a_{i} \in 𝒪_{K}$ , and set $a_{0} = 1$ . Let $p = \sum_{i = 0}^{m - 1} a_{n - i} x^{i}$ , where $m = q^{n - 1} (q - 1) n$ . Apply Lemma 9.7.22 to the reduction of $p$ modulo $π$ , and then lift the result back to $𝒪_{K}$ , obtaining $P = x^{k} p + r$ for some $r \in 𝒪_{K}$ with $r (0) = 1$ and $k \geq \deg r$ . Since $P$ has no multiple zeros modulo $π$ , its roots lie in $K^{ur}$ . Write $P = \prod_{i = 1}^{s} (x - α_{i})$ , where $s = m + k - 1$ .

Now, note that

\prod_{i = 1}^{s} (1 - α_{i} ϖ_{n}) = ϖ_{n}^{s} P (ϖ_{n}^{- 1}) \equiv 1 + \sum_{i = 1}^{\infty} a_{i} ϖ_{n}^{i} \equiv u mod ϖ_{n}^{m},

so there exists $v \in U_{m} (K_{π, n}^{ur})$ with $\prod_{i = 1}^{s} (1 - α_{i} ϖ_{n}) v = u$ . Since $N_{K_{π, n}^{ur} ∕ K^{ur}} (v) \in U_{n} (K^{ur})$ , it suffices to check that $\prod_{i = 1}^{s} N_{K_{π, n}^{ur} ∕ K^{ur}} (1 - α_{i} ϖ_{n}) \in U_{n} (K^{ur})$ . Note that

N_{K_{π, n}^{ur} ∕ K^{ur}} (1 - α_{i} ϖ_{n}) = α_{i}^{q^{n - 1} (q - 1)} \frac{{[π^{n}]}_{f} (α_{i}^{- 1})}{{[π^{n - 1}]}_{f} (α_{i}^{- 1})} .

As $P (0) = 1$ , we then have

\prod_{i = 1}^{s} N_{K_{π, n}^{ur} ∕ K^{ur}} (1 - α_{i} ϖ_{n}) = \prod_{i = 1}^{s} \frac{{[π^{n}]}_{f} (α_{i}^{- 1})}{{[π^{n - 1}]}_{f} (α_{i}^{- 1})} .

As ${[π^{n - 1}]}_{f} (x) \equiv x^{q^{n - 1}} mod π$ , each ${[π^{n - 1}]}_{f} (α_{i}^{- 1})$ is a unit, so it suffices to show that

\prod_{i = 1}^{s} {[π^{n}]}_{f} (α_{i}^{- 1}) \equiv \prod_{i = 1}^{s} {[π^{n - 1}]}_{f} (α_{i}^{- 1}) mod π^{n} 𝒪_{K^{ur}} .

Note that in fact, both sides are contained in $𝒪$ , as the set of $α_{i}$ is a union of Frobenius conjugacy classes, and we have $f (α_{i}^{- 1}) \equiv α_{i}^{- q} \equiv α_{j}^{- 1} mod π 𝒪_{K^{ur}}$ for some $1 \leq j \leq s$ . Thus,

\prod_{i = 1}^{s} f (α_{i}^{- 1}) \equiv \prod_{i = 1}^{s} α_{i}^{- 1} mod π .

We then see that $f$ applied to both sides gives a congruence modulo $π^{2}$ , and recursively we have the desired congruence. □

We then have that the intersection of the norm groups for the $K_{π, n}$ is $⟨ π ⟩$ . The following is then an easy consequence.

Theorem 9.7.24. §

The maximal abelian extension of $K$ is equal to $K^{ur} K_{π, \infty}$ for any uniformizer $π$ of $K$ .

Theorem 9.7.25. §

Let $π$ be a uniformizer of $K$ and $f \in F_{π}$ . The local reciprocity map $ρ_{K}$ for $K$ is the unique map such that

i.: the value $ρ_{K} (π)$ is the Frobenius element in $Gal (K^{ab} ∕ K)$ fixing $K_{π}$ ,
ii.: for $u \in 𝒪^{\times}$ , the value $ρ_{K} (u)$ is the unique element of $Gal (K^{ab} ∕ K^{ur})$ such that $ρ_{K} (u) (ϖ) = {[u^{- 1}]}_{f} (ϖ)$ for all $ϖ \in W_{f, \infty}$ .

Chapter 9 Local class field theory

9.1. The Brauer group of a local field

Notation 9.1.1. §

Definition 9.1.2. §

Lemma 9.1.3. §

Proposition 9.1.4. §

Corollary 9.1.5. §

Corollary 9.1.6. §

Proposition 9.1.7. §

Notation 9.1.8. §

Remark 9.1.9. §

Proposition 9.1.10. §

Corollary 9.1.11. §

Lemma 9.1.12. §

Theorem 9.1.13. §

Definition 9.1.14. §

Theorem 9.1.15. §

9.2. Local reciprocity

Definition 9.2.1. §

Theorem 9.2.2 (Local reciprocity). §

Remark 9.2.3. §

Proposition 9.2.4. §

Remark 9.2.5. §

Remark 9.2.6. §

Proposition 9.2.7. §

Proposition 9.2.8. §

Remark 9.2.9. §

9.3. Norm residue symbols

Notation 9.3.1. §

Definition 9.3.2. §

Remark 9.3.3. §

Proposition 9.3.4. §

Remark 9.3.5. §

Proposition 9.3.6. §

Definition 9.3.7. §

Theorem 9.3.8. §

Remark 9.3.9. §

Lemma 9.3.10. §

Proposition 9.3.11. §

9.4. The existence theorem

Proposition 9.4.1. §

Proposition 9.4.2. §

Proposition 9.4.3. §

Theorem 9.4.4 (Existence theorem of local CFT). §

Theorem 9.4.5 (Uniqueness theorem of local CFT). §

Remark 9.4.6. §

Theorem 9.4.7. §

Lemma 9.4.8. §

Theorem 9.4.9 (Uniqueness of the local reciprocity map). §

Proposition 9.4.10. §

Corollary 9.4.11. §

Remarks 9.4.12. §

9.5. Class field theory over ℚp

Lemma 9.5.1. §

Lemma 9.5.2. §

Lemma 9.5.3. §

Proposition 9.5.4. §

Theorem 9.5.5 (Local Kronecker-Weber). §

Corollary 9.5.6. §

Remark 9.5.7. §

Proposition 9.5.8. §

Theorem 9.5.9. §

9.6. Ramification groups and the unit filtration

Definition 9.6.1. §

Definition 9.6.2. §

Remarks 9.6.3. §

Example 9.6.4. §

Lemma 9.6.5. §

Proposition 9.6.6. §

Proposition 9.6.7. §

Definition 9.6.8. §

Example 9.6.9. §

Theorem 9.6.10 (Hasse). §

Theorem 9.6.11. §

Corollary 9.6.12. §

Definition 9.6.13. §

Remark 9.6.14. §

Example 9.6.15. §

Proposition 9.6.16. §

Chapter 9
Local class field theory

9.5. Class field theory over $ℚ_{p}$