Algebraic Number Theory — Chapter 10 Global class field theory via idèles

Chapter 10
Global class field theory via idèles

10.1. Restricted topological products

We begin by defining the notion of a restricted topological product.

Definition 10.1.1. §

Let $I$ be an indexing set. For each $i \in I$ , let $X_{i}$ be a topological space and $A_{i}$ be an open subset. The restricted topological product of the spaces $X_{i}$ relative to the open subsets $A_{i}$ is the set

\prod_{i \in I} (X_{i}, A_{i}) = {{(x_{i})}_{i \in I} \in \prod_{i \in I} X_{i} ∣ x_{i} \in A_{i} for all but finitely many i \in I},

endowed with the topology which has as a (standard) basis the open sets of the form

{{(x_{i})}_{i \in I} ∣ x_{i} \in U_{i} for i \in J and x_{i} \in A_{i} for i \in I - J},

where $J \subseteq I$ is finite and $U_{j}$ is an open subset of $X_{j}$ for each $j \in J$ . This topology on the set $\prod_{i \in I} (X_{i}, A_{i})$ is referred to as the restricted product topology.

Remark 10.1.2. §

The standard basic open sets of a restricted topological product $\prod_{i \in I} (X_{i}, A_{i})$ have the form

\prod_{j \in J} U_{j} \times \prod_{i \in I - J} A_{i}

with $J \subset I$ finite and $U_{j}$ open in $X_{j}$ for each $j \in J$ . The subspace topology on these open sets is exactly the product topology for the subspace topology from the $X_{i}$ on the sets that form its product.

Remark 10.1.3. §

The restricted topological product $\prod_{i \in I} (X_{i}, A_{i})$ does not in general have the subspace topology of the product topology on $\prod_{i \in I} X_{i}$ . That is, a standard basic open neighborhood of $\prod_{i \in I} X_{i}$ has the form

\prod_{k \in K} V_{k} \times \prod_{i \in I - K} X_{i}

with $K \subset I$ finite and $V_{k}$ open in $X_{k}$ for each $k \in K$ . Any such set contains a basic open neighborhood in the restricted product topology on $\prod_{i \in I} (X_{i}, A_{i})$ . On the other hand, no such set will have its intersection with $\prod_{i \in I} (X_{i}, A_{i})$ contained in any set of the form

\prod_{j \in J} U_{j} \times \prod_{i \in I - J} A_{i}

with $J \subset I$ finite and $U_{j}$ open in $X_{j}$ for each $j \in J$ so long as there are infinitely many $i \in I$ for which $A_{i} \neq X_{i}$ . In other words, the restricted product topology on $\prod_{i \in I} (X_{i}, A_{i})$ is finer and can be strictly finer than the subspace topology for the product topology.

The following simple lemma is quite useful.

Lemma 10.1.4. §

Let $I$ be an indexing set, and for each $i \in I$ , let $X_{i}$ and $Y_{i}$ be topological spaces, let $A_{i}$ be an open subset of $X_{i}$ , let $B_{i}$ be an open subset of $Y_{i}$ , and let $f_{i} : X_{i} \to Y_{i}$ be a continuous function. Suppose that $f_{i} (A_{i}) \subseteq B_{i}$ for all but finitely many $i \in I$ . Then the product of the maps $f_{i}$ restricts to a continuous function

\prod_{i \in I} (X_{i}, A_{i}) \to \prod_{i \in I} (Y_{i}, B_{i}) .

Proof.

For $x = {(x_{i})}_{i \in I} \in X$ , we have that $x_{i} \in A_{i}$ for almost all $i$ , so $f_{i} (x_{i}) \in A_{i}$ for almost all $i$ , and therefore $\prod_{i} f_{i} (x) \in \prod_{i \in I} (Y_{i}, B_{i})$ . Thus, it makes sense to let $f$ denote the restriction of $\prod_{i} f_{i}$ to a map between the restricted topological products.

Consider a finite subset $J$ of $I$ and open subsets $V_{j}$ of $Y_{j}$ for each $j \in J$ , and suppose $f (x) \in \prod_{j \in J} V_{j} \times \prod_{i \in I - J} Y_{i}$ . For each $j \in J$ , there exists an open neighborhood $U_{j}$ of $x_{j}$ in $X$ such that $f_{j} (U_{j}) \subseteq V_{j}$ , as $f_{j}$ is continuous. Consequently, we have that $f (\prod_{j \in J} U_{j} \times \prod_{i \in I - J} X_{i})$ is contained in $\prod_{j \in J} V_{j} \times \prod_{i \in I - J} Y_{i}$ . Thus, $f$ is continuous. □

Lemma 10.1.5. §

Let $I$ be an indexing set. For each $i \in I$ , let $X_{i}$ be a Hausdorff topological space, and let $A_{i}$ be an open subset of $X_{i}$ . Then the restricted topological product $\prod_{i \in I} (X_{i}, A_{i})$ is Hausdorff.

Proof.

This is a consequence of either the discussion of Remark 10.1.3 or Lemma 10.1.5 applied to the case that $Y_{i} = B_{i} = X_{i}$ . The product $\prod_{i \in I} X_{i}$ is Hausdorff, and if distinct $x, y \in \prod_{i \in I} (X_{i}, A_{i})$ are contained in disjoint open neighborhoods in $\prod_{i \in I} X_{i}$ , then the intersection of these neighborhoods with $\prod_{i \in I} (X_{i}, A_{i})$ are disjoint open neighborhoods of $x$ and $y$ in the latter space. □

We are often interested in the case that our sets are topological groups or rings. In the proof, we treat only the case of groups, the case of rings being analogous.

Lemma 10.1.6. §

Let $I$ be an indexing set, and for each $i \in I$ , let $G_{i}$ be a locally compact, Hausdorff topological group (resp., ring), and let $K_{i}$ an open subgroup (subring) of $G_{i}$ that is compact for almost all $i$ . Then the restricted topological product $\prod_{i \in I} (G_{i}, K_{i})$ is a locally compact, Hausdorff topological group (resp., ring).

Proof.

That $𝒢 = \prod_{i \in I} (G_{i}, K_{i})$ is a group is straightforward. That is, clearly $1 = {(1)}_{i} \in G$ , and if $a = {(a_{i})}_{i}$ and $b = {(b_{i})}_{i}$ are elements of $𝒢$ , then $a_{i}, b_{i} \in K_{i}$ for all but finitely many $i \in I$ , so $𝑎𝑏 \in 𝒢$ . Similarly, $a^{- 1} \in 𝒢$ since $a_{i}^{- 1} \in K_{i}$ if $a_{i} \in K_{i}$ . It is then a topological group by applying Lemma 10.1.4 to the multiplication and inverse maps on $𝒢$ .

That $𝒢$ is Hausdorff is Lemma 10.1.5. Let $J$ be a finite subset of $I$ such that $K_{i}$ is compact for $i \in I - J$ , and let $U_{j}$ be a compact neighborhood of $1$ in $G_{j}$ for $j \in J$ . Then $\prod_{j \in J} U_{j} \times \prod_{i \in I - J} K_{i}$ is an compact neighborhood of $1$ in $𝒢$ by Tychonoff’s theorem, so $𝒢$ is locally compact. □

10.2. Adeles

We now define the ring of adeles of a global field $K$ .

Definition 10.2.1. §

Let $K$ be a global field. The ring of adeles (or adele ring) $𝔸_{K}$ of $K$ is the restricted topological product

𝔸_{K} = \prod_{v \in V_{K}} (K_{v}, 𝒪_{v}),

where $V_{K}$ is the set of all places of $K$ , where $K_{v}$ is the completion of $K$ at the place $v$ , and where $𝒪_{v}$ is the valuation ring of $K_{v}$ , which we take to be $K_{v}$ if $v$ is archimedean. An element of $𝔸_{K}$ is referred to as an adele.

Remark 10.2.2. §

Let $α$ be an adele in a global field $K$ . Then $α_{v}$ for some $v \in V_{K}$ shall denote the $v$ -coordinate of $α$ .

Lemma 10.2.3. §

Let $a \in K$ . Then $a \in 𝒪_{v}$ for all but finitely many places $v$ of $K$ .

Proof.

It suffices to check this on the cofinite subset of finite places of $K$ in $V_{K}$ . Only those finitely many finite primes $𝔭$ occur in the factorization of $a 𝒪_{K}$ , so only finitely many have negative valuation $v_{𝔭} (a)$ . The result follows. □

As an immediate consequence of Lemma 10.2.3, we see that every element of a global field gives rise to an element of its adele ring (since units are in particular integers).

Definition 10.2.4. §

The diagonal embedding $δ_{K} : K \to 𝔸_{K}$ is the homomorphism $δ_{K} (a) = {(a)}_{v \in V_{K}}$ .

We note also that we have embeddings of adele rings into adele rings of extension fields.

Definition 10.2.5. §

For $L ∕ K$ finite, the canonical embedding of $𝔸_{K}$ in $𝔸_{L}$ is the map $ι_{L ∕ K} : 𝔸_{K} \to 𝔸_{L}$ given by $ι_{L ∕ K} {(α)}_{w} = α_{v}$ for every place $w$ of $L$ and the place $v$ of $w$ lying below it.

We will show that the diagonal embedding has discrete image. This is rather straightforward for $K = ℚ$ , for example, so we proceed by reduction to this case, using the following result. Note that the diagonal embedding provides $𝔸_{K}$ with the structure of a $K$ -vector space.

Proposition 10.2.6. §

Let $L ∕ K$ be a finite, separable extension of global fields. Then there is a canonical isomorphism of topological $L$ -algebras

κ : L \otimes_{K} 𝔸_{K} \to_{}^{\sim} 𝔸_{L}

given on simple tensors of $b \in L$ and $α \in 𝔸_{K}$ by

κ (b \otimes α) = δ_{L} (b) ι_{L ∕ K} (α) = {(b α_{v})}_{w},

where $v$ is used to denote the place of $K$ lying below a place $w$ of $L$ . Here, a choice of $K$ -basis of $L$ provides an isomorphism $𝔸_{K} \otimes_{K} 𝐿 ≅ 𝔸_{K}^{[L : K]}$ of $𝔸_{K}$ -modules, and we give $𝔸_{K} \otimes_{K} L$ the topology induced via this isomorphism from the product topology on $𝔸_{K}^{[L : K]}$ .

Proof.

Recall that Proposition 6.1.6 says that there is, for each place $v$ of $K$ , an isomorphism

κ_{v} : L \otimes_{K} K_{v} \to_{}^{\sim} \prod_{w ∣ v} L_{w},

and the map $κ$ as defined is the restriction of the product of these to $L \otimes_{K} 𝔸_{K}$ , which has image in $𝔸_{L}$ since both $δ_{L}$ and $ι_{L ∕ K}$ do.

Since the product of the maps $κ_{v}$ is an isomorphism, the map $κ$ is an injection that we claim is surjective. For this, let $b_{1}, \dots, b_{n}$ be a $K$ -basis for $L$ . For all but finitely many finite primes $v$ of $K$ , we have for all places $w$ of $L$ over $v$ both that $b_{i} \in 𝒪_{w}$ for $1 \leq i \leq n$ and that $w (D (b_{1}, \dots, b_{n})) = 0$ . For any such $v$ , the map $κ_{v}$ restricts to a map

κ_{v}^{'} : ⨁_{i = 1}^{n} (1 \otimes 𝒪_{v}) (b_{i} \otimes 1) \to \prod_{w ∣ v} 𝒪_{w}

of free $𝒪_{v}$ -modules of rank $n$ , which by Proposition 6.1.10 is an isomorphism.

Given this, choose a finite set $S$ of places of $K$ containing the infinite places and those finite places $v$ with $b_{i} \notin 𝒪_{w}$ for some $i$ or $w (D (b_{1}, \dots, b_{n})) \neq 0$ for some $w$ dividing $v$ , and let $T$ be the finite set of places of $L$ above it. Since $κ_{v}$ is surjective for all $v \in S$ and $κ_{v}^{'}$ is surjective for all $v \notin S$ , the image of $κ$ contains

\prod_{w \in T} L_{w} \times \prod_{w \in V_{L} - T} 𝒪_{w} .

Since any arbitrary finite set of places of $L$ is contained in some such set $T$ , it follows that $κ$ is surjective.

For continuity of $κ$ , note that each $f_{i} : 𝔸_{K} \to 𝔸_{L}$ defined by $f_{i} (α) = δ_{L} (b_{i}) ι_{L ∕ K} (α)$ is continuous, noting that $ι_{L ∕ K}$ is continuous by Lemma 10.1.4. Then $κ$ viewed as a map

𝔸_{K}^{[L : K]} \to 𝔸_{L}

using this basis is continuous as a sum of the continuous maps $f_{i}$ , since $𝔸_{L}$ is a topological group under addition. □

We next show that $K$ sits discretely in its adele ring.

Proposition 10.2.7. §

The diagonal embedding $δ_{K} : K \to 𝔸_{K}$ has discrete image.

Proof.

If $K$ is a number field, Proposition 10.2.6 for the extension $K ∕ ℚ$ identifies $κ^{- 1} \circ δ_{K}$ for the map $κ$ therein with the map

{id}_{K} \otimes δ_{ℚ} : K \otimes_{ℚ} ℚ \to K \otimes_{ℚ} 𝔸_{ℚ} .

Therefore, it suffices to show that $δ_{ℚ}$ has discrete image. (Similarly, for function fields of characteristic $p$ , it suffices to consider $ι_{𝔽_{p} (t)}$ , for which we omit the proof.)

We identify $ℚ$ with its image in $𝔸_{ℚ}$ under $δ_{ℚ}$ and endow it with the subspace topology. The intersection of the open neighborhood

\prod_{p prime} ℤ_{p} \times {a \in ℝ ∣ | a | < 1}

of $0$ in $𝔸_{ℚ}$ with $ℚ$ consists of elements with nonnegative valuation at $p$ for every $p$ , which to say integers, that also have absolute value less than $1$ . In other words, the intersection is ${0}$ . Since $𝔸_{ℚ}$ is a topological ring and $ℚ$ is a subring, we have by translation that $ℚ$ has discrete image. □

Notation 10.2.8. §

We use the diagonal embedding $δ_{K}$ to identify a global field $K$ with a subring of $𝔸_{K}$ , denoted also by $K$ .

We also have the following.

Proposition 10.2.9. §

Let $K$ be a global field. Then there exists a finite set $S$ of places of $K$ including the archimedean places and positive real numbers $𝜖_{v}$ for each $v \in S$ such that

X = \prod_{v \in V_{K} - S} 𝒪_{v} \times \prod_{v \in S} {a \in K_{v} ∣ | a | \leq 𝜖_{v}} .

(10.2.1)

satisfies $𝔸_{K} = K + X$ .

Proof.

First consider the case $K = ℚ$ . In this case, we claim that the set

Y = \prod_{p prime} ℤ_{p} \times {a \in ℝ ∣ | a | \leq \frac{1}{2}}

satisfies $𝔸_{ℚ} = ℚ + Y$ . Let $α \in 𝔸_{ℚ}$ . Let $p_{1}, \dots, p_{n}$ be the finite list of prime numbers $p$ such that $α_{p} \notin ℤ_{p}$ . Let $k_{i} = - v_{p_{i}} (α_{p_{i}})$ for each $i$ with $1 \leq i \leq n$ . Let $b = p_{1}^{k_{1}} \dots p_{n}^{k_{n}}$ . Let $a \in ℤ$ be the unique integer such that

- \frac{b}{2} < a - b α_{\infty} \leq \frac{b}{2}

and

a \equiv b α_{p_{i}} mod p_{i}^{k_{i}} ℤ_{p_{i}}

for each $i$ . Then $α - \frac{a}{b} \in Y$ , as desired.

For an arbitrary number field, choose a basis $b_{1}, \dots, b_{n}$ of $K$ as a $ℚ$ -vector space, and note that Proposition 10.2.6 tells us that every element $β \in 𝔸_{K}$ has the form

β = \sum_{i = 1}^{n} b_{i} ι_{K ∕ ℚ} (α_{i})

for some $α_{1}, \dots, α_{n} \in 𝔸_{ℚ}$ . In turn, each $α_{i}$ may be written as $α_{i} = y_{i} + c_{i}$ with $y_{i} \in Y$ and $c_{i} \in ℚ$ . We then have

β \in K + \sum_{i = 1}^{n} b_{i} ι_{K ∕ ℚ} (Y),

and we let $X^{'}$ denote the latter sum. We let $S$ be the set consisting of the archimedean places of $K$ and the nonarchimedean places of $K$ such that $b_{i} \notin 𝒪_{v}$ for some $i$ , and we note that every element $α \in X^{'}$ satisfies $α_{v} \in 𝒪_{v}$ for all $v \notin S$ . For each nonarchimedean (resp., archimedean) $v \in S$ , let $𝜖_{v}$ be such that $| b_{i} |_{v} \leq 𝜖_{v}$ (resp., $| b_{i} |_{v} \leq 2 𝜖_{v}$ ) for all $1 \leq i \leq n$ . Then $X$ as defined by (10.2.1) contains $X^{'}$ and satisfies $𝔸_{K} = K + X$ .

The case of a function field is similarly derived from the case $K = 𝔽_{p} (t)$ , and the proof is left to the reader. □

Corollary 10.2.10. §

Let $K$ be a global field. Then the quotient $K$ -vector space $𝔸_{K} ∕ K$ is compact Hausdorff.

Proof.

The compact (Hausdorff) set $X$ of Proposition 10.2.9 maps onto the quotient $𝔸_{K} ∕ K$ under the continuous projection map from $𝔸_{K}$ . The space $𝔸_{K} ∕ K$ is Hausdorff as $K$ is closed in $𝔸_{K}$ , so $𝔸_{K} ∕ K$ is compact as well. □

Since an adele necessarily has valuation less than or equal to $1$ in all but finitely many places, the infinite product in the following definition converges.

Definition 10.2.11. §

Let $K$ be a global field. Let $α \in 𝔸_{K}$ . The content $c_{K} (α)$ of $α$ is defined to be

c_{K} (α) = \prod_{v \in V_{K}} ∥ α_{v} ∥_{v},

where we recall that $∥ α_{v} ∥_{v} = | α_{v} |_{v}$ unless $v$ is complex, in which case $∥ α_{v} ∥_{v} = | α_{v} |_{v}^{2}$ .

Lemma 10.2.12. §

Let $K$ be a global field. Then there exists a positive real number $C$ such that for every $α \in 𝔸_{K}$ with $c_{K} (α) > C$ , there exists an element $a \in K^{\times}$ with $| a |_{v} \leq | α_{v} |_{v}$ for all places $v$ of $K$ .

Sketch of proof.

Since $𝔸_{K}$ is a locally compact abelian group, it has an invariant Haar measure. Letting $S$ denote the set of archimedean places of $K$ , we set

Z = \prod_{v \in V_{K} - S} 𝒪_{v} \times \prod_{v \in S} {\bar{B}}_{1 ∕ 2} (0)

where $B_{1 ∕ 2} (0)$ denotes the closed ball of radius $1 ∕ 2$ around $0$ under the usual absolute value corresponding to $v$ . We normalize our Haar measure so that $Z$ has volume $1$ . As $𝔸_{K} ∕ K$ is compact, it has finite quotient measure, and we take $C$ to be this measure.

Now let $α$ be as in the statement. The set $𝛼𝑍$ has measure $c_{K} (α) > C$ (which we leave to the reader to verify, using uniqueness of Haar measure and noting that local Haar measures will scale by the multiplicative valuation used in defining the content), so it follows that there exist two distinct elements $β$ , $β^{'}$ of $𝛼𝑍$ with the same image in $𝔸_{K} ∕ K$ , which is to say that the difference $a = β - β^{'}$ lies in $K$ . For each $v \in V_{K}$ , we clearly have $| β - β^{'} |_{v} \leq | α |_{v}$ by choice of $Z$ , so the result holds. □

The lemma has the following corollary.

Corollary 10.2.13. §

Let $K$ be a global field and $u$ be a place of $K$ . Let $S$ be a finite set of places of $K$ not including $u$ , and choose a real number $𝜖_{v} > 0$ for each $v \in S$ . Then there exists an element $a \in K^{\times}$ such that $| a |_{v} \leq 𝜖_{v}$ for all $v \in S$ and $| a |_{v} \leq 1$ for all $v \notin S$ with $v \neq u$ .

Proof.

Let $C$ be as in Lemma 10.2.12. Choose elements $α_{v} \in K_{v}$ for each $v$ with $| α_{v} |_{v} \leq 𝜖_{v}$ for each $v \in S$ and $| α_{v} |_{v} = 1$ for all other places $v \neq u$ of $K$ . Let $α_{u} \in K_{u}$ be such that

∥ α_{u} ∥_{u} > C \prod_{v \in S} ∥ α_{v} ∥_{v}^{- 1} .

Setting $α = {(α_{v})}_{v} \in 𝔸_{K}$ , we then have $c_{K} (α) > C$ , so there exists an element $a \in K^{\times}$ with $| a |_{v} \leq | α_{v} |_{v}$ for all $v$ and therefore $| a |_{v} \leq 𝜖_{v}$ for all $v \in S$ and $| a_{v} |_{v} \leq 1$ for $v \notin S \cup {u}$ . □

While any global field $K$ sits discretely in $𝔸_{K}$ , if we exclude one prime from $𝔸_{K}$ , the result is very different.

Theorem 10.2.14 (Strong Approximation). §

Let $K$ be a global field, and let $u$ be a place of $K$ . Set

𝔸_{K}^{\neq u} = \prod_{v \in V_{K} - {u}} (K_{v}, 𝒪_{v}) .

Then $K$ is embedded diagonally as a dense subset of $𝔸_{K}^{\neq u}$ .

Proof.

Let $α \in 𝔸_{K}^{\neq u}$ , let $δ > 0$ , and let $T$ be a finite subset of $V_{k} - {u}$ that includes its archimedean places and places with $α_{v} \notin 𝒪_{v}$ . We claim that there exists $a \in K$ such that $| a - α_{v} |_{v} \leq δ$ for all $v \in T$ and $| a |_{v} \leq 1$ for all $v \notin T$ , which will prove the result.

By Proposition 10.2.9, we have a set $X$ of the form in (10.2.1) such that $𝔸_{K} = K + X$ . We use the notation $S$ and $𝜖_{v}$ for $v \in S$ found therein. Setting $𝜖_{v} = 1$ for $v \notin S$ , there exists by Corollary 10.2.13 an element $b \in K^{\times}$ with

| b |_{v} \leq {\begin{matrix} 𝜖_{v}^{- 1} δ & if v \in T, \\ 𝜖_{v}^{- 1} & if v \notin T \cup {u} . \end{matrix}

Note that

𝔸_{K} = b 𝔸_{K} = 𝑏𝑋 + K .

Viewing $𝔸_{K}^{\neq u}$ as the subset of $𝔸_{K}$ of elements with $u$ -coordinate $0$ , we may then write $α = 𝑏𝑥 + a$ for some $x \in X$ and $a \in K$ . Since $| b x_{v} |_{v} \leq δ$ for all $v \in T$ and $| b x_{v} |_{v} \leq 1$ for all $v \notin T \cup {u}$ , the element $a = α_{v} - b x_{v}$ for any $v \neq u$ has the desired properties. □

Let us record the rephrasing of the strong approximation theorem found in its proof. The statement is more clearly a direct generalization of weak approximation.

Corollary 10.2.15. §

Let $K$ be a global field, and let $u$ be a place of $K$ . Let $α_{v} \in K_{v}$ for each place $v \neq u$ of $K$ and suppose that $α_{v} \in 𝒪_{v}$ for all but finitely many such $v$ . Then for every $𝜖 > 0$ and finite set of primes $S$ of $K$ with $u \notin S$ , there exists $a \in K$ such that $| a - α_{v} |_{v} < 𝜖$ for all $v \in S$ and $| a |_{v} \leq 1$ for all finite places $v \notin S$ with $v \neq u$ .

We next investigate how adele rings behave in extensions. For a finite extension of $L$ of $K$ , we identify $𝔸_{K}$ with a closed subgroup of $𝔸_{L}$ via the canonical embedding.

Lemma 10.2.16. §

Let $L ∕ K$ be a finite Galois extension of global fields. Then $𝔸_{L}^{Gal (L ∕ K)} = 𝔸_{K}$ .

Proof.

The Galois group $G = Gal (L ∕ K)$ permutes the places of $L$ lying over a place $v$ of $K$ . Let

{(α_{w})}_{w} \in \prod_{\begin{array}{c} w \in V_{L} \\ w ∣ v \end{array}} L_{w}

be $G$ -invariant. Since the decomposition group $G_{w}$ at $w ∣ v$ preserves the $w$ -coordinate, it fixes $α_{w}$ . Thus, $α_{w} \in K_{v}$ , and this holds for all $w ∣ v$ . Moreover, $G$ acts on $\prod_{w ∣ v} K_{v}$ by permuting the coordinates, and the action of $G$ on the set of places is transitive, so all of the $α_{w}$ must be equal. That is, ${(α_{w})}_{w}$ is in the image of some $a \in K_{v}$ in the product $\prod_{w ∣ v} L_{w}$ . If, moreover, ${(α_{w})}_{w} \in \prod_{w ∣ v} 𝒪_{w}$ , then clearly $a \in 𝒪_{v}$ .

With respect to inclusion maps, we have

𝔸_{K} = \underset{S \subset V_{K}}{lim_{\to}} (\prod_{v \in S} K_{v} \times \prod_{v \in V_{K} - S} 𝒪_{v}),

where $S$ runs over the finite sets of places of $K$ . For such a set $S$ , let $S_{L}$ be the subset of $V_{L}$ of places lying over places in $S$ . Every finite set of places of $L$ is contained in some $S_{L}$ , so

𝔸_{L}^{G} = \underset{S \subset V_{K}}{lim_{\to}} {(\prod_{w \in S_{L}} L_{w} \times \prod_{w \in V_{L} - S_{L}} 𝒪_{w})}^{G} = \underset{S \subset V_{K}}{lim_{\to}} (\prod_{v \in S} (\prod_{\begin{array}{c} w \in V_{L} \\ w ∣ v \end{array}} L_{w})^{G} \times \prod_{v \in V_{K} - S} (\prod_{\begin{array}{c} w \in V_{L} \\ w ∣ v \end{array}} 𝒪_{w})^{G}) = 𝔸_{K} .

□

Let us consider norm and trace maps on adeles.

Definition 10.2.17. §

Let $L ∕ K$ be an extension of global fields.

a.

The norm map $N_{L ∕ K} : 𝔸_{L} \to 𝔸_{K}$ is the multiplicative function defined by

N_{L ∕ K} (β) = (\prod_{w ∣ v} N_{L ∕ K} (β_{w}))_{v}

on $β \in 𝔸_{L}$ .

b.

The trace map ${Tr}_{L ∕ K} : 𝔸_{L} \to 𝔸_{K}$ is the homomorphism

{Tr}_{L ∕ K} (β) = (\sum_{w ∣ v} {Tr}_{L ∕ K} (β_{w}))_{v} .

on $β \in 𝔸_{L}$ .

Remarks 10.2.18. §

Let $L ∕ K$ be a finite extension of global fields.

a.

That the norm and trace for $L ∕ K$ on adeles have images inside the adeles follows from the fact that $β \in 𝔸_{L}$ has $β_{w}$ in the valuation ring of $L_{w}$ for all but finitely many $w$ , and hence for all $w$ dividing $v$ for all but finitely many places $v$ of $K$ , and therefore $N_{L ∕ K} (β_{w})$ and ${Tr}_{L ∕ K} (β_{w})$ lie in the valuation ring of $K_{v}$ for all $w$ dividing $v$ for all but finitely many $v$ .

b.

In the notation of Definition 6.1.8, the norm and trace maps on adeles have $v$ -coordinates on $β \in 𝔸_{L}$ given by

\begin{array}{rcl} N_{L ∕ K} {(β)}_{v} = N_{L ∕ K}^{v} ({(β_{w})}_{w}) & and & {Tr}_{L ∕ K} {(β)}_{v} = {Tr}_{L ∕ K}^{v} ({(β_{w})}_{w}), \end{array}

where $w$ runs over the places dividing $v$ .

c.

It follows form Lemma 10.1.4 that $N_{L ∕ K}$ and ${Tr}_{L ∕ K}$ are continuous maps on adeles.

10.3. Idèles

In this section, we define the idèles, the elements of which are the units in the adeles. We continue to let $K$ denote a global field.

Definition 10.3.1. §

Let $K$ be a global field. The group of idèles (or idèle group) $𝕀_{K}$ of $K$ is the restricted topological product

𝕀_{K} = \prod_{v \in V_{K}} (K_{v}^{\times}, 𝒪_{v}^{\times}),

where $V_{K}$ is the set of all places of $K$ , where $K_{v}$ is the completion of $K$ at the place $v$ , and where $𝒪_{v}$ is the valuation ring of $K_{v}$ , for which we set $𝒪_{v}^{\times} = K_{v}^{\times}$ if $v$ is archimedean. An element of $𝕀_{K}$ is referred to as an idèle.

Remark 10.3.2. §

Note that $𝕀_{K} = 𝔸_{K}^{\times}$ as sets, but $𝕀_{K}$ does not have the subspace topology from $𝔸_{K}$ . For each finite prime $v$ , fix a uniformizer $π_{v}$ in $K_{v}$ , and let $α_{v}$ be the adele that is $π_{v}$ in its $v$ -coordinate and $1$ in every other coordinate. Then every open neighborhood of $1$ in $𝔸_{K}$ contains all but finitely many $α_{v}$ . On the other hand, the basic open neighborhood

\prod_{v \in V_{K}} 𝒪_{v}^{\times}

of $1$ in $𝕀_{K}$ contains not a single $α_{v}$ . On the other hand, the intersection of a basic open neighborhood of $𝔸_{K}$ with $𝕀_{K}$ is an open neighborhood of $𝕀_{K}$ , so the topology on $𝕀_{K}$ is strictly finer than the subspace topology from $𝔸_{K}$ .

We leave it to the reader to check the following.

Lemma 10.3.3. §

The restricted product topology on the idèle group $𝕀_{K}$ of a global field agrees with the subspace topology induced by the injection

𝕀_{K} \to 𝔸_{K} \times 𝔸_{K}, α \mapsto (α, α^{- 1}) .

Remark 10.3.4. §

Note that the content $c_{K} (α)$ of an idèle $α$ is a positive real number, as all but finitely many coordinates of $α$ will have multiplicative valuation $1$ and the valuations of the other coordinates will be nonzero. In fact, the property that of having nonzero content characterizes the idèles as a subset of the adeles, as the reader may quickly check.

We may then make the following definition.

Definition 10.3.5. §

Let $K$ be a global field. The content homomorphism $c_{K} : 𝕀_{K} \to ℝ_{> 0}$ is the function that takes an idèle $α$ to its content

c_{K} (α) = \prod_{v \in V_{K}} ∥ α_{v} ∥_{v} .

Let $𝕀_{K}^{1}$ denote the kernel of $c_{K}$ .

Proposition 10.3.6. §

Let $K$ be a global field. Then the content homomorphism $c_{K} : 𝕀_{K} \to ℝ_{> 0}$ is continuous.

Proof.

We leave it to the reader to verify the following simple claim, which implies the statement. For any $𝜖 > 0$ , there exists a sufficiently small $δ > 0$ such that $c_{K}^{- 1} ((1 - 𝜖, 1 + 𝜖))$ contains $\prod_{v \in V_{K} - S} 𝒪_{v}^{\times} \times \prod_{v \in S} B_{δ} (1)$ , where $S$ is the set of archimedean places of $K$ and $B_{δ} (1) \subset K_{v}^{\times}$ is a ball of radius $δ$ about $1$ . □

Corollary 10.3.7. §

The group $𝕀_{K}^{1}$ of idèles of content $1$ is a closed subgroup of $𝕀_{K}$

We have the following result on the kernel of the content homomorphism.

Lemma 10.3.8. §

The topology on $𝕀_{K}^{1}$ from $𝕀_{K}$ agrees with its subspace topology from $𝔸_{K}$ .

Proof.

By Remark 10.3.2, the subspace topology on $𝕀_{K}^{1}$ from $𝕀_{K}$ is finer than the subspace topology from $𝔸_{K}$ , so we need only show that the intersection of a basic open neighborhood of $1$ in $𝕀_{K}$ with $𝕀_{K}^{1}$ contains the intersection of an open neighborhood of $1$ in $𝔸_{K}$ with $𝕀_{K}^{1}$ .

Let $S$ be a finite set of places of $K$ containing the archimedean places, and for each $v \in S$ , let $U_{v}$ be an open subset of $K_{v}^{\times}$ containing $1$ . Then

U = \prod_{v \in S} U_{v} \times \prod_{v \in V_{K} - S} 𝒪_{v}^{\times}

is a basic open in $𝕀_{K}$ containing $1$ . We may suppose that the sets $U_{v}$ are chosen to be balls of sufficiently small radius such that the products of the (modified) valuations of any elements in $\prod_{v \in S} U_{v}$ is less than $2$ . Since every element $𝒪_{v} - 𝒪_{v}^{\times}$ has valuation at most $\frac{1}{2}$ , we then have

U \cap 𝕀_{K}^{1} = (\prod_{v \in S} U_{v} \times \prod_{v \in V_{K} - S} 𝒪_{v}) \cap 𝕀_{K}^{1},

and $\prod_{v \in S} U_{v} \times \prod_{v \in V_{K} - S} 𝒪_{v}$ is a basic open neighborhood of $1$ in $𝔸_{K}$ . □

Note that we may think of $K^{\times}$ as a subgroup of $𝕀_{K}$ via the diagonal embedding. By the product formula for valuations on global fields, every element of $K^{\times}$ lies in $𝕀_{K}^{1}$ .

Proposition 10.3.9. §

The image of $K^{\times}$ in $𝕀_{K}^{1}$ is discrete, and $𝕀_{K}^{1} ∕ K^{\times}$ is compact Hausdorff.

Proof.

The first statement follows from Corollary 10.2.10, since Lemma 10.3.8 tells us that $𝕀_{K}^{1}$ has the subspace topology from $𝔸_{K}$ . For the second statement, let $α \in 𝔸_{K}$ be an adele with $c_{K} (α) > C$ , where $C$ is as in the statement of Lemma 10.2.12. We define a compact subset of $𝔸_{K}$ by

X = {β \in 𝕀_{K}^{1} ∣ | β_{v} |_{v} \leq | α_{v} |_{v} for all v \in V_{K}},

where here we use the fact that $𝕀_{K}^{1}$ is closed in $𝔸_{K}$ . For an arbitrary $γ \in 𝕀_{K}^{1}$ , Lemma 10.2.12 tells us that there exists $a \in K^{\times}$ with $| a |_{v} \leq | γ_{v}^{- 1} α_{v} |_{v}$ for all places $v$ of $K$ . We then have $𝛾𝑎 \in X$ , so $X$ surjects onto the Hausdorff space $𝕀_{K}^{1} ∕ K^{\times}$ , and therefore the latter quotient is compact. □

Definition 10.3.10. §

The principal idèles of a global field $K$ are the elements of $𝕀_{K}$ that lie in $K^{\times}$ (under its diagonal embedding).

Definition 10.3.11. §

Let $K$ be a global field. Then the idèle class group $ℂ_{K}$ of $K$ is the quotient topological group $𝕀_{K} ∕ K^{\times}$ . The image $[α]$ of $α \in 𝕀_{K}$ in $ℂ_{K}$ is the idèle class of $α$ .

Notation 10.3.12. §

For a global field $K$ , we shall use $V_{K, f}$ to denote its set of finite places.

Definition 10.3.13. §

Let $K$ be a global field. The fractional ideal of $𝒪_{K}$ defined by an idèle $α$ of $K$ is the finite product

\prod_{v \in V_{K, f}} 𝔭_{v}^{v (α)},

where $𝔭_{v}$ denotes the prime corresponding to a finite place $v$ of $K$ .

Proposition 10.3.14. §

Let $K$ be a number field. Let $π_{K} : 𝕀_{K} \to I_{K}$ be the homomorphism that takes an idèle to the fractional ideal it defines. Then $π_{K} (𝕀_{K}^{1}) = I_{K}$ , and $π_{K}$ is continuous if we endow $I_{K}$ with the discrete topology.

Proof.

For the first statement, we need only show that every nonzero prime $𝔭$ is the image of an element of $𝕀_{K}^{1}$ . We may take an idèle $α$ of content $1$ that is a uniformizer $π_{𝔭}$ in the coordinate corresponding to $𝔭$ , that in a fixed archimedean place $w$ satisfies $∥ α_{w} ∥_{w} = | π_{𝔭} |_{𝔭}^{- 1}$ , and which is $1$ in all other coordinates.

For the second statement, we need only note that

π_{K}^{- 1} ({(1)}) = \prod_{v \in V_{K}} 𝒪_{v}^{\times}

is open in $𝕀_{K}$ . □

Remark 10.3.15. §

Proposition 10.3.14 enables us to give a second proof that ${Cl}_{K}$ is finite. That is, since $π_{K}$ in the proposition takes $K^{\times}$ onto $P_{K}$ , we have an induced continuous, surjective map $𝕀_{K}^{1} ∕ K^{\times} \to {Cl}_{K}$ . It follows that ${Cl}_{K}$ is both compact as the continuous image of a compact space and discrete as a quotient of $I_{K}$ , and therefore ${Cl}_{K}$ is finite.

Notation 10.3.16. §

For a finite extension $L ∕ K$ of global fields, we use $ι_{L ∕ K} : ℂ_{K} \to ℂ_{L}$ also to denote the map induced by the canonical embedding $ι_{L ∕ K} : 𝕀_{K} \to 𝕀_{L}$ .

Lemma 10.3.17. §

For a finite extension $L ∕ K$ of global fields, the map $ι_{L ∕ K} : ℂ_{K} \to ℂ_{L}$ is injective.

Proof.

Identifying $𝕀_{K}$ and $L^{\times}$ with their images in $𝕀_{L}$ , we need only see that $𝕀_{K} \cap L^{\times} = K^{\times}$ . Let $M$ be a finite Galois extension of $K$ containing $L$ . We claim that $𝕀_{K} \cap M^{\times} = K^{\times}$ , which will prove the result. But Lemma 10.2.16 gives us the first equality in

𝕀_{K} \cap M^{\times} = 𝕀_{M}^{Gal (M ∕ K)} \cap M^{\times} = {(M^{\times})}^{Gal (M ∕ K)} = K^{\times},

the second equality following from the compatibility of the Galois action on $M^{\times}$ and $𝕀_{M}$ under the diagonal embedding. □

Notation 10.3.18. §

We identify $ℂ_{K}$ with a (closed) subgroup of $ℂ_{L}$ via the embedding $ι_{L ∕ K}$ , noting Lemma 10.3.17.

As a consequence of Lemma 10.2.16, we have that $𝕀_{L}^{Gal (L ∕ K)} = 𝕀_{K}$ for any finite Galois extension $L ∕ K$ . We claim that the same holds for idèle class groups.

Lemma 10.3.19. §

Let $L ∕ K$ be a finite Galois extension of global fields. Then $ℂ_{L}^{Gal (L ∕ K)} = ℂ_{K}$ .

Proof.

We have an exact sequence of modules for $G = Gal (L ∕ K)$ given by

0 \to L^{\times} \to 𝕀_{L} \to ℂ_{L} \to 0,

and this gives rise to a long exact sequence starting

0 \to K^{\times} \to 𝕀_{K} \to ℂ_{L}^{G} \to H^{1} (G, L^{\times}) .

Since the latter group is zero by Hilbert’s Theorem 90, the resulting short exact sequence yields the result. □

Since the idèles are the units in the adele ring, the norm map on adele ring is immediately seen to define a norm map on the idèle group. Continuity of the norm follows from Lemma 10.1.4, as with adeles.

Definition 10.3.20. §

Let $L ∕ K$ be an extension of global fields. The norm map $N_{L ∕ K} : 𝕀_{L} \to 𝕀_{K}$ is the homomorphism that is the restriction of $N_{L ∕ K} : 𝔸_{L} \to 𝔸_{K}$ .

Since the norms of principal idèles are principal, we may make the following definition.

Definition 10.3.21. §

The norm map $N_{L ∕ K} : ℂ_{L} \to ℂ_{K}$ is the map induced on quotient groups by the corresponding norm map on idèle groups.

Remark 10.3.22. §

The norm map is continuous on idèle groups and idèle class groups.

10.4. Statements

The reciprocity map in the idèle-theoretic approach to global class field theory is constructed out of the local reciprocity maps of the completions of the global fields in question. We provide the preliminary results to its construction.

Lemma 10.4.1. §

Let $K$ be a global field, let $L$ be a finite abelian extension of $K$ , and let $α \in 𝕀_{K}$ . Then $ρ_{L_{w} ∕ K_{v}} (α_{v}) = 1$ for all places $w$ lying over $v$ for all but finitely many places $v$ of $K$ .

Proof.

All but finitely many $v$ are unramified in $L ∕ K$ and for all but finitely many $v$ , the valuation $v$ is nonarchimedean and $α_{v}$ is a unit in its valuation ring $𝒪_{v}$ . Since $ρ_{L_{w} ∕ K_{v}} (𝒪_{v}^{\times})$ is contained in the inertia subgroup of $Gal (L_{w} ∕ K_{v})$ for any $v$ and this inertia subgroup is trivial in an unramified extension, we have the result. □

Remark 10.4.2. §

Suppose we start with a global field $K$ and a place $v$ of $K$ . Consider the canonical map from $G_{K_{v}}^{ab}$ to a decomposition group $D_{v}$ in $G_{K}^{ab}$ at a place $w$ over $v$ . (Note that, while the map $G_{K_{v}} ↪ G_{K}$ identifying $G_{K_{v}}$ with the decomposition group at a place of $K^{sep}$ is injective, the map $G_{K_{v}}^{ab} \to G_{K}^{ab}$ it induces may not be.) The resulting map $G_{K_{v}}^{ab} \to G_{K}^{ab}$ is independent of the choice of $w$ since conjugation by an element of $G_{K}^{ab}$ is a trivial automorphism of $G_{K}^{ab}$ . We may then view the local reciprocity map as producing global elements via the composition

K_{v}^{\times} \to_{}^{ρ_{K_{v}}} G_{K_{v}}^{ab} \to_{}^{\sim} D_{v} ↪ G_{K}^{ab}

that takes $α \in K_{v}^{\times}$ to $ρ_{K_{v}} (α) |_{K^{ab}}$ , and since all of the maps in the composition are independent of $w$ , this map is as well.

The following lemma is now an immediate consequence of Remark 10.4.2.

Lemma 10.4.3. §

Let $L ∕ K$ be a finite abelian extension of global fields, let $v$ be a place of $K$ , and let $α \in K_{v}^{\times}$ . The quantity

ρ_{L_{w} ∕ K_{v}} (α) |_{L} \in Gal (L ∕ K)

for a place $w$ of $L$ lying over $v$ is independent of $w$ .

Lemmas 10.4.1 and 10.4.3 allow us to define the reciprocity map for a finite abelian extension of global fields.

Definition 10.4.4. §

Let $L ∕ K$ be a finite abelian extension of global fields. The (global) reciprocity map for $L ∕ K$ is the homomorphism $Φ_{L ∕ K} : 𝕀_{K} \to Gal (L ∕ K)$ defined by

Φ_{L ∕ K} (α) = \prod_{v \in V_{K}} ρ_{L_{w} ∕ K_{v}} (α_{v}) |_{L},

where for each valuation $v$ of $K$ , we have chosen a valuation $w$ of $L$ lying over $v$ .

We note the following compatibility among the global reciprocity maps.

Lemma 10.4.5. §

Let $K$ be a global field, and let $L$ and $M$ be finite abelian extensions of $K$ with $L \subseteq M$ . For every $α \in 𝕀_{K}$ , we have $Φ_{L ∕ K} (α) = Φ_{M ∕ K} (α) |_{L}$ .

Proof.

For each $v \in V_{K}$ , we choose $w \in V_{L}$ lying over $v$ and $u \in P_{M}$ lying over $w$ . By property (ii) of local reciprocity, we have

ρ_{L_{w} ∕ K_{v}} (α_{v}) = ρ_{M_{u} ∕ K_{v}} (α_{v}) |_{L_{w}},

and the result is then an immediate consequence of the definition of the global reciprocity map. □

Corollary 10.4.6. §

Let $K$ be a global field. Then for each $α \in 𝕀_{K}$ , the quantity

\underset{L}{lim_{\leftarrow}} Φ_{L ∕ K} (α),

with the inverse limit taken over finite abelian extensions $L$ of $K$ with respect to restriction maps, is well-defined.

We may therefore make the following definition.

Definition 10.4.7. §

Let $K$ be a global field. The (global) reciprocity map for $K$ is the homomorphism $Φ_{K} : 𝕀_{K} \to G_{K}^{ab}$ given by

Φ_{K} = \underset{L}{lim_{\leftarrow}} Φ_{L ∕ K},

where the inverse limit is taken over finite abelian extensions $L$ of $K$ with respect to restriction maps.

Remark 10.4.8. §

For $α \in 𝕀_{K}$ , we have

Φ_{K} (α) = \underset{L}{lim_{\leftarrow}} Φ_{L ∕ K} (α) = \underset{L}{lim_{\leftarrow}} \prod_{v \in V_{K}} ρ_{L_{w} ∕ K_{v}} (α_{v}) |_{L} = \prod_{v \in V_{K}} (\underset{L_{w}}{lim_{\leftarrow}} ρ_{L_{w} ∕ K_{v}} (α_{v})) |_{K^{ab}} = \prod_{v \in V_{K}} ρ_{K_{v}} (α_{v}) |_{K^{ab}},

where the first two inverse limits run over the finite abelian extensions of $K$ and the third runs over the completions of the finite abelian extensions of $K$ at a fixed prime of $K^{ab}$ over $v$ .

Our key result is now a reworking of Artin reciprocity.

Theorem 10.4.9 (Global reciprocity). §

Let $K$ be a global field.

a.: We have $Φ_{K} (a) = 1$ for all $a \in K^{\times}$ .
b.: For every finite abelian extension $L$ of $K$ , the reciprocity map $Φ_{L ∕ K}$ is surjective with kernel $K^{\times} N_{L ∕ K} (𝕀_{L})$ .

In other words, the global reciprocity map factors through the idèle class group.

Definition 10.4.10. §

Let $K$ be a global field.

a.: The global reciprocity map $ϕ_{K} : ℂ_{K} \to G_{K}^{ab}$ is the homomorphism induced on the quotient $ℂ_{K}$ of $𝕀_{K}$ by the global reciprocity map $Φ_{K}$ on idèles.
b.: The global reciprocity map $ϕ_{L ∕ K} : ℂ_{K} \to Gal (L ∕ K)$ for $L ∕ K$ is the composition of $ϕ_{K}$ with restriction to $Gal (L ∕ K)$ .

The following is then just a rewording of global reciprocity.

Theorem 10.4.11. §

Let $L ∕ K$ be a finite abelian extension of global fields. The global reciprocity map $ϕ_{L ∕ K}$ induces an isomorphism

ℂ_{K} ∕ N_{L ∕ K} ℂ_{L} \to_{}^{\sim} Gal (L ∕ K) .

Remark 10.4.12. §

For a number field $K$ , the reciprocity map $ϕ_{K}$ is surjective, and its kernel is the connected component $ℂ_{K}^{\circ}$ of $1$ in $ℂ_{K}$ . This connected component is the closure of the image of the subgroup of $𝕀_{K}$ consisting of idèles that are zero in all nonarchimedean coordinates and positive in all real coordinates.

We have the following compatibilities between reciprocity maps, which are quickly derived from the analogous result in local reciprocity.

Proposition 10.4.13. §

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

a.

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

Diagram description: Global 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: blackboard C subscript (L); column 2: G subscript (L) superscript (ab).
Row 2, from left to right: column 1: blackboard C subscript (K); column 2: G subscript (K) superscript (ab).

Arrows and lines:

An arrow from blackboard C subscript (L) to G subscript (L) superscript (ab), labelled phi subscript (L).
An arrow from blackboard C subscript (L) to blackboard C subscript (K), 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 blackboard C subscript (K) to G subscript (K) superscript (ab), labelled phi subscript (K).

where $R_{L ∕ K}$ is the restriction map on Galois groups,

b.

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

Diagram description: Global 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: blackboard C subscript (K); column 2: G subscript (K) superscript (ab).
Row 2, from left to right: column 1: blackboard C subscript (L); column 2: G subscript (L) superscript (ab).

Arrows and lines:

An arrow from blackboard C subscript (K) to G subscript (K) superscript (ab), labelled phi subscript (K).
An arrow from blackboard C subscript (K) to blackboard C subscript (L), labelled iota 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 blackboard C subscript (L) to G subscript (L) superscript (ab), labelled phi subscript (L).

where the map $ι_{L ∕ K}$ is induced by the natural injection map $𝕀_{K} \to 𝕀_{L}$ , and

c.

for any embedding $σ : L ↪ K^{sep}$ ,

Global reciprocity: conjugation. A full diagram description follows.

Diagram description: Global reciprocity: conjugation

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

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

Arrows and lines:

An arrow from blackboard C subscript (L) to G subscript (L) superscript (ab), labelled phi subscript (L).
An arrow from blackboard C subscript (L) to blackboard C subscript (sigma (L)), labelled sigma.
An arrow from G subscript (L) superscript (ab) to G subscript (sigma (L)) superscript (ab), labelled sigma superscript (star).
An arrow from blackboard C subscript (sigma (L)) to G subscript (sigma (L)) superscript (ab), labelled phi subscript (sigma (L)).

where the map $σ^{*}$ is induced by conjugation by $σ$ .

Proof.

We prove only part a, in that parts b and c is similar. For $β \in ℂ_{L}$ , we have

\begin{matrix} ϕ_{L} (β) |_{K^{ab}} = \prod_{w \in V_{L}} ρ_{L_{w}} (β_{w}) |_{K^{ab}} = \prod_{v \in V_{K}} \prod_{w ∣ v} ρ_{L_{w}} (β_{w}) |_{K^{ab}} = \prod_{v \in V_{K}} \prod_{w ∣ v} ρ_{K_{v}} (N_{L_{w} ∕ K_{v}} β_{w}) |_{K^{ab}} \\ = \prod_{v \in V_{K}} ρ_{K_{v}} (N_{L ∕ K}^{v} ({(β_{w})}_{w ∣ v})) |_{K^{ab}} = ϕ_{K} (N_{L ∕ K} β), \end{matrix}

where Remark 10.4.8 is used in the first and last equality and Proposition 9.2.7 is used in the third equality. □

Much as in the case of local class field theory, we have a one-to-one correspondence between norm subgroups of $ℂ_{K}$ and finite abelian extensions of $K$ . That norm subgroups are open follows by the same arguments as in the case of local fields.

Theorem 10.4.14 (Existence theorem of global CFT). §

The open subgroups of $ℂ_{K}$ of finite index are exactly the norm subgroups $N_{L ∕ K} ℂ_{L}$ with $L$ a finite abelian extension of $K$ .

The following consequences of the global reciprocity law and existence theorem can then be obtained much as before.

Proposition 10.4.15. §

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

a.: $N_{L ∕ K} ℂ_{L} \cap N_{M ∕ K} ℂ_{M} = N_{𝐿𝑀 ∕ K} ℂ_{𝐿𝑀}$ ,
b.: $N_{L ∕ K} ℂ_{L} \cdot N_{M ∕ K} ℂ_{M} = N_{(L \cap M) ∕ K} ℂ_{L \cap M}$ , and
c.: $N_{M ∕ K} ℂ_{M} \subseteq N_{L ∕ K} ℂ_{L}$ if and only if $L \subseteq M$ .

Theorem 10.4.16 (Uniqueness theorem of global CFT). §

Let $L$ and $M$ be distinct finite abelian extensions of a global field $K$ . Then $N_{L ∕ K} ℂ_{L} \neq N_{M ∕ K} ℂ_{M}$ .

10.5. Comparison of the approaches

In this section, we compare the ideal-theoretic and idèle-theoretic approaches to global class field theory for number fields. For this, we begin by comparing ideal groups with groups of idèles. This requires a good deal of notation.

Notation 10.5.1. §

Let $K$ be a number field, and let $𝔪$ be a modulus for $K$ . For a finite place $v$ such that the associated prime ideal $𝔭$ divides $𝔪_{f}$ , set $m_{v} = v (𝔪_{f})$ . For a real place $v$ such that the associated absolute value divides $𝔪_{\infty}$ , set $m_{v} = 1$ , and let $U_{1} (K_{v})$ denote the positive real numbers in $K_{v} = ℝ$ . For exactly those $v$ of either sort, we say that $v$ divides $𝔪$ and write $v ∣ 𝔪$ .

Definition 10.5.2. §

Let $K$ be a number field and $𝔪$ a modulus for $K$ .

a.: The $𝔪$ -idèle group is the open subgroup $𝕀_{K}^{𝔪}$ of $𝕀_{K}$ given by
$𝕀_{K}^{𝔪} = {α \in 𝕀_{K} ∣ α_{v} \in U_{m_{v}} (K_{v}) for all v ∣ 𝔪} .$
b.: The congruence subgroup of $𝕀_{K}^{𝔪}$ with modulus $𝔪$ is the open subgroup
$W_{𝔪} = \prod_{\begin{array}{c} v \in V_{K} \\ v ∣ 𝔪 \end{array}} U_{m_{v}} (K_{v}) \times \prod_{\begin{array}{c} v \in V_{K} \\ v ∤ 𝔪 \end{array}} 𝒪_{v}^{\times} .$

Proposition 10.5.3. §

Let $K$ be a number field and $𝔪$ be a modulus for $K$ .

a.

The homomorphism $π_{K}^{𝔪} : 𝕀_{K}^{𝔪} \to I_{K}^{𝔪}$ given by

π_{K}^{𝔪} (α) = \prod_{\begin{array}{c} v \in V_{K, f} \\ v ∤ 𝔪 \end{array}} 𝔭_{v}^{v (α_{v})} .

is surjective with kernel $W_{𝔪}$ .

b.

The inclusion of $𝕀_{K}^{𝔪}$ in $𝕀_{K}$ induces an isomorphism

ι_{K}^{𝔪} : 𝕀_{K}^{𝔪} ∕ K_{𝔪, 1} \to_{}^{\sim} ℂ_{K} .

Proof.

That $π_{K}^{𝔪}$ is surjective is immediate from the definitions, since elements of $𝕀_{K}^{𝔪}$ have arbitrary coodinates (with all but finitely many unit coordinates) for $v ∤ 𝔪$ . The kernel is also clearly $W_{𝔪}$ , as the requirement that an element of $𝕀_{K}^{𝔪}$ lie in the kernel is exactly that it be a unit in all finite $v$ not dividing $𝔪$ . This proves part a.

As for part b, note that $K^{\times} \cap 𝕀_{K}^{𝔪} = K_{𝔪, 1}$ , since the condition that $a \in K^{\times}$ lie in $𝕀_{K}^{𝔪}$ is exactly that it lie in each $U_{m_{v}} (K_{v})$ for $v ∣ 𝔪$ , which is to say that it lies in $K_{𝔪, 1}$ . For $α \in 𝕀_{K}$ , we may choose $b \in K^{\times}$ such that $α_{v} b^{- 1} \in U_{m_{v}} (K_{v})$ for all $v$ dividing $𝔪$ by weak approximation. It follows that $α b^{- 1} \in 𝕀_{K}^{𝔪}$ and therefore that $α \in 𝕀_{K}^{𝔪} K^{\times}$ . Since this tells us that $𝕀_{K} = 𝕀_{K}^{𝔪} K^{\times}$ , the map $ι_{K}^{𝔪}$ is onto. □

Notation 10.5.4. §

Let us use $η_{K}^{𝔪}$ to denote the composition

η_{K}^{𝔪} = {\bar{π}}_{K}^{𝔪} \circ {(ι_{K}^{𝔪})}^{- 1} : ℂ_{K} \to {Cl}_{K}^{𝔪},

where ${\bar{π}}_{K}^{𝔪} : 𝕀_{K}^{𝔪} ∕ K_{𝔪, 1} \to {Cl}_{K}^{𝔪}$ is the surjection induced by $π_{K}^{𝔪}$ , and where $π_{K}^{𝔪}$ and $ι_{K}^{𝔪}$ are as in Proposition 10.5.3.

The following now gives the comparison between the Artin map $ψ_{L ∕ K}^{𝔪} : {Cl}_{K}^{𝔪} \to Gal (L ∕ K)$ for an abelian extension $L ∕ K$ with defining modulus $𝔪$ and the global reciprocity map $ϕ_{L ∕ K} : ℂ_{K} \to Gal (L ∕ K)$ .

Theorem 10.5.5. §

Let $L ∕ K$ be a finite abelian extension of number fields and $𝔪$ a defining modulus for $L ∕ K$ . Then

ϕ_{L ∕ K} = ψ_{L ∕ K}^{𝔪} \circ η_{K}^{𝔪}

as maps from $ℂ_{K}$ to $Gal (L ∕ K)$ .

Proof.

Choose an idèle class in $ℂ_{K}$ , let $α \in 𝕀_{K}^{𝔪}$ represent it, and let $𝔞 \in I_{K}^{𝔪}$ be the product

𝔞 = \prod_{\begin{array}{c} v \in V_{K, f} \\ v ∤ 𝔪 \end{array}} 𝔭_{v}^{v (α_{v})},

which implies that ${[𝔞]}_{𝔪} = η_{K}^{𝔪} ([α])$ . We have

ψ_{L ∕ K}^{𝔪} ({[𝔞]}_{𝔪}) = \prod_{\begin{array}{c} v \in V_{K, f} \\ v ∤ 𝔪 \end{array}} ψ_{L ∕ K}^{𝔪} {({[𝔭_{v}]}_{𝔪})}^{v (α_{v})} = \prod_{\begin{array}{c} v \in V_{K, f} \\ v ∤ 𝔪 \end{array}} {(𝔭_{v}, L ∕ K)}^{v (α_{v})} = \prod_{\begin{array}{c} v \in V_{K, f} \\ v ∤ 𝔪 \end{array}} ρ_{L_{w} ∕ K_{v}} (α_{v}) .

(10.5.1)

Now, note that the conductor $𝔣_{L ∕ K}$ divides $𝔪$ and that, by Proposition 7.4.21, we have $𝔣_{L ∕ K, f} 𝒪_{v} = 𝔣_{L_{w} ∕ K_{v}}$ . Since for any finite $v$ dividing $𝔪$ and $w$ a place of $v$ lying over it we have by choice of $α$ that $α_{v} \equiv 1 mod 𝔣_{L_{w} ∕ K_{v}}$ , the definition of the local conductor tells us that $ρ_{L_{w} ∕ K_{v}} (α_{v}) = 1$ . Moreover, if $τ$ is a real embedding such that $| |_{τ}$ divides $𝔪$ , we have that $τ (α_{v}) > 0$ , so $ρ_{L_{w} ∕ K_{v}} (α_{v}) = 1$ as well. Therefore, we have

\prod_{\begin{array}{c} v \in V_{K, f} \\ v ∤ 𝔪 \end{array}} ρ_{L_{w} ∕ K_{v}} (α_{v}) = \prod_{v \in V_{K}} ρ_{L_{w} ∕ K_{v}} (α_{v}) = Φ_{L ∕ K} (α),

and this together with (10.5.1) yields the result. □

Remark 10.5.6. §

The global reciprocity map is defined a product of local reciprocity maps and computes the Artin maps for all finite abelian extensions. The Artin maps avoid much of the difficulty of ramification, as they arise from ideal groups that exclude ramified primes. The connection with local reciprocity is then much weaker, as one only sees the local maps for unramified extensions, whereby any uniformizer is taken to the unique Frobenius in the Galois group of the local extension. In that sense, the Artin maps then miss much of the complexity of the maps of local class field theory, which on the other hand is seen in the idèlic viewpoint.

Let us end this section by examining the case of class field theory over $ℚ$ .

Example 10.5.7. §

Let us verify the global reciprocity law for $ℚ$ via our computation of the local reciprocity map over $ℚ_{p}$ . The computation is given in Proposition 9.5.8, and we use it repeatedly.

Let $p$ be a prime and $k \geq 1$ . By the Kronecker-Weber theorem, it suffices to demonstrate that each $Φ_{ℚ}$ carries $- 1$ and each prime number $ℓ$ to Galois elements that act trivially on a primitive $p^{k}$ th root of unity $ζ_{p^{k}}$ .

For all primes $q \notin {p, ℓ}$ , we have that $ρ_{ℚ_{q}} (ℓ)$ fixes $ζ_{p^{k}}$ . Also, since $ℓ > 0$ , we have $ρ_{ℝ} (ℓ) = 1$ . If $ℓ = p$ , then we have

Φ_{ℚ} (ℓ) (ζ_{p^{k}}) = ρ_{ℚ_{ℓ}} (ℓ) (ζ_{p^{k}}) = ζ_{p^{k}} .

On the other hand, if $ℓ \neq p$ , then we have

Φ_{ℚ} (ℓ) (ζ_{p^{k}}) = ρ_{ℚ_{ℓ}} (ℓ) \cdot ρ_{ℚ_{p}} (ℓ) (ζ_{p^{k}}) = ρ_{ℚ_{ℓ}} (ℓ) (ζ_{p^{k}}^{ℓ^{- 1}}) = ζ_{p^{k}},

As for $- 1$ , we have that $ρ_{ℚ_{q}} (- 1)$ fixes $ζ_{p^{k}}$ for all $q \neq p$ , that $ρ_{ℚ_{p}} (- 1)$ inverts $ζ_{p^{k}}$ , and that $ρ_{ℝ} (- 1)$ is complex conjugation, so

Φ_{ℚ} (- 1) (ζ_{p^{k}}) = ρ_{ℝ} (- 1) \cdot ρ_{ℚ_{p}} (- 1) (ζ_{p^{k}}) = ρ_{ℝ} (- 1) (ζ_{p^{k}}^{- 1}) = ζ_{p^{k}} .

Remark 10.5.8. §

Recall that we did not actually prove Theorem 9.5.9 that the map constructed in Proposition 9.5.8 equals the local reciprocity map $ρ_{ℚ_{p}}$ . Via the argument of the last lemma, noting that local reciprocity maps for unramified extensions will be trivial on units and take uniformizers to the Frobenius element, we can actually use the global reciprocity law to give a short proof of this theorem.

Example 10.5.9. §

We refer to the subspace

𝕀_{ℚ}^{f} = \prod_{p prime} (ℚ_{p}^{\times}, ℤ_{p}^{\times})

of $𝕀_{ℚ}$ as the group of finite idèles. Note that

𝕀_{ℚ} = 𝕀_{ℚ}^{f} \times ℝ^{\times} .

Since $ℝ_{> 0}$ is the kernel of $ρ_{ℝ}$ , the map $Φ_{ℚ}$ factors through $𝕀_{ℚ}^{f} \times {\pm 1}$ . We have

𝕀_{ℚ} ∕ ℚ^{\times} ℝ_{> 0} ≅ (𝕀_{ℚ}^{f} \times {\pm 1}) ∕ ℚ^{\times} ≅ 𝕀_{ℚ}^{f} ∕ ℚ_{> 0} .

Now, consider the map $𝕀_{ℚ}^{f} \to ℚ_{> 0}$ that takes a finite idèle to the unique positive generator of the ideal to which it gives rise: that is, $α$ is taken to $\prod_{p} p^{v_{p} (α_{p})}$ . This is surjective with kernel

{\hat{ℤ}}^{\times} = \prod_{p} ℤ_{p}^{\times}

and splits the natural inclusion $ℚ_{> 0} \to 𝕀_{ℚ}^{f}$ . In other words, we have an identification

𝕀_{ℚ} ∕ ℚ^{\times} ℝ_{> 0} ≅ {\hat{ℤ}}^{\times},

and the global reciprocity map $ϕ_{ℚ} : ℂ_{ℚ} \to G_{ℚ}^{ab}$ factors through ${\hat{ℤ}}^{\times}$ .

The resulting map $ϕ : {\hat{ℤ}}^{\times} \to G_{ℚ}^{ab}$ is the inversion of the inverse map to the cyclotomic character $χ : G_{ℚ}^{ab} \to {\hat{ℤ}}^{\times}$ . That is, we have

ϕ (a) (ζ) = ζ^{a^{- 1}}

for all $a = {(a_{p})}_{p} \in {\hat{ℤ}}^{\times}$ and roots of unity $ζ$ . To see this, note that if $ζ_{p^{k}}$ is a primitive $p^{k}$ th root of unity for some prime $p$ and $k \geq 1$ , then Theorem 9.5.9 and our definition of $ϕ$ imply that

ϕ (a) (ζ_{p^{k}}) = ρ_{ℚ_{p}} (a_{p}) (ζ_{p^{k}}) = ζ_{p^{k}}^{a^{- 1}},

where we may make sense of $a^{- 1}$ modulo $p^{k}$ . In particular, $ϕ$ is an isomorphism, so the image of $ℝ_{> 0}$ is the kernel of $ϕ_{ℚ}$ .

10.6. Cohomology of the idèles

Let $L ∕ K$ be a finite Galois extension of global fields with Galois group $G$ . For a place $w$ of $L$ , let $G_{w}$ denote the decomposition group of $w$ in $G$ .

Lemma 10.6.1. §

Let $v$ be a place of $K$ and $u$ denote a fixed place of $L$ lying over it. We have isomorphisms of $G$ -modules

\begin{array}{rcl} {Ind}_{G_{u}}^{G} (L_{u}^{\times}) ≅ \prod_{w ∣ v} L_{w}^{\times} & and & {Ind}_{G_{u}}^{G} (𝒪_{u}^{\times}) ≅ \prod_{w ∣ v} 𝒪_{w}^{\times} . \end{array}

Proof.

The elements $σ \in G$ induce isomorphisms $σ : L_{u} \to L_{σ (u)}$ of $K$ -algebras that in particular preserve valuations. The maps are then the restriction of the map

{Ind}_{G_{u}}^{G} (L_{u}) = ℤ [G] \otimes_{ℤ [G_{u}]} L_{u} \to \prod_{w ∣ v} L_{w}

induced by the biadditive map taking $(σ, β)$ for $σ \in G$ and $β \in L_{u}$ to $σ (β) \in L_{σ (u)}$ . That this is well-defined follows from the fact that if $σ = σ^{'} τ$ for some $τ \in G_{u}$ , then $(σ, β)$ and $(σ^{'}, τ (β))$ map to the same element. It is then easily seen to be an isomorphism of $G$ -modules, and then the restrictions are isomorphisms as well. □

By Lemma 10.6.1 and Shapiro’s lemma, we have isomorphisms

\begin{array}{rcl} {\hat{H}}^{i} (G_{u}, L_{u}^{\times}) ≅ {\hat{H}}^{i} (G, \prod_{w ∣ v} L_{w}^{\times}) & and & {\hat{H}}^{i} (G_{u}, 𝒪_{u}^{\times}) ≅ {\hat{H}}^{i} (G, \prod_{w ∣ v} 𝒪_{w}^{\times}) \end{array}

for all $i \in ℤ$ . Together, these enable us to describe the Tate cohomology groups of the idèle group $𝕀_{L}$ .

Notation 10.6.2. §

For a finite set of places $S$ of $K$ , we set

𝕀_{L, S} = \prod_{v \in S} \prod_{w ∣ v} L_{w}^{\times} \times \prod_{v \notin S} \prod_{w ∣ v} 𝒪_{w}^{\times} .

Remark 10.6.3. §

We have $𝕀_{L} = {lim_{\to}}_{S} 𝕀_{L, S}$ , where $S$ runs over all finite subsets of $V_{K}$ , with the injective maps induced by inclusions of sets of places. We can of course use any cofinal set of finite sets of places here.

Proposition 10.6.4. §

Let $S$ be a finite set of places of $K$ containing the archimedean places and those places that ramify in $L ∕ K$ . We have isomorphisms

{\hat{H}}^{i} (G, 𝕀_{L, S}) ≅ ⨁_{v \in S} {\hat{H}}^{i} (G_{w}, L_{w}^{\times})

for all $i \in Z$ , where $w$ denotes a choice of place over $v \in S$ .

Proof.

As Tate cohomology commutes with products by construction, we have

\begin{array}{l} {\hat{H}}^{i} (G, 𝕀_{L, S}) & ≅ \prod_{v \in S} {\hat{H}}^{i} (G, \prod_{w ∣ v} L_{w}^{\times}) \times \prod_{v \notin S} {\hat{H}}^{i} (G, \prod_{w ∣ v} 𝒪_{w}^{\times}) \\ ≅ \prod_{v \in S} {\hat{H}}^{i} (G_{w}, L_{w}^{\times}) \times \prod_{v \notin S} {\hat{H}}^{i} (G_{w}, 𝒪_{w}^{\times}), \end{array}

where in the latter step we have fixed a place $w$ over each place $v \in V_{K}$ . Since each $v \notin S$ is unramified in $L$ , we have ${\hat{H}}^{i} (G_{w}, 𝒪_{w}^{\times}) = 0$ for all $i \in ℤ$ by Lemma 9.1.6. The result follows. □

By (9.2.1), Hilbert’s Theorem 90 and via the isomorphisms given by the local invariant maps, we have the following.

Corollary 10.6.5. §

Let $S$ be a finite set of places of $K$ containing the archimedean places and those places that ramify in $L ∕ K$ . We have $| {\hat{H}}^{0} (G, 𝕀_{L, S}) | = \prod_{v \in S} | G_{w} |$ , the group $H^{1} (G, 𝕀_{L, S})$ is trivial, and

H^{2} (G, 𝕀_{L, S}) = ⨁_{v \in S} \frac{1}{| G_{w} |} ℤ ∕ ℤ

for any choice $w$ of place of $L$ over $v \in S$ .

Proposition 10.6.6. §

We have

{\hat{H}}^{i} (G, 𝕀_{L}) ≅ ⨁_{v \in V_{K}} {\hat{H}}^{i} (G_{w}, L_{w}^{\times})

for all $i \in ℤ$ , where $w$ denotes a choice of place over $v \in V_{K}$ .

Proof.

It follows from Remark 10.6.3 that

{\hat{H}}^{i} (G, 𝕀_{L}) ≅ \underset{S}{lim_{\to}} {\hat{H}}^{i} (G, 𝕀_{L, S}),

where $S$ runs over the finite sets of places of $K$ containing the archimedean places of $K$ and those places that ramify in $L ∕ K$ . By Proposition 10.6.4, we then have

{\hat{H}}^{i} (G, 𝕀_{L}) ≅ \underset{S}{lim_{\to}} \prod_{v \in S} {\hat{H}}^{i} (G_{w}, L_{w}^{\times}) ≅ ⨁_{v \in V_{K}} {\hat{H}}^{i} (G_{w}, L_{w}^{\times}) .

□

Again, we have the following.

Corollary 10.6.7. §

We have $H^{1} (G, 𝕀_{L}) = 0$ and

H^{2} (G, 𝕀_{L}) ≅ ⨁_{v \in V_{K}} \frac{1}{| G_{w} |} ℤ ∕ ℤ .

10.7. The first inequality

We will restrict our proofs of the reciprocity laws of global class field theory to number fields. We remark that the proofs we give carry over with little-to-no change to the function field setting for extensions of degree prime to the characteristic, but the proof of the second inequality in the case of equal characteristic requires some additional work.

So, we now let $L ∕ K$ be a finite Galois extension of number fields, still with Galois group $G$ . Our interest is in the $G$ -cohomology of $ℂ_{L}$ , in that we would like to define invariant maps that make it into a class formation. Recall that we have a surjection $𝕀_{L} \to I_{L}$ taking an idèle to the fractional ideal it defines, and this map induces a surjection $ℂ_{L} \to {Cl}_{L}$ .

We will use $S$ to denote a finite set of places of $K$ , which we consistently suppose contains the archimedean places of $K$ . (In the function field setting, $S$ should be taken to be nonempty and the class group considered below should be replaced by a certain divisor class group.)

Lemma 10.7.1. §

Suppose that $S$ contains a set of finite places generating the ideal class group of $𝒪_{L}$ . Then $𝕀_{K} = 𝕀_{K, S} K^{\times}$ .

Proof.

The kernel of the surjection $𝕀_{K} \to I_{K}$ is generated by the product of the local units at finite places and local multiplicative groups at infinite places, so the kernel of $ℂ_{K} \to {Cl}_{K}$ is as well. We then note that the class group is generated by the classes of the chosen set of finite representatives $𝔭$ , and these are the images of idèles in $𝕀_{K, S}$ that are $1$ in places but that for $𝔭$ and the uniformizer at the prime. Thus $𝕀_{K} ∕ 𝕀_{K, S} K^{\times} = 0$ . (Since we take $𝒪_{w}^{\times} = L_{w}^{\times}$ for archimedean places $w$ , it is not strictly necessary to include these places in our set.) □

Definition 10.7.2. §

The ring of $S$ -integers $𝒪_{K, S}$ of $K$ is the set of elements of $K$ that lie in the valuation ring at all nonarchimedean places of $K$ not in $S$ . The $S$ -unit group of $K$ is $𝒪_{K, S}^{\times}$ .

Remark 10.7.3. §

We have $𝕀_{K, S} \cap K^{\times} = 𝒪_{K, S}^{\times}$ .

Notation 10.7.4. §

We use $𝒪_{L, S}$ to denote the $S_{L}$ -integer ring of $𝒪_{L}$ , where $S_{L}$ denotes the set of places of $L$ lying over those in $S$ .

We have the following extension of Dirichlet’s unit theorem. It also holds for function fields, though we restrict to the case of number fields.

Proposition 10.7.5. §

For a finite set $S$ of places of a number field $K$ containing its archimedean places, we have

{rank}_{ℤ} 𝒪_{K, S}^{\times} = | S | - 1 .

Proof.

By Dirichlet’s unit theorem, we know that ${rank}_{ℤ} 𝒪_{K} = r_{1} (K) + r_{2} (K) - 1$ , one less than the number of archimedean places of $K$ . We have an exact sequence

1 \to 𝒪_{K}^{\times} \to 𝒪_{K, S}^{\times} \to_{}^{\sum_{v \in S_{f}} v} ⨁_{v \in S_{f}} ℤ,

where $S_{f}$ denotes the set of finite places in $S$ . It then suffices to exhibit an $S$ -unit with nonzero additive valuation at a given $v \in S_{f}$ and trivial valuation at all other finite places of $K$ . For this, note that some power of the prime $𝔭$ corresponding to $v$ is principal, and any generator is then an $S$ -unit with the desired property. □

As with local class field theory, much can be gained from the study of cyclic extensions.

Theorem 10.7.6. §

If $L ∕ K$ is cyclic, then $h (ℂ_{L}) = [L : K]$ , for $G = Gal (L ∕ K)$ .

Proof.

Let $S$ contain the ramified places in $L ∕ K$ and a set of finite places lying below primes generating the ideal class group of $𝒪_{L}$ . We have $ℂ_{L} ≅ 𝕀_{L, S} ∕ 𝒪_{L, S}^{\times}$ by Lemma 10.7.1. Thus, we have

h (ℂ_{L}) = \frac{h (𝕀_{L, S})}{h (𝒪_{L, S}^{\times})} .

Now, consider the $ℝ$ -vector space $V$ with basis the elements of the set $S_{L}$ of places of $L$ over those in $S$ . Let $G$ act on $V$ by its canonical permutation of the standard basis. Consider its $ℤ [G]$ -submodule $A$ generated over $ℤ$ by the standard basis of $V$ . We have $𝐴 ≅ \oplus_{v \in S} {Ind}_{G_{w}}^{G} (ℤ)$ , where $w$ is again used to denote a place over $v$ . Then

h (A) ≅ \prod_{v \in S} h (G_{w}, ℤ) = \prod_{v \in S} n_{v},

where $n_{v}$ denotes the local degree of $L ∕ K$ at a prime over $v$ .

We define a second lattice as follows. We have the homomorphism $ℓ_{L, S} : 𝒪_{L, S}^{\times} \to V$ given by

ℓ_{L, S} (β) = {(\log ∥ β ∥_{w})}_{w \in S_{L}} .

By Corollary 4.4.2, we have that $\ker ℓ_{L, S}$ is finite and, by the product formula, the image $B^{0}$ of $ℓ_{L, S}$ is contained in the hyperplane $V^{0}$ of elements that sum to zero. The $𝒪_{L, S}^{\times}$ is the rank of $𝒪_{L}^{\times}$ plus the number of finite places in $S$ , as some power of any finite prime is principal, and from this and Theorem 10.7.5, we see that $B^{0}$ must be a complete lattice in the hyperplane $V^{0}$ .

Let $x = {(1)}_{w \in S_{L}} \in V^{G}$ , and set

B = ℤ𝑥 + B^{0},

which is a complete lattice in $V$ . We have an exact sequence of $G$ -modules

0 \to B^{0} \to B \to ℤ𝑥 \to 0,

so $h (B) = h (B^{0}) h (ℤ) = 𝑛h (𝒪_{L, S}^{\times})$ . On the other hand, any two complete lattices in a finite-dimensional $ℝ$ -vector space are isomorphic upon tensor product with $ℚ$ , from which one can see that their Herbrand quotients are equal. Thus we have $h (A) = h (B)$ , which tells us upon application of Corollary 10.6.5 that

h (𝒪_{L, S}^{\times}) = \frac{1}{n} h (A) = \frac{1}{n} \prod_{v \in S} n_{v} .

Combining this with our computation of $h (𝕀_{L, S})$ yields the theorem. □

As a corollary of this, we obtain what is known as the first inequality of global class field theory.

Corollary 10.7.7 (The first inequality). §

For any finite cyclic extension $L ∕ K$ , we have $[ℂ_{K} : N_{L ∕ K} ℂ_{L}] \geq [L : K]$ .

Proof.

The quantity on the left-hand side of the inequality is the order of ${\hat{H}}^{0} (G, ℂ_{L})$ , which is a multiple of the Herbrand quotient. □

The following is a simple consequence of the much stronger Čebotarev density theorem. We prove it using the first inequality.

Corollary 10.7.8. §

Let $L ∕ K$ be finite abelian and $S$ be a finite set of places of $K$ containing the archimedean places and the ramified places in $L ∕ K$ . Then $G$ is generated by the Frobenius elements in $G$ of places not in $S$ .

Proof.

We may by enlarging $S$ suppose that it contains a set of representatives of the class group of $K$ . Let $E$ be the fixed field of the subgroup of $G$ generated by the Frobenius elements of nonarchimedean places not in $S$ . Then for any $v \notin S$ and place $w$ lying over $v$ in $G$ , we have that the local extension $E_{w} ∕ K_{v}$ is trivial, and in particular that $N_{E_{w} ∕ K_{v}} E_{w}^{\times} = K_{v}^{\times}$ . Thus, we have that $N_{E ∕ K} 𝕀_{E, S} = 𝕀_{K, S}$ , and by Lemma 10.7.1 we have that $K^{\times} 𝕀_{K, S} = 𝕀_{K}$ , so $N_{E ∕ K} ℂ_{E} = ℂ_{K}$ . This implies the same equality with $E$ replaced by any cyclic subextension, and then by the first inequality, such an extension must be trivial, so $E = K$ . □

We leave it to the reader to prove the following additional consequences in a similar fashion, using Corollary 10.7.8.

Corollary 10.7.9. §

Let $L ∕ K$ be cyclic of prime power degree. Then there exist infinitely many primes of $K$ that remain inert in $L ∕ K$ .

Corollary 10.7.10. §

Let $L_{1}, \dots, L_{t}$ be cyclic extensions of $K$ of prime degree $p$ such that each $L_{i}$ is disjoint from the compositum of the $L_{j}$ for $j \neq i$ . Then there are infinitely many primes of $K$ that are inert in $L_{1}$ and split completely in $L_{i}$ for $i \geq 2$ .

10.8. The second inequality

We turn to the opposite inequality, known as the second inequality, for general Galois extensions of number fields, beginning with Kummer extensions of prime exponent.

For now, let us fix $n \geq 1$ . For a subset $T$ of $S$ , we set

I_{T} = \prod_{v \in S - T} K_{v}^{\times} \times \prod_{v \in T} K_{v}^{\times n} \times \prod_{v \notin S} 𝒪_{v}^{\times} .

Lemma 10.8.1. §

Suppose that $μ_{n} \subset K$ and $S$ contains the primes dividing $n$ . Let $T$ be a finite subset of $S$ , and set $Δ = K^{\times} \cap I_{T}$ and $L = K (Δ^{1 ∕ n})$ . Then $L ∕ K$ is unramified outside of the places in $S - T$ and completely split at the places in $T$ .

Proof.

It suffices to see that for all $a \in Δ$ , the extension $K_{v} (a^{1 ∕ n}) ∕ K_{v}$ is unramified if $v \in S - T$ and trivial if $v \in T$ . If $v \notin S$ , then $a \in 𝒪_{v}^{\times}$ , so $K_{v} (a^{1 ∕ n}) ∕ K_{v}$ is tamely ramified, being of prime-to- $p$ degree. Its is moreover unramified as the group $⟨ a ⟩ K_{v}^{\times n}$ contains no $m$ th power of a uniformizer of $K_{v}$ for $m$ properly dividing $n$ . If $v \in T$ , then $a \in K_{v}^{\times n}$ , so clearly $K_{v} (a^{1 ∕ n}) = K_{v}$ . □

The following simple group-theoretic lemma will be of use to us shortly.

Lemma 10.8.2. §

For subgroups $A$ , $B$ , and $C$ of a group with $A \leq B$ of finite index, we have

[A : B] = [𝐴𝐶 : 𝐵𝐶] [A \cap C : B \cap C] .

Proof.

By the second and third isomorphism theorems and the fact that $A \leq B$ , we have

\frac{𝐴𝐶}{𝐵𝐶} ≅ \frac{𝐴𝐶 ∕ C}{𝐵𝐶 ∕ C} ≅ \frac{A ∕ (A \cap C)}{(A \cap 𝐵𝐶) ∕ (A \cap C)} ≅ \frac{A}{(A \cap C) B} ≅ \frac{A ∕ B}{(A \cap C) B ∕ B},

and $(A \cap C) B ∕ 𝐵 ≅ (A \cap C) ∕ (B \cap C)$ . □

Lemma 10.8.3. §

Suppose that $μ_{n} \subset K$ and $S$ contains the primes over $n$ and a set of representatives for ${Cl}_{K}$ . Let $S_{1}$ be a subset of $S$ and $S_{2} = S - S_{1}$ . Let $Δ_{i} = K^{\times} \cap I_{S_{i}}$ and $L_{i} = L (Δ_{i}^{1 ∕ n})$ for $i \in {1, 2}$ . Then

a.: $I_{S_{1}} \subseteq N_{L_{2} ∕ K} 𝕀_{L_{2}}$ and $I_{S_{2}} \subseteq N_{L_{1} ∕ K} 𝕀_{L_{1}}$ , and
b.: $[ℂ_{K} : K^{\times} I_{S_{1}}] [ℂ_{K} : K^{\times} I_{S_{2}}] = [L_{1} : K] [L_{2} : K]$ .

Proof.

For part a, let $α = {(α_{v})}_{v} \in I_{S_{i}}$ with $i \in {1, 2}$ . Let $j$ be such that ${i, j} = {1, 2}$ , and let $E = L_{j}$ for brevity. For $v \in S_{i}$ , we have $α_{v} \in K_{v}^{\times n}$ . By the the local reciprocity law, the quotient of $K_{v}^{\times}$ by the norm group $N_{E_{w} ∕ K_{v}} E_{w}^{\times}$ for $w ∣ v$ has exponent dividing $n$ , so therefore $α_{v}$ lies in it. Any $v \in S_{j}$ splits completely in $E ∕ K$ , so $α_{v}$ is automatically a local norm for all places $w ∣ v$ . For all $v \notin S$ , the extension $E ∕ K$ is unramified at $v$ , and $α_{v} \in 𝒪_{v}^{\times}$ . Since the local extension is unramified, its norm group contains $𝒪_{v}^{\times}$ (and in fact is generated by it and the uniformizer of $K_{v}$ to the power of the residue degree of the extension). Thus, $α \in N_{E ∕ K} 𝕀_{E}$ . That is, we have $I_{S_{i}} \subseteq N_{E ∕ K} 𝕀_{E}$ .

As for part b, by Lemma 10.7.1 and the group-theoretic equality of indices that is Lemma 10.8.2, we have

[𝕀_{K} : K^{\times} I_{S_{1}}] = [K^{\times} 𝕀_{K, S} : K^{\times} I_{S_{1}}] = \frac{[𝕀_{K, S} : I_{S_{1}}]}{[K^{\times} \cap 𝕀_{K, S} : K^{\times} \cap I_{S_{1}}]} = \frac{[𝕀_{K, S} : I_{S_{1}}]}{[𝒪_{K, S}^{\times} : Δ_{1}]} .

Note that

𝕀_{K, S} ∕ I_{S_{1}} ≅ \prod_{v \in S_{1}} K_{v}^{\times} ∕ K_{v}^{\times n},

For a place $v$ over $p$ we have

[K_{v}^{\times} : K_{v}^{\times n}] = n^{2} \cdot ∥ n ∥_{v}^{- 1},

since $K_{v}^{\times} ≅ ℤ \times μ (K_{v}) \times ℤ_{p}^{[K_{v} : ℚ_{p}]}$ by Propositions 6.3.4 and 6.3.9 and $μ_{n} \subseteq μ (K_{v})$ . In fact, for any place $v$ , we have we still have the equality, noting that the only archimedean places for $n \geq 3$ are complex. Letting $s = | S |$ , we then have

[𝕀_{K, S} : I_{S_{1}}] [𝕀_{K, S} : I_{S_{2}}] = n^{2 s} \prod_{v \in S} ∥ n ∥_{v}^{- 1} = n^{2 s} \prod_{v \notin S} ∥ n ∥_{v} = n^{2 s}

by the product formula and the fact that every place that divides $n$ and all archimedean places lie in $S$ .

We also have $[𝒪_{K, S} : 𝒪_{K, S}^{\times n}] = n^{s}$ by Proposition 10.7.5 (which says that $𝒪_{K, S}^{\times} ≅ ℤ^{s - 1} \times μ (K)$ ), and $[Δ_{1} : 𝒪_{K, S}^{\times n}] = [L_{1} : K]$ by Kummer theory. Thus,

[𝒪_{K, S}^{\times} : Δ_{1}] [𝒪_{K, S}^{\times} : Δ_{2}] = n^{2 s} {([L_{1} : K] [L_{2} : K])}^{- 1},

and we then have

[𝕀_{K} : K^{\times} I_{S_{1}}] [𝕀_{K} : K^{\times} I_{S_{2}}] = \frac{[𝕀_{K, S} : I_{S_{1}}] [𝕀_{K, S} : I_{S_{2}}]}{[𝒪_{K, S}^{\times} : Δ_{1}] [𝒪_{K, S}^{\times} : Δ_{2}]} = [L_{1} : K] [L_{2} : K],

as claimed. □

We now specialize to the case that $n$ equals a prime $p$ .

Proposition 10.8.4. §

Let $K$ be a number field that contains the $p$ th roots of unity for a prime $p$ . Let $L ∕ K$ be finite abelian of exponent $p$ . Then $[ℂ_{K} : N_{L ∕ K} ℂ_{L}]$ divides $[L : K]$ .

Proof.

Let $k$ be such that $[L : K] = p^{k}$ , and let $a_{1}, \dots, a_{k} \in K^{\times}$ be such that $L = K (a_{1}^{1 ∕ p}, \dots, a_{k}^{1 ∕ p})$ . We aim to construct finite disjoint sets $S_{1}$ and $S_{2}$ of primes such that $S = S_{1} \cup S_{2}$ satisfies the conditions of Lemma 10.8.3 with $K^{\times} I_{S_{2}} = 𝕀_{K}$ and $L_{2} = L$ . We will then have $[ℂ_{K} : N_{L_{1} ∕ K} ℂ_{L_{1}}] = 1$ , which by the first inequality forces $L_{1} = K$ , and Lemma 10.8.3 then implies the desired divisibility.

To start, choose $S_{1}$ to consist of the archimedean places, the primes over $p$ , a set of representatives of ${Cl}_{K}$ , and every finite place $v$ such that $v (a_{i}) \neq 0$ for $1 \leq i \leq k$ . Then $K^{\times} 𝕀_{K, S_{1}} = 𝕀_{K}$ by Lemma 10.7.1, and $a_{i} \in 𝒪_{K, S_{1}}^{\times}$ for $1 \leq i \leq k$ . Let $b_{1}, \dots, b_{t} \in 𝒪_{K, S_{1}}^{\times}$ be such that the images of $a_{1}, \dots, a_{k}, b_{1}, \dots, b_{t}$ form a basis of $𝒪_{K, S_{1}}^{\times} ∕ 𝒪_{K, S_{1}}^{\times p}$ . Now, by Corollary 10.7.10, we may choose $S_{2} = {v_{1}, \dots, v_{t}}$ , where for each $v_{i}$ splits completely in $L ∕ K$ , remains inert in $K (b_{i}^{1 ∕ p}) ∕ K$ , and splits completely in $K (b_{j}^{1 ∕ p}) ∕ K$ for $j \neq i$ .

Recalling that we have set $n = p$ , we have

𝕀_{K, S_{1}} \cap I_{S_{2}} = \prod_{v \in S_{1}} K_{v}^{\times} \times \prod_{v \in S_{2}} 𝒪_{v}^{\times p} \times \prod_{v \notin S} 𝒪_{v}^{\times} .

We then have

𝕀_{K, S_{1}} ∕ (𝕀_{K, S_{1}} \cap I_{S_{2}}) ≅ \prod_{i = 1}^{t} 𝒪_{v_{i}}^{\times} ∕ 𝒪_{v_{i}}^{\times p},

and since the residue characteristic of $K_{v_{i}}$ is not $p$ and $μ_{p} \subset K_{v_{i}}^{\times}$ , we have $𝒪_{v_{i}}^{\times} ∕ 𝒪_{v_{i}}^{\times p} ≅ ℤ ∕ 𝑝ℤ$ . Note that $b_{i} \in 𝕀_{K, S_{1}}$ , and for $1 \leq i, j \leq t$ , we have $b_{i} \notin 𝒪_{v_{j}}^{\times p}$ if and only if $i = j$ . Thus, the images of the $b_{i}$ generate $𝕀_{K, S_{1}} ∕ (𝕀_{K, S_{1}} \cap I_{S_{2}})$ . Since the $b_{i}$ lie in $K$ , this tells us that $K^{\times} I_{S_{2}} = K^{\times} 𝕀_{K, S_{1}} = 𝕀_{K}$ , as desired.

Next, note that

I_{S_{2}} \subseteq 𝕀_{K, S_{1}} 𝕀_{K}^{p} = 𝕀_{K, S_{1}} {(K^{\times} 𝕀_{K, S_{1}})}^{p} = 𝕀_{K, S_{1}} K^{\times p}

by what we have just shown. In particular,

Δ_{2} = I_{S_{2}} \cap K^{\times} \subseteq 𝒪_{K, S_{1}}^{\times} K^{\times p},

and $Δ_{2} K^{\times p} ∕ K^{\times p}$ is generated by the $S_{1}$ -units that are locally $p$ th powers at all $v \in S_{2}$ . Recall that $𝒪_{K, S_{1}}^{\times} K^{\times p} ∕ K^{\times p}$ is generated by the images of $a_{1}, \dots, a_{k}, b_{1}, \dots, b_{t}$ . Since each $v_{j}$ splits completely in the subfield $K (a_{i}^{1 ∕ p})$ of $L$ , we have that $a_{i} \in 𝒪_{v_{j}}^{\times p}$ , so $a_{i} \in Δ_{2}$ for $1 \leq i \leq k$ . On the other hand, any non- $p$ th-power in $⟨ b_{1}, \dots, b_{t} ⟩$ has nontrivial image in $𝒪_{v_{j}}^{\times} ∕ 𝒪_{v_{j}}^{\times p}$ for some $j$ , so does not lie in $Δ_{2} K^{\times p}$ . Thus, the images of the $a_{i}$ generate $Δ_{2} K^{\times p} ∕ K^{\times p}$ , which is to say that $L_{2} = L$ . □

We now turn to more general extensions, no longer supposing $μ_{n} \subset K$ .

Lemma 10.8.5. §

For any finite extension $L ∕ K$ , the index $[ℂ_{K} : N_{L ∕ K} ℂ_{L}]$ is finite and divisible only by primes dividing $[L : K]$ .

Proof.

Once we have finiteness, the divisibility statement follows from the fact that for any $α \in ℂ_{K}$ , we have $α^{[L : K]} \in N_{L ∕ K} ℂ_{L}$ . For finiteness, we may suppose that $L ∕ K$ is Galois, since the norms of idèle classes from the Galois closure of $L$ to $K$ will also be norms from $L$ . By Lemma 10.7.1, we may find a finite set of primes $S$ of $K$ containing the primes that ramify in $L$ such that $𝕀_{L} = 𝕀_{L, S} L^{\times}$ and $𝕀_{K} = 𝕀_{K, S} K^{\times}$ . Then

[ℂ_{K} : N_{L ∕ K} ℂ_{L}] = [𝕀_{K} : K^{\times} N_{L ∕ K} 𝕀_{L}] = [K^{\times} 𝕀_{K, S} : K^{\times} N_{L ∕ K} 𝕀_{L, S}] \leq [𝕀_{K, S} : N_{L ∕ K} 𝕀_{L, S}] = \prod_{v \in S} [L_{w} : K_{v}],

where $w$ is any place of $L$ over $v$ , with the last equality by Proposition 10.6.5. □

For brevity, for a finite extension $E ∕ F$ of number fields, we let $n_{E ∕ F} = [ℂ_{F} : N_{E ∕ F} ℂ_{E}]$ .

Lemma 10.8.6. §

Let $M ∕ K$ be a finite Galois extension and $L$ an intermediate field. Then $n_{M ∕ K}$ divides $n_{M ∕ L} n_{L ∕ K}$ .

Proof.

Note that

n_{M ∕ K} = [N_{L ∕ K} ℂ_{L} : N_{M ∕ K} ℂ_{M}] n_{L ∕ K} .

The map $N_{L ∕ K}$ induces a surjective map

ℂ_{L} ∕ N_{M ∕ L} ℂ_{M} \to N_{L ∕ K} ℂ_{L} ∕ N_{M ∕ K} ℂ_{M}

so $[N_{L ∕ K} ℂ_{L} : N_{M ∕ K} ℂ_{M}]$ divides $n_{M ∕ L}$ . □

The following is then immediate from the multiplicativity of degrees of field extensions.

Corollary 10.8.7. §

Let $M ∕ K$ be a finite Galois extension and $L$ an intermediate field. If $n_{M ∕ L} ∣ [M : L]$ and $n_{L ∕ K} ∣ [L : K]$ , then $n_{M ∕ K} ∣ [M : K]$ .

Theorem 10.8.8. §

Let $L ∕ K$ be a finite Galois extension of number fields with Galois group $G$ . Then ${\hat{H}}^{0} (G, ℂ_{L})$ and $H^{2} (G, ℂ_{L})$ have order dividing $[L : K]$ , and $H^{1} (G, ℂ_{L}) = 0$ .

Proof.

By Lemma 9.1.12 applied in the cases $(i, r) = (0, 1), (1, 0), (2, 1)$ in that order, the result follows for arbitrary Galois extensions from the case of cyclic extensions of prime degree. So, suppose that $L ∕ K$ is cyclic of degree a prime $p$ . If we can show that $n_{L ∕ K} = | {\hat{H}}^{0} (G, ℂ_{L}) |$ divides $p$ , then the $2$ -periodicity of Tate cohomology and Theorem 10.7.6 give the result.

By Lemma 8.2.3, we have that $n_{L ∕ K} ∣ n_{L (μ_{p}) ∕ K}$ , and by Lemma 10.8.6, we have that

n_{L (μ_{p}) ∕ K} ∣ n_{L (μ_{p}) ∕ K (μ_{p})} n_{K (μ_{p}) ∕ K} .

Since $n_{K (μ_{p}) ∕ K}$ is prime to $p$ and $n_{L ∕ K}$ is a power of $p$ by Lemma 10.8.5, we have that $n_{L ∕ K}$ divides $n_{L (μ_{p}) ∕ K (μ_{p})}$ , which is $p$ by Proposition 10.8.4. □

Corollary 10.8.9 (The second inequality). §

For any finite Galois extension $L ∕ K$ , we have

[ℂ_{K} : N_{L ∕ K} ℂ_{L}] \leq [L : K] .

From the fact that $H^{1} (G, ℂ_{L}) = 0$ , we obtain the interesting consequence that in cyclic extensions, global elements that are local norms everywhere are global norms.

Corollary 10.8.10. §

Suppose that $L ∕ K$ is cyclic. If $a \in K^{\times}$ and $a \in N_{L_{w} ∕ K_{v}} L_{w}^{\times}$ for some $w ∣ v$ for all places $v \in V_{K}$ , then $a \in N_{L ∕ K} L^{\times}$ .

Proof.

Since $H^{1} (G, ℂ_{L}) = 0$ , the map

H^{2} (G, L^{\times}) \to H^{2} (G, 𝕀_{L})

is an injection. Since $G$ is cyclic, we have that the corresponding map on $0$ th Tate cohomology groups is injective as well. In other words, the map

K^{\times} ∕ N_{L ∕ K} L^{\times} \to ⨁_{v \in V_{K}} K^{\times} ∕ N_{L_{w} ∕ K_{v}} L_{w}^{\times}

is injective, noting that the norm group for $L_{w} ∕ K_{v}$ is independent of the choice of $w ∣ v$ (as $L ∕ K$ is Galois). This is exactly what was claimed. □

10.9. The reciprocity law

We continue to let $K$ denote a number field and $S$ a finite set of places of $K$ containing the archimedean places. In this section, we use $L$ to denote a finite abelian extension of $K$ with Galois group $G$ .

As noted in the proof of the Corollary 10.8.10, the triviality of $H^{1} (G, ℂ_{L})$ implies that the map

Br (L ∕ K) = H^{2} (G, L^{\times}) \to H^{2} (G, 𝕀_{L}) ≅ ⨁_{v \in S} Br (L_{w} ∕ K_{v})

is injective. In the direct limit over all Galois extensions $L$ of $K$ , we obtain an injective map

Br (K) ↪ ⨁_{v \in V_{K}} Br (K_{v}) .

Let us use ${inv}_{v} : Br (K) \to ℚ ∕ ℤ$ to denote the composition of the map $Br (K) \to Br (K_{v})$ with the local invariant map ${inv}_{K_{v}}$ . (Here, ${inv}_{ℝ}$ is the unique injection of the group $Br (ℝ)$ of order $2$ in $ℚ ∕ ℤ$ .) We see from the fact that $Br (K)$ maps to the direct sum that

\sum_{v \in V_{K}} {inv}_{v} : Br (K) \to ℚ ∕ ℤ

is well-defined. We will show that this map is zero.

In the following, we also use the notation ${inv}_{v}$ to denote the composition

{inv}_{v} : H^{2} (G, 𝕀_{L}) \to Br (K_{v}) \to_{}^{{inv}_{K_{v}}} ℚ ∕ ℤ .

Lemma 10.9.1. §

For $α \in 𝕀_{K}$ and $χ \in Hom (G, ℚ ∕ ℤ)$ , we have

\sum_{v \in V_{K}} {inv}_{v} (\bar{α} \cup 𝛿𝜒) = χ (Φ_{L ∕ K} (α)),

where $δ : H^{1} (G, ℚ ∕ ℤ) \to H^{2} (G, ℤ)$ is the connecting homomorphism for $0 \to ℤ \to ℚ \to ℚ ∕ ℤ \to 0$ , and $\bar{α}$ denotes the class of $α$ in ${\hat{H}}^{0} (G, 𝕀_{L})$ .

Proof.

By definition and the compatibility of cup products with restriction, we have

{inv}_{v} (\bar{α} \cup 𝛿𝜒) = {inv}_{K_{v}} (\bar{α_{v}} \cup δ χ_{v}),

where $\bar{α_{v}}$ denotes the image of $α_{v}$ in ${\hat{H}}^{0} (G_{v}, L_{w}^{\times})$ for a place $w$ over $v$ , where $χ_{v} \in H^{1} (G_{v}, ℚ ∕ ℤ)$ is the restriction of $χ$ to the decomposition group $G_{w}$ , and where $δ$ continues to denote the corresponding connecting homomorphism. By Proposition 8.1.10, we have that

{inv}_{K_{v}} (\bar{α_{v}} \cup δ χ_{v}) = χ_{v} (ρ_{L_{w} ∕ K_{v}} (α_{v})) .

By definition of $Φ_{L ∕ K}$ and the fact that $χ$ is a homomorphism, we have that

χ (Φ_{L ∕ K} (α)) = \sum_{v \in V_{K}} χ_{v} (ρ_{L_{w} ∕ K_{v}} (α_{v})),

hence the result. □

From the global recirpocity law for $ℚ$ , we may easily prove the global reciprocity law for cyclotomic extensions.

Lemma 10.9.2. §

Let $L$ be an extension of $K$ contained in $K (μ_{N})$ for some $N \geq 1$ . Then $Φ_{L ∕ K} (a) = 1$ for all $a \in K^{\times}$ .

Proof.

For $E = ℚ (μ_{N})$ , we have $Φ_{L ∕ K} (a) |_{E} = Φ_{E ∕ ℚ} (N_{K ∕ ℚ} (a))$ by part a of Proposition 10.4.13. Since the restriction map $G \to Gal (E ∕ ℚ)$ is injective, we are reduced to the already proven reciprocity law for $ℚ$ . □

Lemma 10.9.3. §

For any $n \geq 1$ , let $S$ denote the set of places of $K$ dividing $n$ and all real places. Then there exists a cyclic extension $L$ of $K$ contained in $K (μ_{N})$ for some $N \geq 1$ such that the local degree of $L ∕ K$ is divisible by $n$ for all $v \in S_{f}$ and is equal to $2$ for all real places of $K$ .

Proof.

Without loss of generality, we may suppose that $2$ divides $n$ . Let $n = p_{1}^{r_{1}} \dots p_{k}^{r_{k}}$ be the prime factorization of $n$ . If $p_{i}$ is odd, let $L_{i}$ be the maximal pro- $p$ subextension of the field given by adjoining to $K$ all $p_{i}$ -power roots of unity. Otherwise, let $L_{i}$ be the extension of $K$ given by adjoining $ζ - ζ^{- 1}$ for all $2$ -power roots of unity $ζ$ . The unique degree $2$ subextension of the latter field has no real places. For each $i$ , the completions of the fields $L_{i}$ at places $v \in S_{f}$ are infinite pro- $p_{i}$ procyclic extensions. In particular, the compositum $L$ of the fields $L_{i}$ is a procyclic extension of $K$ that contains a finite degree extension that is the desired subfield. □

We require the following simple cohomological lemma, the proof of which is left to the reader.

Lemma 10.9.4. §

Let $G$ be a finite cyclic group of order $n$ and $χ : G \to ℚ ∕ ℤ$ be an injective character. Let $δ$ be the connecting map for $0 \to ℤ \to ℚ \to ℚ ∕ ℤ \to 0$ . Let $g$ be a generator of $G$ , and let $u_{g} \in {\hat{H}}^{- 2} (G, ℤ)$ be as in Proposition 1.10.3. Viewing ${\hat{H}}^{0} (G, ℤ)$ and $ℤ ∕ 𝑛ℤ$ , and letting $\tilde{χ} : G \to ℤ ∕ 𝑛ℤ$ be the isomorphism obtained from $χ$ by multiplication by $n$ , we have

u_{g} \cup 𝛿𝜒 = \tilde{χ} (g) \in ℤ ∕ 𝑛ℤ .

In other words, we have the following, the map being inverse to cup product with $u_{g}$ for a generator $g \in G$ with $χ (g) = \frac{1}{n}$ .

Corollary 10.9.5. §

Let $G$ be a finite cyclic group of order $n$ and $χ : G \to ℚ ∕ ℤ$ be an injective character. Let $δ$ be the connecting map for $0 \to ℤ \to ℚ \to ℚ ∕ ℤ \to 0$ . For any $G$ -module $A$ and and $i \in ℤ$ , the map

{\hat{H}}^{i} (G, A) \to {\hat{H}}^{i + 2} (G, A), c \mapsto 𝛿𝜒 \cup c

is an isomorphism.

Proposition 10.9.6. §

The map $\sum_{v \in V_{K}} {inv}_{v} : Br (K) \to ℚ ∕ ℤ$ is trivial.

Proof.

Let $β \in Br (K)$ , and let $n$ be the least common multiple of the orders of the elements ${inv}_{v} (β)$ for $v \in V_{K}$ . Let $S$ contain the places where ${inv}_{v} (β)$ is nonzero, and let $L$ be as in Lemma 10.9.3. Then for each $v \in S$ , the group $Br (L_{w} ∕ K_{v})$ sits in $Br (K_{v})$ as as a cyclic subgroup of order a multiple of $n$ , and hence it contains the image of $β$ . It follows that $β \in Br (L ∕ K)$ . Since $L ∕ K$ is cyclic, we have an injective character $χ : G \to ℚ ∕ ℤ$ . Corollary 10.9.5 then tells us that there exists $b \in K^{\times}$ such that such that $b \cup 𝛿𝜒 = β$ . By Lemma 10.9.1, we have

\sum_{v \in V_{K}} {inv}_{v} (β) = χ (Φ_{L ∕ K} (b)),

and $Φ_{L ∕ K} (b) = 0$ by Lemma 10.9.2. □

We can now prove that the global reciprocity map factors through $ℂ_{K}$ .

Corollary 10.9.7. §

We have $Φ_{K} (a) = 1$ for all $a \in K^{\times}$ .

Proof.

It suffices to show that $Φ_{L ∕ K} (a) = 1$ for all finite abelian extensions $L ∕ K$ , and for this, it suffices to show that $χ (Φ_{L ∕ K} (a)) = 0$ for all characters $χ : G \to ℚ ∕ ℤ$ for all such $L$ . By Lemma 10.9.1, the latter quantity equals $\sum_{v \in V_{K}} {inv}_{v} (\bar{a} \cup 𝛿𝜒)$ , but this is zero by Proposition 10.9.6. □

Note that for any finite extension $L$ of $K$ , we have an injection $ℂ_{K} ↪ ℂ_{L}$ by Lemma 10.3.17. We aim to construct an invariant map $inv : H^{2} (G_{K}, ℂ_{K^{sep}}) \to ℚ ∕ ℤ$ to show that $ℂ_{K^{sep}} = {lim_{\to}}_{L} ℂ_{L}$ together with the invariant maps associated for finite separable extensions of $K$ forms a class formation.

Since since $H^{1} (G, ℂ_{L}) = 0$ and

H^{2} (G, 𝕀_{L}) ≅ ⨁_{v \in V_{K}} Br (L_{w} ∕ K_{v}),

the latter by Proposition 10.6.6, we have an exact sequence

0 \to Br (L ∕ K) \to ⨁_{v \in V_{K}} Br (L_{w} ∕ K_{v}) \to H^{2} (G, ℂ_{L}) .

Recall that we have an isomorphism

{inv}_{L_{w} ∕ K_{v}} : Br (L_{w} ∕ K_{v}) \to_{}^{\sim} \frac{1}{| G_{w} |} ℤ ∕ ℤ,

where $G_{w}$ is the decomposition group at any $w ∣ v$ . The sum of these local invariant maps

\sum_{v \in V_{K}} {inv}_{L_{w} ∕ K_{v}} : ⨁_{v \in S} Br (L_{w} ∕ K_{v}) \to \frac{1}{| G |} ℤ ∕ ℤ

is surjective due to the existence of an inert prime $w$ over some $v \in V_{K}$ . Let

{\tilde{inv}}_{L ∕ K} : H^{2} (G, 𝕀_{L}) ↠ \frac{1}{| G |} ℤ ∕ ℤ

denote the composite map. By Proposition 10.9.6, we have that $Br (L ∕ K)$ is contained in the kernel of ${\tilde{inv}}_{L ∕ K}$ . That is, ${\tilde{inv}}_{L ∕ K}$ factors through a surjective global invariant map

{inv}_{L ∕ K} : B_{L ∕ K} ↠ \frac{1}{| G |} ℤ ∕ ℤ

from the image $B_{L ∕ K}$ of $H^{2} (G, 𝕀_{L}) \to H^{2} (G, ℂ_{L})$ . We aim to show that $H^{2} (G, 𝕀_{L}) \to H^{2} (G, ℂ_{L})$ is surjective so $B_{L ∕ K} = H^{2} (G, ℂ_{K})$ , and ${inv}_{L ∕ K}$ is an injective. We start with cyclic extensions.

Lemma 10.9.8. §

Let $L ∕ K$ be finite cyclic. Then $H^{2} (G, 𝕀_{L}) \to H^{2} (G, ℂ_{L})$ is surjective, and the invariant map

{inv}_{L ∕ K} : H^{2} (G, ℂ_{L}) \to_{}^{\sim} \frac{1}{| G |} ℤ ∕ ℤ

is an isomorphism.

Proof.

We have $H^{3} (G, L^{\times}) ≅ H^{1} (G, L^{\times}) = 0$ by the periodicty of Tate cohomology, so $B_{L ∕ K} = H^{2} (G, ℂ_{L})$ . Since $H^{2} (G, ℂ_{L})$ has order $| G |$ by the first and second inequalities, the result follows. □

The next lemma shows that global invariant maps behave as expected under restriction.

Lemma 10.9.9. §

Let $E$ be a finite extension of $K$ contained in $L$ , and set $H = Gal (L ∕ E)$ . The diagram

Restriction of global invariant maps. A full diagram description follows.

Diagram description: Restriction of global invariant maps

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: H superscript (2)(G,blackboard I subscript (L)); column 2: fraction (1) over (|G|) blackboard Z / blackboard Z.
Row 2, from left to right: column 1: H superscript (2)(H,blackboard I subscript (L)); column 2: fraction (1) over (|H|) blackboard Z / blackboard Z.

Arrows and lines:

An arrow from H superscript (2)(G,blackboard I subscript (L)) to fraction (1) over (|G|) blackboard Z / blackboard Z, labelled tilde of (inv) subscript (L / K).
An arrow from H superscript (2)(G,blackboard I subscript (L)) to H superscript (2)(H,blackboard I subscript (L)), labelled Res.
An arrow from fraction (1) over (|G|) blackboard Z / blackboard Z to fraction (1) over (|H|) blackboard Z / blackboard Z, labelled [E:K].
An arrow from H superscript (2)(H,blackboard I subscript (L)) to fraction (1) over (|H|) blackboard Z / blackboard Z, labelled tilde of (inv) subscript (L / E).

commutes.

Proof.

Since the global invariant maps are sums of local invariant maps, this reduces to the commutativity of the diagram

Restriction and sums of local invariant maps. A full diagram description follows.

Diagram description: Restriction and sums of local invariant maps

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: H superscript (2)(G, product subscript (w divides v) L subscript (w) superscript (times)); column 2: fraction (1) over (|G|) blackboard Z / blackboard Z.
Row 2, from left to right: column 1: H superscript (2)(H, product subscript (w divides v) L subscript (w) superscript (times)); column 2: fraction (1) over (|H|) blackboard Z / blackboard Z.

Arrows and lines:

An arrow from H superscript (2)(G, product subscript (w divides v) L subscript (w) superscript (times)) to fraction (1) over (|G|) blackboard Z / blackboard Z, labelled sum subscript (w divides v) inv subscript (L subscript (w) / K subscript (v)).
An arrow from H superscript (2)(G, product subscript (w divides v) L subscript (w) superscript (times)) to H superscript (2)(H, product subscript (w divides v) L subscript (w) superscript (times)), labelled Res.
An arrow from fraction (1) over (|G|) blackboard Z / blackboard Z to fraction (1) over (|H|) blackboard Z / blackboard Z, labelled [E:K].
An arrow from H superscript (2)(H, product subscript (w divides v) L subscript (w) superscript (times)) to fraction (1) over (|H|) blackboard Z / blackboard Z, labelled sum subscript (w divides v) inv subscript (L subscript (w) / K subscript (v)).

where $v \in V_{K}$ and $w$ denotes a place of $L$ over $v$ . By Lemma 10.6.1 and Shapiro’s lemma, this reduces to the commutativity of

Restriction on relative local Brauer groups. A full diagram description follows.

Diagram description: Restriction on relative local 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 (L subscript (w subscript (0)) / K subscript (v)); column 2: fraction (1) over (|G|) blackboard Z / blackboard Z.
Row 2, from left to right: column 1: direct sum subscript (u divides v) Br (L subscript (w) / E subscript (u)); column 2: fraction (1) over (|H|) blackboard Z / blackboard Z.

Arrows and lines:

An arrow from Br (L subscript (w subscript (0)) / K subscript (v)) to fraction (1) over (|G|) blackboard Z / blackboard Z, labelled inv subscript (L subscript (w subscript (0)) / K subscript (v)).
An arrow from Br (L subscript (w subscript (0)) / K subscript (v)) to direct sum subscript (u divides v) Br (L subscript (w) / E subscript (u)), labelled Res.
An arrow from fraction (1) over (|G|) blackboard Z / blackboard Z to fraction (1) over (|H|) blackboard Z / blackboard Z, labelled [E:K].
An arrow from direct sum subscript (u divides v) Br (L subscript (w) / E subscript (u)) to fraction (1) over (|H|) blackboard Z / blackboard Z, labelled sum subscript (u divides v) inv subscript (L subscript (w) / E subscript (u)).

where $u$ runs over the places over $v$ in $E$ , we use $w$ to denote a place over $u$ , and $w_{0}$ denotes a fixed place of $L$ over $v$ . Here, each $w$ is conjugate to $w_{0}$ over $K$ , and the $u$ -coordinate fo the restriction map $Res$ is induced by conjugation by $σ \in G$ with $σ (w_{0}) = w$ followed by restriction. We remark that ${inv}_{L_{w} ∕ K_{v}} \circ σ^{*} = {inv}_{L_{w_{0}} ∕ K_{v}}$ by definition. The local invariant maps have the property that

{inv}_{L_{w} ∕ E_{u}} \circ {Res}_{E_{u} ∕ K_{v}} = [E_{u} : K_{v}] {inv}_{L_{w} ∕ K_{v}},

so we have

(\sum_{u ∣ v} {inv}_{L_{w} ∕ E_{u}}) \circ {Res}_{E_{u} ∕ K_{v}} = \sum_{u ∣ v} [E_{u} : K_{v}] {inv}_{L_{w} ∕ K_{v}} = [E : K] {inv}_{L_{w} ∕ K_{v}},

the latter step as the sum of local degrees is the global degree. □

We next treat the general case.

Proposition 10.9.10. §

The map $H^{2} (G, 𝕀_{L}) \to H^{2} (G, ℂ_{L})$ is surjective, and the invariant map

{inv}_{L ∕ K} : H^{2} (G, ℂ_{L}) \to_{}^{\sim} \frac{1}{| G |} ℤ ∕ ℤ

is an isomorphism.

Proof.

Since $H^{1} (G, A_{L}) = 0$ , where $A_{L} \in {L^{\times}, 𝕀_{L}, ℂ_{L}}$ , the inflation maps

Inf : H^{2} (G, A_{L}) \to H^{2} (G_{K}, A_{K^{sep}})

are injective, being part of the inflation-restriction sequences. The direct limit of the maps ${\tilde{inv}}_{L ∕ K}$ over Galois extensions $L ∕ K$ provide a surjective map

{\tilde{inv}}_{K} : H^{2} (G_{K}, 𝕀_{K^{sep}}) \to ℚ ∕ ℤ .

which factors through a surjective map

{inv}_{K} : B_{K} \to ℚ ∕ ℤ,

where $B_{K} = {lim_{\to}}_{L} B_{L ∕ K}$ . By Lemma 10.9.3,

H^{2} (G_{K}, 𝕀_{K^{sep}}) ≅ ⨁_{v \in V_{K}} Br (K_{v})

is the union of its subgroups $H^{2} (Gal (F ∕ K), 𝕀_{F})$ , where $F$ runs over the set $E$ of cyclic cyclotomic extensions of $K$ . From the map of exact sequences

Brauer groups and cohomology of ideles and idele classes. A full diagram description follows.

Diagram description: Brauer groups and cohomology of ideles and idele classes

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: union subscript (F in script E) Br (F / K); column 3: union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard I subscript (F)); column 4: union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard C subscript (F)); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: Br (K); column 3: H superscript (2)(G subscript (K),blackboard I subscript (K superscript (sep))); column 4: H superscript (2)(G subscript (K),blackboard C subscript (K superscript (sep))).

Arrows and lines:

An arrow from 0 (row 1, column 1) to union subscript (F in script E) Br (F / K), without a label.
An arrow from union subscript (F in script E) Br (F / K) to union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard I subscript (F)), without a label.
A hooked arrow from union subscript (F in script E) Br (F / K) to Br (K), without a label.
An arrow from union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard I subscript (F)) to union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard C subscript (F)), without a label.
An arrow from union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard I subscript (F)) to H superscript (2)(G subscript (K),blackboard I subscript (K superscript (sep))), labelled isomorphism symbol.
An arrow from union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard C subscript (F)) to 0 (row 1, column 5), without a label.
A hooked arrow from union subscript (F in script E) H superscript (2)( Gal (F / K),blackboard C subscript (F)) to H superscript (2)(G subscript (K),blackboard C subscript (K superscript (sep))), without a label.
An arrow from 0 (row 2, column 1) to Br (K), without a label.
An arrow from Br (K) to H superscript (2)(G subscript (K),blackboard I subscript (K superscript (sep))), without a label.
An arrow from H superscript (2)(G subscript (K),blackboard I subscript (K superscript (sep))) to H superscript (2)(G subscript (K),blackboard C subscript (K superscript (sep))), without a label.

we see that $Br (K) = ⋃_{F \in E} Br (F ∕ K)$ is the kernel of ${\tilde{inv}}_{K}$ and $B_{K} = ⋃_{F \in E} H^{2} (G, ℂ_{F})$ . In particular, we have an induced isomorphism ${inv}_{K} : B_{K} \to ℚ ∕ ℤ$ .

Since $B_{L ∕ K}$ injects into $B_{K} ≅ ℚ ∕ ℤ$ with image $\frac{1}{| G |} ℤ ∕ ℤ$ , we see that $B_{L ∕ K}$ has order $[L : K]$ . On the other hand, $B_{L ∕ K}$ divides the order of $H^{2} (G, ℂ_{L})$ , which divides $[L : K]$ by Theorem 10.8.8. So $B_{L ∕ K} = H^{2} (G, ℂ_{L})$ is mapped isomorphically to $\frac{1}{| G |} ℤ ∕ ℤ$ by ${inv}_{L ∕ K}$ , as asserted. □

We have now constructed our global class formation.

Theorem 10.9.11. §

Given a number field $K$ , the pair $(ℂ_{K^{sep}}, inv)$ , where $inv$ is the collection of invariant maps

{inv}_{E} : H^{2} (G_{E}, ℂ_{K^{sep}}) \to ℚ ∕ ℤ

for finite separable extensions $E$ of $K$ in $K^{sep}$ forms a class formation for $K$ . Moreover, the reciprocity map defined by this class formation is the map $ϕ_{K} : ℂ_{K} \to G_{K}^{ab}$ induced by the product $Φ_{K}$ of local reciprocity maps on the idèles.

Proof.

That we have a global class formation is an immediate corollary of Proposition 10.9.9 and Lemma 10.9.10. Let $ϕ_{L ∕ K}$ be the resulting reciprocity map. We know by Proposition 8.1.10 that

{inv}_{L ∕ K} (\bar{a} \cup δ (χ)) = χ (ϕ_{L ∕ K} (a))

for all $χ \in Hom (G, ℚ ∕ ℤ)$ and $a \in ℂ_{K}$ . On the other hand, we showed in Lemma 10.9.1 that

{\tilde{inv}}_{L ∕ K} (\bar{α} \cup 𝛿𝜒) = χ (Φ_{L ∕ K} (α)),

for all such $χ$ and $α \in 𝕀_{K}$ . Since $ϕ_{L ∕ K}$ and $Φ_{L ∕ K}$ are completely determined by their compositions with characters of $G$ , we must have that $Φ_{L ∕ K}$ factors through $ℂ_{K}$ and induces $ϕ_{L ∕ K}$ on the quotient. □

10.10. Power reciprocity laws

In this section, we consider higher reciprocity laws that generalize quadratic reciprocity. For this, we introduce the notion of an $n$ th power residue symbol for a number field.

Notation 10.10.1. §

In this section, $n$ will denote a positive integer, and $K$ will denote a number field that contains the full group $μ_{n}$ of $n$ th roots of unity.

Definition 10.10.2. §

The $n$ th power residue symbol for $K$ is a function with values

{(\frac{a}{𝔟})}_{n, K} \in μ_{n}

defined on pairs $(a, 𝔟)$ consisting of a nonzero element $a$ of $𝒪_{K}$ and a nonzero ideal $𝔟$ of $𝒪_{K}$ such that $𝔟$ is relatively prime to $(𝑛𝑎)$ defined as follows. For a prime $𝔭$ not dividing $(𝑛𝑎)$ , its value is the unique $n$ th root of unity in $K^{\times}$ satisfying the congruence

{(\frac{a}{𝔭})}_{n, K} \equiv a^{(𝑁𝔭 - 1) ∕ n} mod 𝔭,

and for an arbitrary $𝔟$ with prime factorization $𝔟 = 𝔭_{1}^{r_{1}} \dots 𝔭_{k}^{r_{k}}$ , its value is given by

{(\frac{a}{𝔟})}_{n, K} = \prod_{i = 1}^{k} {(\frac{a}{𝔭_{i}})}_{n, K}^{r_{i}} .

Notation 10.10.3. §

If $a$ and $b$ are nonzero elements of $𝒪_{K}$ such that $(b)$ is relatively prime to $(𝑛𝑎)$ , then we set

{(\frac{a}{b})}_{n, K} = {(\frac{a}{(b)})}_{n, K} .

Remark 10.10.4. §

The $2$ nd power residue symbol for $ℚ$ is none other than the Jacobi symbol, since $N (p) = p$ for a prime $(p)$ of $ℤ$ .

We will derive an $n$ th power reciprocity law for the power residue symbols, generalizing quadratic reciprocity. To begin with, we have the following.

Lemma 10.10.5. §

Suppose that $𝔭$ is a nonzero prime of $K$ not dividing $n$ and that $a \in 𝒪_{K}$ with $v_{𝔭} (a) = 0$ . Let $π_{𝔭}$ be a unifomizer of $K_{𝔭}$ . Then

{(\frac{a}{𝔭})}_{n, K} = {(a, π_{𝔭})}_{n, K_{𝔭}} .

Proof.

Since $𝑁𝔭$ is the order of the residue field of $K_{𝔭}$ and $a$ is a unit in the valuation ring of $K_{𝔭}$ , that the two sides are equal are an immediate consequence of our formula for the tame symbol. □

Corollary 10.10.6. §

Let $a, b \in 𝒪_{K}$ be nonzero, and suppose that $(b)$ is relatively prime to $(𝑛𝑎)$ . Then

{(\frac{a}{b})}_{n, K} = \prod_{𝔭 ∣ (b)} {(a, b)}_{n, K_{𝔭}},

where the product is over nonzero primes of $𝒪_{K}$ dividing $(b)$ .

Proof.

Write $(b) = 𝔭_{1}^{r_{1}} \dots 𝔭_{k}^{r_{k}}$ for distinct primes $𝔭_{1}, \dots, 𝔭_{k}$ and positive integers $r_{i}$ for some $k \geq 0$ . Letting $π_{𝔭_{i}}$ denote a uniformizer for $K_{𝔭_{i}}$ , we have by Lemma 10.10.5 that

{(\frac{a}{b})}_{n, K} = \prod_{i = 1}^{k} {(\frac{a}{𝔭_{i}})}_{n, K}^{r_{i}} = \prod_{i = 1}^{k} {(a, π_{𝔭_{i}}^{r_{i}})}_{n, K_{𝔭_{i}}} .

Since $b$ is $π_{𝔭_{i}}^{r_{i}}$ times a unit in the valuation ring of $K_{𝔭_{i}}$ , we have (for instance by the formula for the tame symbol) that

{(a, π_{𝔭_{i}}^{r_{i}})}_{n, K_{𝔭_{i}}} = {(a, b)}_{n, K_{𝔭_{i}}}

for each $i$ , and the result follows. □

Note also that global reciprocity gives us the following product formula for norm residue symbols.

Lemma 10.10.7. §

For every $a, b \in K^{\times}$ , we have

\prod_{v \in V_{K}} {(a, b)}_{n, K_{v}} = 1 .

Proof.

We have $ρ_{K_{v} (a^{1 ∕ n}) ∕ K_{v}} (b) = 1$ outside of a finite set $S = {v_{1}, \dots, v_{t}}$ of places of $K$ , so the product is finite, and global reciprocity then says that

\prod_{v \in S} ρ_{K_{v} (a^{1 ∕ n}) ∕ K_{v}} (b) = 1 .

It follows that

\begin{matrix} \prod_{v \in V_{K}} {(a, b)}_{n, K_{v}} = \prod_{v \in S} {(a, b)}_{n, K_{v}} = \prod_{v \in S} \frac{ρ_{K_{v}} (b) (a^{1 ∕ n})}{a^{1 ∕ n}} \\ = \prod_{i = 1}^{t} ρ_{K_{v_{1}}} (b) \dots ρ_{K_{v_{i - 1}}} (b) (\frac{ρ_{K_{v_{i}}} (b) (a^{1 ∕ n})}{a^{1 ∕ n}}) = \frac{(\prod_{v \in S} ρ_{K_{v} (a^{1 ∕ n}) ∕ K_{v}} (b)) (a^{1 ∕ n})}{a^{1 ∕ n}} = 1 . \end{matrix}

□

We are now in a position to prove the $n$ th power reciprocity law for $K$ .

Theorem 10.10.8 (Higher reciprocity law). §

Let $K$ be a number field containing the group $μ_{n}$ of $n$ th roots of unity. Let $a, b \in 𝒪_{K}$ elements relatively prime to each other and to $n$ . We then have

{(\frac{a}{b})}_{n, K} {(\frac{b}{a})}_{n, K}^{- 1} = \prod_{v ∣ 𝑛∞} {(b, a)}_{n, K_{v}},

where the product is over the places of $K$ extending a prime dividing $n$ or the real infinite place of $ℚ$ . Moreover, if $c \in 𝒪_{K}$ is relatively prime to $a$ and divisible only by primes dividing $n$ , then

{(\frac{c}{b})}_{n, K} = \prod_{v ∣ 𝑛∞} {(b, c)}_{n, K_{v}} .

Proof.

Corollary 10.10.6 tells us that

{(\frac{a}{b})}_{n, K} {(\frac{b}{a})}_{n, K}^{- 1} = \prod_{𝔭 ∣ (b)} {(a, b)}_{n, K_{𝔭}} \cdot \prod_{𝔭 ∣ (a)} {(b, a)}_{n, K_{𝔭}}^{- 1} = \prod_{𝔭 ∣ (𝑎𝑏)} {(a, b)}_{n, K_{𝔭}} .

Note that ${(a, b)}_{n, K_{𝔭}} = 1$ unless the prime $𝔭$ of $𝒪_{K}$ divides one of $a$ , $b$ , or $n$ , since otherwise the extension $K_{𝔭} (a^{1 ∕ n}) ∕ K_{𝔭}$ is unramified and $b$ is a unit in $K_{𝔭}$ . Applying Lemma 10.10.7, we then have

\prod_{𝔭 ∣ (𝑎𝑏)} {(a, b)}_{n, K_{𝔭}} = (\prod_{v ∣ 𝑛∞} {(a, b)}_{n, K_{v}})^{- 1} = \prod_{v ∣ 𝑛∞} {(b, a)}_{n, K_{v}},

finishing the proof in this case. In the remaining case, the same argument, but now noting that ${(a, b)}_{n, K_{𝔭}} = 1$ unless $𝔭$ divides $b$ or $n$ , we have

{(\frac{c}{b})}_{n, K} = \prod_{𝔭 ∣ (b)} {(c, b)}_{n, K_{𝔭}} = \prod_{v ∣ 𝑛∞} {(b, c)}_{n, K_{𝔭}} .

□

Example 10.10.9. §

Take the case that $K = ℚ$ and $n = 2$ . Let $a$ and $b$ be positive, odd integers. Then Theorem 10.10.8 implies that

(\frac{a}{b}) (\frac{b}{a}) = {(b, a)}_{2, ℚ_{p}} {(b, a)}_{2, ℝ} .

The first symbol is ${(- 1)}^{(a - 1) (b - 1) ∕ 4}$ by Proposition 9.3.6, and the second symbol is trivial by Remark 9.3.9. Similarly, we have

\begin{array}{rcl} (\frac{- 1}{b}) = {(- 1)}^{(b - 1) ∕ 2} & and & (\frac{2}{b}) = {(- 1)}^{(p^{2} - 1) ∕ 8} . \end{array}

Thus, the power reciprocity law for $K = ℚ$ and $n = 2$ is simply quadratic reciprocity.

The following special case of a result of the author serves as an entertaining example of the use of higher reciprocity laws.

Proposition 10.10.10. §

Let $p$ be an odd prime number, and let $m$ be an integer. Suppose that $ℓ = Φ_{p} (𝑝𝑚)$ is a prime number. Then every divisor of $m$ is a $p$ th power residue modulo $ℓ$ .

Proof.

Let $a$ be a divisor of $m$ . Note that

ℓ = N_{ℚ (μ_{p}) ∕ ℚ} (1 - 𝑝𝑚 ζ_{p})

and can only be prime if $m$ is nonzero. The definition of the $p$ th power residue symbol says that

{(\frac{a}{1 - 𝑝𝑚 ζ_{p}})}_{p, ℚ (μ_{p})} \equiv a^{(ℓ - 1) ∕ p} mod (1 - 𝑝𝑚 ζ_{p}),

(10.10.1)

Since $a$ is an integer, this implies that the symbol in (10.10.1) is the unique $p$ th root of unity congruent to $a^{(ℓ - 1) ∕ p}$ modulo $ℓ$ . Thus, it will be trivial if and only if $a$ is a $p$ th power residue modulo $ℓ$ .

So, we compute the symbol. We have

{(\frac{a}{1 - 𝑝𝑚 ζ_{p}})}_{p, ℚ (μ_{p})} = {(\frac{1 - 𝑝𝑚 ζ_{p}}{a})}_{p, ℚ (μ_{p})} {(1 - 𝑝𝑚 ζ_{p}, a)}_{p, ℚ_{p} (μ_{p})} = {(1 - 𝑝𝑚 ζ_{p}, a)}_{p, ℚ_{p} (μ_{p})} .

If $p$ divides $m$ , then $1 - 𝑝𝑚 ζ_{p}$ is a $p$ th power in $ℚ_{p} (μ_{p})$ , and we are done. If $a$ is a $p$ th power in $ℤ_{p}^{\times}$ , we are done as well. So, we may assume that $a$ is not a $p$ th power in $ℤ_{p}^{\times}$ . Then $ℚ_{p} (ζ_{p}, a^{1 ∕ p}) = ℚ_{p} (ζ_{p}, {(1 - p)}^{1 ∕ p})$ , and the conductor of the latter extension of $ℚ_{p} (μ_{p})$ is ${(1 - ζ_{p})}^{2}$ by Proposition 9.6.16. Since $1 - 𝑝𝑚 ζ_{p} \in U_{p - 1} (ℚ_{p} (μ_{p}))$ and $p - 1 \geq 2$ , we then have that ${(1 - 𝑝𝑚 ζ_{p}, a)}_{p, ℚ_{p} (μ_{p})} = 1$ , which completes the proof. □