Iwasawa Theory — Chapter 1 Class groups and units

Chapter 1
Class groups and units

1.1. Notation and background

Throughout, we will let $F$ be a number field. We recall a number of objects attached to $F$ and finite Galois extensions thereof and results regarding them.

Notation 1.1.1. §

To a number field, we attach the following objects:

the ring of integers $𝒪_{F}$ of $F$ ,
the unit group $𝒪_{F}^{\times}$ of $F$ , which is to say the unit group of $𝒪_{F}$ ,
the ideal group $I_{F}$ of $F$ , i.e., the group of nonzero finitely generated $𝒪_{F}$ -submodules of $F$ ,
the principal ideal group $P_{F}$ of $F$ , i.e., those $𝒪_{F}$ -submodules $(α)$ of $F$ generated by a single element $α \in F^{\times}$ , and
the class group ${Cl}_{F} = I_{F} ∕ P_{F}$ of $F$ .

Remark 1.1.2. §

The class group ${Cl}_{F}$ is a finite abelian group.

These objects fit into the following nice commutative diagram

Units, principal ideals, ideals, and the class group. A full diagram description follows.

Diagram description: Units, principal ideals, ideals, and the class group

The structural squares and triangles displayed here commute. The lower row is exact; the map from the multiplicative group to ideals factors through the principal ideals in the upper row.

Objects, listed by row and column:

Row 1, from left to right: column 6: P subscript (F).
Row 2, from left to right: column 1: 1; column 3: script O subscript (F) superscript (times); column 5: F superscript (times); column 7: I subscript (F); column 9: Cl subscript (F); column 11: 0.

Arrows and lines:

A hooked arrow from P subscript (F) to I subscript (F), without a label.
An arrow from 1 to script O subscript (F) superscript (times), without a label.
An arrow from script O subscript (F) superscript (times) to F superscript (times), without a label.
An arrow from F superscript (times) to I subscript (F), without a label.
A double-headed arrow from F superscript (times) to P subscript (F), without a label.
An arrow from I subscript (F) to Cl subscript (F), without a label.
An arrow from Cl subscript (F) to 0, without a label.

in which the lower row is exact.

Definition 1.1.3. §

The absolute norm $𝑁𝔞$ of a nonzero ideal $𝔞$ of $𝒪_{F}$ is the index $𝑁𝔞 = [𝒪_{F} : 𝔞]$ .

Notation 1.1.4. §

The number of real places of $F$ is denoted $r_{1} (F)$ , and the number of complex places of $F$ is denoted $r_{2} (F)$ .

Remark 1.1.5. §

Since each complex place consists of a pair of complex conjugate embeddings of $F$ , the degree formula tells us that $r_{1} (F) + 2 r_{2} (F) = [F : ℚ]$ .

Theorem 1.1.6 (Dirichlet’s unit theorem). §

The unit group $𝒪_{F}^{\times}$ is a finitely generated abelian group of rank $r_{1} (F) + r_{2} (F) - 1$ with torsion subgroup the group $μ (F)$ of roots of unity in $F$ .

Next, we turn quickly to the zeta function of a number field.

Definition 1.1.7. §

The Dedekind $ζ$ -series $ζ_{F}$ of a number field $F$ is

ζ_{F} (s) = \sum_{𝔞 \subseteq 𝒪_{F}} \frac{1}{{(𝑁𝔞)}^{s}},

for $s \in ℂ$ with real part $Re s > 1$ , where the sum is taken over nonzero ideals of $𝒪_{F}$ .

Theorem 1.1.8. §

The Dedekind $ζ$ -series of $F$ converges absolutely on $s$ with $Re s > 1$ . It has a unique meromorphic continuation to $ℂ$ which is holomorphic outside $1$ and has a simple pole at $s = 1$ .

With this in hand, we define the Dedekind zeta function to be the meromorphic continuation of $ζ_{F}$ to $ℂ$ .

Definition 1.1.9. §

The Dedekind zeta function $ζ_{F}$ of a number field $F$ is the meromorphic continuation to $ℂ$ of the Dedekind $ζ$ -series $ζ_{F}$ .

The Dedekind zeta function has the following functional equation relating its values at $s$ and $1 - s$ .

Theorem 1.1.10. §

Let

Λ_{F} (s) = {(2^{- r_{2} (F)} π^{- [F : ℚ] ∕ 2} | d_{F} |^{1 ∕ 2})}^{s} Γ {(s ∕ 2)}^{r_{1} (F)} Γ {(s)}^{r_{2} (F)} ζ_{F} (s),

where $Γ$ is the gamma function and $d_{F}$ denotes the discriminant of $F$ . Then $Λ_{F} (s)$ is meromorphic on $ℂ$ with simple poles at $0$ and $1$ and satisfies

Λ_{F} (s) = Λ_{F} (1 - s) .

For any Galois extension $E ∕ F$ and prime $𝔭$ of $F$ , let $φ_{𝔓}$ denote a Frobenius at a prime $𝔓$ over $𝔭$ . We have

φ_{𝔓} (α) \equiv α^{𝑁𝔭} mod 𝔓

for $α \in 𝒪_{E}$ . If $E ∕ F$ is unramified at $𝔭$ , the conjugacy class of $φ_{𝔓}$ in $Gal (E ∕ F)$ depends only on $𝔭$ , and let us denote it by $[φ_{𝔭}]$ . If $E ∕ F$ is abelian and unramified, we denote the unique Frobenius more simply by $φ_{𝔭}$ .

We will denote the class of a fractional ideal $𝔞 \in I_{F}$ by $[𝔞] \in {Cl}_{F}$ . The class group has another description in terms of the Hilbert class field $H_{F}$ of $F$ , which is to say the maximal unramified abelian extension of $F$ . We recall the following classical result of class field theory.

Theorem 1.1.11. §

The Artin map

ϕ_{F} : {Cl}_{F} \to Gal (H_{F} ∕ F),

defined by $ϕ_{F} ([𝔭]) = φ_{𝔭}$ for all primes $𝔭$ of $F$ , is an isomorphism.

1.2. Regulators

Let $F$ be a number field. We will shorten our notation for units slightly as follows.

Notation 1.2.1. §

We set $E_{F} = 𝒪_{F}^{\times}$ .

Definition 1.2.2. §

We say that a set of $r$ units of $F$ is independent if it generates a subgroup of $E_{F}$ isomorphic to $ℤ^{r}$ .

We will use the following notation.

Notation 1.2.3. §

Set $r = r_{1} (F) + r_{2} (F) - 1$ . Let $σ_{1}, \dots, σ_{r_{1} (F)} : F ↪ ℝ$ be the real embeddings of $F$ and $σ_{r_{1} (F) + 1}, \dots, σ_{r + 1} : F ↪ ℂ$ be representatives of the distinct complex conjugacy classes of complex embeddings of $F$ . Let $V = \oplus_{i = 1}^{r_{1} (F) + r_{2} (F)} ℝ σ_{i}$ and

V_{0} = {\sum_{i = 1}^{r + 1} a_{i} σ_{i} \in V ∣ \sum_{i = 1}^{r + 1} a_{i} = 0} .

We define an $ℝ$ -linear homomorphism $κ : F^{\times} \otimes_{ℤ} ℝ \to V$ by

κ (α) = \sum_{i = 1}^{r + 1} c_{i} \log | σ_{i} (α) | σ_{i},

where

c_{i} = {\begin{matrix} 1 & if σ_{i} real \\ 2 & if σ_{i} complex . \end{matrix}

Let $κ_{0}$ denote the restriction of $κ$ to $E_{F}$ . As in the proof of Dirichlet’s unit theorem, the map $κ_{0} : E_{F} \otimes_{ℤ} ℝ \to V_{0}$ is an isomorphism, the image landing in $V_{0}$ by the product formula.

Definition 1.2.4. §

The regulator $R_{F} (α_{1}, α_{2}, \dots, α_{r})$ of a set ${α_{1}, α_{2}, \dots, α_{r}}$ of $r$ independent units is $| \det ℜ |$ , where $ℜ = ℜ (α_{1}, α_{2}, \dots, α_{r})$ is the $r$ -by- $r$ matrix with $(i, j)$ -entry $c_{i} \log | σ_{i} (α_{j}) |$ .

Remark 1.2.5. §

Exactly one archimedean place is omitted in the definition of the regulator. For any $α \in E_{F}$ , one has

\sum_{i = 1}^{r + 1} c_{i} \log | σ_{i} (α) | = \log \prod_{i = 1}^{r + 1} | σ_{i} (α) |^{c_{i}} = 0

by the product formula, so the rows of the matrix determining the regulator sum to the negative of what would have been the row corresponding to the embedding that is omitted. The choice of $σ_{i}$ and their ordering are then seen by the usual rules for the effect of row operations on determinants to not affect the absolute value of the determinant of the matrix in question.

In particular, we have the following.

Lemma 1.2.6. §

For a set ${α_{1}, α_{2}, \dots, α_{r}}$ of $r = {rank}_{ℤ} E_{F}$ independent units, $R_{F} (α_{1}, α_{2}, \dots, α_{r})$ is the absolute value of the determinant of the linear transformation $κ_{0} : E_{F} \otimes_{ℤ} ℝ \to V_{0}$ relative to the basis of $E_{F} \otimes_{ℤ} ℝ$ given by the $α_{i}$ and the basis of $V_{0}$ given by $σ_{j} - σ_{j + 1}$ for $1 \leq j \leq r$ .

Definition 1.2.7. §

Let $A$ and $B$ be subgroups of an abelian group.

a.: We say that $A$ and $B$ are commensurable if $A$ and $B$ are of finite index in $A + B$ .
b.: If $A$ and $B$ are commensurable, then we define the relative index of $A$ in $B$ by
$(B : A) = [A + B : A] \cdot {[A + B : B]}^{- 1} .$

The following is easily verified.

Lemma 1.2.8. §

Let $A$ and $B$ be commensurable complete lattices in a finite-dimensional real vector space $V$ . There exists an $ℝ$ -linear automorphism $T$ of $V$ such that $T (A) = B$ , and for any such $T$ , we have $(A : B) = | \det (T) |$ .

Lemma 1.2.9. §

Suppose that ${α_{1}, α_{2}, \dots, α_{r}}$ and ${β_{1}, β_{2}, \dots, β_{r}}$ are independent sets of $r$ units in $F$ , where $r = {rank}_{ℤ} E_{F}$ . Let

\begin{array}{rcl} A = μ (F) \cdot ⟨ α_{1}, α_{2}, \dots, α_{r} ⟩ & and & B = μ (F) \cdot ⟨ β_{1}, β_{2}, \dots, β_{r} ⟩ . \end{array}

Then

\frac{R_{F} (β_{1}, β_{2}, \dots, β_{r})}{R_{F} (α_{1}, α_{2}, \dots, α_{r})} = (A : B) .

Proof.

Let $V = E_{F} \otimes_{ℤ} ℝ$ . There exists an automorphism $T$ of $V$ carrying the image of $A$ in $V$ to the image of $B$ . Since both $A$ and $B$ contain $μ (F)$ , we have $(A : B) = | \det T |$ . On the other hand, $κ_{0}$ is an $ℝ$ -linear isomorphism, so

\frac{R_{F} (β_{1}, β_{2}, \dots, β_{r})}{R_{F} (α_{1}, α_{2}, \dots, α_{r})} = \frac{| \det (κ_{0} \circ T) |}{| \det κ_{0} |} = | \det T |,

with the determinant taken relative to the bases of Lemma 1.2.6. □

Corollary 1.2.10. §

If ${α_{1}, α_{2}, \dots, α_{r}}$ and ${β_{1}, β_{2}, \dots, β_{r}}$ are independent sets in $E_{F}$ with images generating the same subgroup of $E_{F} ∕ μ (F)$ , then

R_{F} (α_{1}, α_{2}, \dots, α_{r}) = R_{F} (β_{1}, β_{2}, \dots, β_{r}) .

We may then make the following definitions.

Definition 1.2.11. §

A fundamental set of units of a number field $F$ is a set ${α_{1}, \dots, α_{r}}$ of $r$ units in $E_{F}$ such that

E_{F} = μ (F) \cdot ⟨ α_{1}, α_{2}, \dots, α_{r} ⟩ .

Definition 1.2.12. §

The regulator $R_{F}$ of a number field $F$ is $R_{F} (α_{1}, α_{2}, \dots, α_{r})$ for any fundamental set of units ${α_{1}, α_{2}, \dots, α_{r}}$ of $F$ .

We also need the following notation.

Notation 1.2.13. §

Let $w_{F}$ denote the number of roots of unity in a number field $F$ .

Now that we have defined the regulator, we can describe the residue at $s = 1$ of the Dedekind zeta function.

Theorem 1.2.14 (Analytic class number formula). §

For a number field $F$ , one has

\lim_{s \to 1} (s - 1) ζ_{F} (s) = \frac{2^{r_{1} (F)} {(2 π)}^{r_{2} (F)} h_{F} R_{F}}{w_{F} | d_{F} |^{1 ∕ 2}} .

1.3. Finite Galois extensions

Suppose that $E ∕ F$ is a finite Galois extension, and let $G = Gal (E ∕ F)$ . Then ${Cl}_{E}$ becomes a $G$ -module via the action $σ ([𝔞]) = [𝜎𝔞]$ for $σ \in G$ and $𝔞 \in I_{E}$ , where

𝜎𝔞 = {σ (a) ∣ a \in 𝔞} \in I_{E} .

The Galois group $Gal (H_{E} ∕ E)$ is a $G$ -module too, but to see this requires a little bit of work.

Proposition 1.3.1. §

Let $E$ be a finite Galois extension of $F$ . Then $H_{E} ∕ F$ is Galois.

Proof.

Let $\tilde{H_{E}}$ denote the Galois closure of $H_{E}$ as an extension of $F$ . Let $G = Gal (E ∕ F)$ . For $σ \in G$ , let $\tilde{σ}$ denote a lift of $σ$ to $Gal (\tilde{H_{E}} ∕ F)$ . Note that the field $\tilde{σ} (H_{E})$ is independent of the choice of lift $\tilde{σ}$ of $σ$ , as any element in $Gal (\tilde{H_{E}} ∕ E)$ necessarily preserves the subfield $H_{E}$ of $\tilde{H_{E}}$ , as $H_{E} ∕ E$ is Galois.

Next, we claim that

\tilde{H_{E}} = \prod_{σ \in G} \tilde{σ} (H_{E}) .

It suffices to show that $\prod_{σ \in G} \tilde{σ} (H_{E}) ∕ F$ is Galois by the minimality of $\tilde{H_{E}}$ as a Galois extension of $F$ . For this, note that for any $δ \in Gal (\tilde{H_{E}} ∕ F)$ , one has

δ \tilde{σ} (H_{E}) = \tilde{σ^{'}} (H_{E}),

where $σ^{'} = δ |_{E} σ \in G$ , by the independence of conjugates of $H_{E}$ from the choice of lift. This proves the claim. It then follows that $\tilde{H_{E}} ∕ E$ is abelian, since each $\tilde{σ} (H_{E}) ∕ E$ is. That is, any $τ \in Gal (\tilde{σ} (H_{E}) ∕ E)$ has the property that $\tilde{σ} |_{H_{E}}^{- 1} τ \tilde{σ} |_{H_{E}} \in Gal (H_{E} ∕ E)$ , and the latter group is abelian.

Now, let $I_{v}$ be the inertia group at a prime $v$ of $E$ in the abelian extension $Gal (\tilde{H_{E}} ∕ E)$ . If $I_{v}$ is nontrivial, then its image in some $Gal (\tilde{σ} (H_{E}) ∕ E)$ must be as well. Then ${\tilde{σ}}^{- 1} I_{v} \tilde{σ}$ has nontrivial image in $Gal (H_{E} ∕ E)$ . Since the former group equals the inertia group $I_{σ^{- 1} v}$ and $H_{E} ∕ E$ is unramified, this image must be trivial. Therefore $I_{v} = 0$ , and so $\tilde{H_{E}} ∕ E$ is an unramified abelian extension of $E$ containing $H_{E}$ . By the maximality of $H_{E}$ , we have $\tilde{H_{E}} = H_{E}$ , as desired. □

Consequently, if $E ∕ F$ is finite Galois with group $G$ , then $Gal (H_{E} ∕ E)$ becomes a $G$ -module for the conjugation action: $σ \in G$ acts on $τ \in Gal (H_{E} ∕ E)$ by sending it to $𝜎𝜏 σ^{- 1}$ . The following is then a consequence of class field theory.

Proposition 1.3.2. §

For $E ∕ F$ finite Galois with Galois group $G$ , then Artin map $ϕ_{E}$ is $G$ -equivariant (i.e., a $G$ -module homomorphism), which is to say that

ϕ_{E} ([𝜎𝔞]) = σ ϕ_{E} ([𝔞]) σ^{- 1}

for all $σ \in G$ and $𝔞 \in I_{E}$ .

Definition 1.3.3. §

For $E ∕ F$ finite Galois, we define a map

j_{E ∕ F} : {Cl}_{F} \to {Cl}_{E}, j_{E ∕ F} ([𝔞]) = [𝔞 𝒪_{E}]

and the norm map

N_{E ∕ F} : {Cl}_{E} \to {Cl}_{F}, N_{E ∕ F} ([𝔞]) = [(\prod_{σ \in G} σ (𝔞)) \cap F] .

Our goal in this section will be to study these maps.

Notation 1.3.4. §

For a number field $K$ and a fixed prime $p$ , we use $A_{K}$ to denote the $p$ -part of the class group of $K$ .

Lemma 1.3.5. §

Let $p$ be a prime. If the order of $G$ is prime to $p$ , then the maps

\begin{array}{rcl} A_{F} \to A_{E}^{G} & and & {(A_{E})}_{G} \to A_{F} \end{array}

defined by $j_{E ∕ F}$ and $N_{E ∕ F}$ , respectively, are isomorphisms.

Proof.

Note that $N_{E ∕ F} \circ j_{E ∕ F} = | G |$ , so $j_{E ∕ F}$ is injective on $A_{F}$ and the image of $N_{E ∕ F}$ contains $A_{F}$ . Let $ℤ^{'} = ℤ [| G |^{- 1}]$ , and note that $A_{E}$ is a $ℤ^{'} [G]$ -module. Define an idempotent

𝜀_{G} = \frac{1}{| G |} N_{G} \in ℤ^{'} [G] .

Since $𝜀_{G} A_{E} = A_{E}^{G}$ , the group $A_{E}^{G}$ is both a submodule and a quotient of $A_{E}$ . Note that $| G | 𝜀_{G} = j_{E ∕ F} \circ N_{E ∕ F}$ , which forces $j_{E ∕ F}$ to have image $A_{E}^{G}$ on $A_{F}$ . The map $A_{F} \to A_{E}^{G}$ induced by $j_{E ∕ F}$ is therefore an isomorphism. As $A_{E}$ is finite, both $A_{E}^{G}$ and ${(A_{E})}_{G}$ have the same order, so therefore now the same order as $A_{F}$ . Since $N_{E ∕ F}$ induces a surjective map ${(A_{E})}_{G} \to A_{F}$ , that map must also be an isomorphism. □

For instance, we have that $A_{ℚ (μ_{p})}^{Δ} = 0$ for any prime $p$ , as $p$ is prime to $[ℚ (μ_{p}) : ℚ] = p - 1$ .

Proposition 1.3.6. §

Let $E ∕ F$ be a finite Galois extension of number fields with Galois group $G$ . There is a canonical exact sequence

0 \to \ker j_{E ∕ F} \to H^{1} (G, 𝒪_{E}^{\times}) \to I_{E}^{G} ∕ I_{F} \to {Cl}_{E}^{G} ∕ j_{E ∕ F} ({Cl}_{F}) \to H^{1} (G, P_{E}) .

Proof.

By Hilbert’s Theorem 90, we have a commutative diagram

Diagram description: Hilbert theorem 90 and invariant principal ideals

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: script O subscript (F) superscript (times); column 3: F superscript (times); column 4: P subscript (F); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: script O subscript (F) superscript (times); column 3: F superscript (times); column 4: P subscript (E) superscript (G); column 5: H superscript (1)(G,script O subscript (E) superscript (times)); column 6: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to script O subscript (F) superscript (times) (row 1, column 2), without a label.
An arrow from script O subscript (F) superscript (times) (row 1, column 2) to F superscript (times) (row 1, column 3), without a label.
Equality joins script O subscript (F) superscript (times) (row 1, column 2) and script O subscript (F) superscript (times) (row 2, column 2), without a label.
An arrow from F superscript (times) (row 1, column 3) to P subscript (F), without a label.
Equality joins F superscript (times) (row 1, column 3) and F superscript (times) (row 2, column 3), without a label.
An arrow from P subscript (F) to 0 (row 1, column 5), without a label.
An arrow from P subscript (F) to P subscript (E) superscript (G), without a label.
An arrow from 0 (row 2, column 1) to script O subscript (F) superscript (times) (row 2, column 2), without a label.
An arrow from script O subscript (F) superscript (times) (row 2, column 2) to F superscript (times) (row 2, column 3), without a label.
An arrow from F superscript (times) (row 2, column 3) to P subscript (E) superscript (G), without a label.
An arrow from P subscript (E) superscript (G) to H superscript (1)(G,script O subscript (E) superscript (times)), without a label.
An arrow from H superscript (1)(G,script O subscript (E) superscript (times)) to 0 (row 2, column 6), without a label.

and it provides an isomorphism $H^{1} (G, 𝒪_{E}^{\times}) ≅ P_{E}^{G} ∕ P_{F}$ . Noting this and applying the snake lemma to the commutative diagram

Diagram description: Invariant ideals and class groups

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: P subscript (F); column 3: I subscript (F); column 4: Cl subscript (F); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: P subscript (E) superscript (G); column 3: I subscript (E) superscript (G); column 4: Cl subscript (E) superscript (G); column 5: H superscript (1)(G,P subscript (E)).

Arrows and lines:

An arrow from 0 (row 1, column 1) to P subscript (F), without a label.
An arrow from P subscript (F) to I subscript (F), without a label.
An arrow from P subscript (F) to P subscript (E) superscript (G), without a label.
An arrow from I subscript (F) to Cl subscript (F), without a label.
An arrow from I subscript (F) to I subscript (E) superscript (G), without a label.
An arrow from Cl subscript (F) to 0 (row 1, column 5), without a label.
An arrow from Cl subscript (F) to Cl subscript (E) superscript (G), without a label.
An arrow from 0 (row 2, column 1) to P subscript (E) superscript (G), without a label.
An arrow from P subscript (E) superscript (G) to I subscript (E) superscript (G), without a label.
An arrow from I subscript (E) superscript (G) to Cl subscript (E) superscript (G), without a label.
An arrow from Cl subscript (E) superscript (G) to H superscript (1)(G,P subscript (E)), without a label.

we obtain the desired exact sequence. □

Note that the map $\ker j_{E ∕ F} \to H^{1} (G, 𝒪_{E}^{\times})$ is given explicitly by taking $[𝔞]$ to a cocycle $σ \mapsto α^{σ - 1}$ , where $α \in E^{\times}$ satisfies $(α) = 𝔞 𝒪_{E}$ . The map $H^{1} (G, 𝒪_{E}^{\times}) \to I_{E}^{G} ∕ I_{F}$ is given by taking a cocycle $f$ to the image of an element $(α) \in I_{E}^{G}$ with $f (σ) = α^{σ - 1}$ for all $σ \in G$ .

Definition 1.3.7. §

An ideal in $I_{F}$ with class in the kernel of $j_{E ∕ F}$ is said to capitulate in the extension $E ∕ F$ . The kernel of $j_{E ∕ F}$ is known as the capitulation kernel.

Lemma 1.3.8. §

Let $E ∕ F$ be a Galois extension of number fields with Galois group $G$ . The cokernel of $N_{E ∕ F}$ is canonically isomorphic to the Galois group of the maximal unramified abelian subextension of $F$ inside $E$ .

Proof.

The norm map on ideal classes factors through the $G$ -coinvariant group ${({Cl}_{E})}_{G}$ of ${Cl}_{F}$ . We consider the complex

{({Cl}_{E})}_{G} \to_{}^{N_{E ∕ F}} {Cl}_{F} \to coker N_{E ∕ F} \to 0 .

Using the Artin map, we may write the latter complex as

Gal {(H_{E} ∕ E)}_{G} \to Gal (H_{F} ∕ F) \to coker N_{E ∕ F} \to 0,

where the first map is restriction, and therefore has image $Gal (H_{F} ∕ E \cap H_{F})$ . It follows that

coker N_{E ∕ F} ≅ Gal (H_{F} \cap E ∕ F),

as desired. □

Lemma 1.3.8 has the following immediate corollary.

Corollary 1.3.9. §

If $E ∕ F$ is totally ramified at any prime, then $N_{E ∕ F}$ is surjective.

Now suppose that $E ∕ F$ is abelian. Let $I_{v}$ denote the inertia group in $G = Gal (E ∕ F)$ at a prime $v$ of $F$ , and let

Σ_{E ∕ F} : ⨁_{v} I_{v} \to G

denote the map that is the product of the natural inclusions.

Proposition 1.3.10. §

Let $E ∕ F$ be an abelian extension of number fields with Galois group $G$ . Then there is an exact sequence

\ker Σ_{E ∕ F} \to Q_{E ∕ F} \to_{}^{N_{E ∕ F}} {Cl}_{F} \to coker Σ_{E ∕ F} \to 0,

where $Q_{E ∕ F}$ is the quotient of ${Cl}_{E}$ by its subgroup that is taken to the commutator subgroup of $Gal (H_{E} ∕ F)$ under the Artin map.

Proof.

The exactness outside of $Q_{E ∕ F}$ follows from Lemma 1.3.8, since $coker Σ_{E ∕ F} ≅ Gal (H_{F} \cap E ∕ F)$ by definition.

We define

\ker Σ_{E ∕ F} \to Q_{E ∕ F}

as follows. Employing the Artin map, we have canonical isomorphisms

Q_{E ∕ F} ≅ Gal (H_{E} ∕ E) ∕ [Gal (H_{E} ∕ F), Gal (H_{E} ∕ F)] ≅ Gal (L ∕ E),

where $L$ is the maximal unramified extension of $E$ that is abelian over $F$ . Let $J_{v}$ denote the inertia group at a prime $w$ over $v$ in $Gal (L ∕ F)$ . As $L ∕ E$ is unramified, $J_{v}$ maps isomorphically to $I_{v}$ under restriction. We have a map

⨁_{v} J_{v} \to Gal (L ∕ F)

given by the product of the canonical inclusions, and the map from $\oplus_{v} I_{v}$ is then given by the identifications $I_{v} ≅ J_{v}$ . Since $\ker Σ_{E ∕ F}$ lands in $Gal (L ∕ E)$ under this map, we have the desired map.

It remains to check exactness at $Q_{E ∕ F}$ . Again, we use the Artin isomorphism to see that the kernel of the map to ${Cl}_{F}$ is precisely $Gal (L ∕ E \cdot H_{F})$ . On the other hand, the image of $\ker Σ_{E ∕ F}$ in $Gal (L ∕ E)$ is the intersection with $Gal (L ∕ E)$ of the subgroup of $Gal (L ∕ F)$ generated by its inertia groups. As $L ∕ F$ is abelian and $H_{F}$ is the Hilbert class field of $F$ , this is precisely $Gal (L ∕ E \cdot H_{F})$ . □

We have the following interesting corollary.

Corollary 1.3.11. §

Let $E ∕ F$ be a cyclic $p$ -extension, and suppose that there is at most one prime of $F$ that ramifies in it. Then the map ${({Cl}_{E})}_{G} \to {Cl}_{F}$ induced by $N_{E ∕ F}$ is injective.

Proof.

Since $E ∕ F$ is cyclic, the quotient $Q_{E ∕ F}$ of Proposition 1.3.10 equals ${({Cl}_{E})}_{G}$ . Since $\oplus_{v \in S} I_{v}$ is either $I_{v}$ at the unique ramified prime, or $0$ if there is no ramified prime, the map $Σ_{E ∕ F}$ is injective. □

In the case that $| G |$ is a power of $p$ , we can give another nice consequence.

Lemma 1.3.12. §

Suppose that $E ∕ F$ is a finite Galois $p$ -extension ramified at no more than one prime of $F$ . If $E ∕ F$ is nonabelian, suppose further that, if there is such a prime, that it is nonsplit in the extension. Then if $A_{F} = 0$ , we have $A_{E} = 0$ as well.

Proof.

We begin with the case that $G = Gal (E ∕ F)$ is cyclic. By Corollary 1.3.11, the map ${(A_{E})}_{G} \to A_{F}$ is injective, and therefore ${(A_{E})}_{G} = 0$ . Thus, we have

A_{E} ∕ 𝔪_{G} A_{E} = {(A_{E})}_{G} ∕ p {(A_{E})}_{G} = 0 .

Noting Proposition 2.3.7, Nakayama’s lemma then tells us that $A_{E} = 0$ .

Since any finite $p$ -group has a finite filtration with cyclic graded quotients, the result in general follows from the cyclic case by recursion, noting that there is at most one prime that ramifies in each intermediate extension. That is, in the abelian setting, $A_{F} = 0$ forces a single ramified prime to be totally ramified in $E ∕ F$ , so in particular it does not split in any intermediate subfield, and in the non-abelian setting, the latter statement holds by assumption. □

We illustrate the use of this with the following interesting example.

Example 1.3.13. §

Let $F = ℚ (μ_{p})$ and $E = F (p^{1 ∕ p})$ . Then this extension is totally ramified of degree $p$ at the unique prime above $p$ in $F$ , which is $(1 - ζ_{p})$ for a primitive $p$ th root of unity $ζ_{p}$ . Therefore, we have that ${(A_{E})}_{G} ≅ A_{F}$ via the norm map. For a prime $p$ such that $A_{F} = 0$ , which is known as a regular prime (e.g., all primes less than $37$ ), Lemma 1.3.12 implies that $A_{E} = 0$ . When $p = 37$ , it turns out that $A_{F} = {Cl}_{F} ≅ ℤ ∕ 37 ℤ$ , and in fact we have that $A_{E}$ is isomorphic to $ℤ ∕ 37 ℤ$ as well.

To go even further, it is useful to restrict to the case of a cyclic extension. We begin with the following useful result.

Proposition 1.3.14. §

Let $E ∕ F$ be a cyclic extension of number fields. Then

N_{E ∕ F} E^{\times} = F^{\times} \cap ⋂_{v} N_{E_{w} ∕ F_{v}} E_{w}^{\times},

where $v$ runs over all primes of $F$ and $w$ is some prime of $E$ above $v$ .

Proof.

Let $G_{v}$ denote the decomposition group in $G$ at any $w$ over $v$ . Choose a set $S$ of representatives of $G_{v} ∖ G$ . For $a \in E^{\times}$ , we have

N_{E ∕ F} (a) = N_{E_{w} ∕ F_{v}} (\prod_{σ \in S} 𝜎𝑎),

so every global norm is a local norm everywhere.

Recall the following exact sequence for the Brauer group of $E ∕ F$ :

0 \to H^{2} (G, E^{\times}) \to ⨁_{v} H^{2} (G_{v}, E_{w}^{\times}) \to \frac{1}{| G |} ℤ ∕ ℤ,

where $G_{v}$ denotes the decomposition group in $G$ at any $w$ over $v$ . By the periodicity of Tate cohomology of a cyclic group, this becomes

0 \to {\hat{H}}^{0} (G, E^{\times}) \to ⨁_{v} {\hat{H}}^{0} (G_{v}, E_{w}^{\times}) \to \frac{1}{| G |} ℤ ∕ ℤ .

In particular, we have an injection

F^{\times} ∕ N_{E ∕ F} E^{\times} ↪ ⨁_{v} F_{v}^{\times} ∕ N_{E_{w} ∕ F_{v}} E_{w}^{\times} .

Therefore, if $a \in F^{\times}$ is a local norm everywhere, it is a global norm, as desired. □

Note that an element $a \in F^{\times}$ is automatically a local norm at any prime where $E ∕ F$ is unramified and the valuation of $a$ at that prime is trivial. Hence, there are actually only finitely many places to check that $a$ is a local norm to see that it is a global one.

We now derive a nine-term exact sequence that gives us information on the behavior of class groups in cyclic extensions. A proof is possible by making use of Tate cohomology, as found in the appendix to [HS], but we give a more explicit proof.

Theorem 1.3.15. §

Let $E ∕ F$ be a cyclic extension of number fields, and let $G$ be its Galois group. Let $I_{v}$ denote the inertia group in $G$ at a place $v$ of $F$ , and let

Σ_{E ∕ F} : ⨁_{v} I_{v} \to G

denote the map that is the product of the natural inclusions. Then we have an exact sequence

\begin{matrix} 0 \to \ker j_{E ∕ F} \to H^{1} (G, 𝒪_{E}^{\times}) \to I_{E}^{G} ∕ I_{F} \to {Cl}_{E}^{G} ∕ j_{E ∕ F} ({Cl}_{F}) \\ \to 𝒪_{F}^{\times} ∕ N_{E ∕ F} 𝒪_{E}^{\times} \to \ker Σ_{E ∕ F} \to {({Cl}_{E})}_{G} \to_{}^{N_{E ∕ F}} {Cl}_{F} \to coker Σ_{E ∕ F} \to 0 . \end{matrix}

Moreover, the group $I_{E}^{G} ∕ I_{F}$ is noncanonically isomorphic to the direct sum of $I_{v}$ over all finite primes $v$ .

Proof.

By Proposition 1.3.6, we have an exact sequence

0 \to \ker j_{E ∕ F} \to H^{1} (G, 𝒪_{E}^{\times}) \to I_{E}^{G} ∕ I_{F} \to {Cl}_{E}^{G} ∕ j_{E ∕ F} ({Cl}_{F}) \to H^{1} (G, P_{E})

including the first row. By Proposition 1.3.10, the final part of the sequence beginning with $\ker Σ_{E ∕ F}$ is exact.

Let $σ$ be a generator of $G$ . Define

{Cl}_{E}^{G} ∕ j_{E ∕ F} ({Cl}_{F}) \to_{}^{\partial} 𝒪_{F}^{\times} ∕ N_{E ∕ F} 𝒪_{E}^{\times}

as the map that takes image of an ideal class $[𝔞] \in {Cl}_{E}^{G}$ to the image of $N_{E ∕ F} α$ , where $α$ is any generator of $𝔞^{σ - 1}$ . To see that this is well-defined, note that if $α$ is replaced by another generator $α^{'}$ , then $α^{'} = 𝛼𝑢$ with $u \in 𝒪_{E}^{\times}$ , and

N_{E ∕ F} α^{'} \cdot {(N_{E ∕ F} α)}^{- 1} \in N_{E ∕ F} 𝒪_{E}^{\times} .

Moreover, if $𝔞$ is replaced by an ideal $𝔞^{'}$ with the same class, then $𝔞^{'} = 𝔞 \cdot b$ for some $b \in E^{\times}$ . We then have

{(𝔞𝑏)}^{σ - 1} = 𝔞^{σ - 1} b^{σ - 1} = (α b^{σ - 1}) .

It follows that

N_{E ∕ F} (α b^{σ - 1}) = N_{E ∕ F} (α) .

Finally, if $𝔟 \in j_{E ∕ F} ({Cl}_{F})$ , then $𝔟^{σ - 1} = (1)$ , so $\partial$ takes $[𝔟]$ to $1$ .

We check exactness at ${Cl}_{E}^{G} ∕ j_{E ∕ F} ({Cl}_{F})$ . If $𝔟 \in I_{E}^{G}$ , then again $𝔟^{σ - 1} = (1)$ , so $\partial$ maps $[𝔟]$ to $1$ . On the other hand, if $\partial$ takes the image of $[𝔞]$ to $N_{E ∕ F} α = 1$ , and therefore $α = β^{σ - 1}$ with $σ \in G$ . We then have

{(𝔞 β^{- 1})}^{σ - 1} = (1),

which means $𝔞 β^{- 1} \in I_{E}^{G}$ . As $[𝔞 β^{- 1}] = [𝔞]$ , we have that the image of $[𝔞]$ is in the image of the map from $I_{E}^{G} ∕ I_{F}$ .

Next, we define

𝒪_{F}^{\times} ∕ N_{E ∕ F} 𝒪_{E}^{\times} \to \ker (⨁_{v} I_{v} \to G)

by the direct sum of the local reciprocity maps $ρ_{E_{w} ∕ F_{v}}$ . (We remark that $I_{v} = 0$ for all but finitely many $v$ , so the map $Σ_{E ∕ F}$ makes sense.) Since the product of the reciprocity maps at all places on a global element is trivial, the image of this map is indeed contained in $\ker Σ_{E ∕ F}$ Also, the map is well-defined since every global norm is a local norm. Note that the image of $\partial$ is the set of $N_{E ∕ F} α \in 𝒪_{F}^{\times}$ with $α \in E^{\times}$ . Again, such elements are local norms, and map to zero under each $ρ_{E_{w} ∕ F_{v}}$ . Conversely, if $c \in 𝒪_{F}^{\times}$ satisfies $ρ_{E_{w} ∕ F_{v}} (c) = 1$ for every $v$ , then $c \in N_{E_{w} ∕ F_{v}} E_{w}^{\times}$ for all $v$ , since $c$ is a unit. By Theorem 1.3.14, we have that $c = N_{E ∕ F} α$ for some $α \in F^{\times}$ with $(α) = 𝔞^{σ - 1}$ for some $𝔞 \in I_{E}$ and $σ \in G$ . In other words, $c$ is the image of the image of the class of $[𝔞]$ under $\partial$ .

We check exactness at $\ker Σ_{E ∕ F}$ . Let $L$ denote the maximal abelian extension of $E$ that is abelian over $F$ . For $c \in 𝒪_{F}^{\times}$ , we have

\prod_{v} ρ_{L_{w^{'}} ∕ F_{v}} (c) = 1,

with $w^{'}$ lying over $w$ . Since $ρ_{L_{w^{'}} ∕ F_{v}} (c) |_{E} = ρ_{E_{w} ∕ F_{v}} (c)$ , the image of $c$ in $J_{v}$ is $ρ_{L_{w^{'}} ∕ F_{v}} (c)$ , and the resulting product in $Gal (L ∕ E)$ is trivial. On the other hand, suppose that ${\tilde{σ}}_{v} \in J_{v}$ lifts some $σ_{v} \in I_{v}$ and

\prod_{v} {\tilde{σ}}_{v} = 1 .

Then there exist local units $c_{v} \in 𝒪_{F_{v}}^{\times}$ for each $v$ with $ρ_{L_{w^{'}} ∕ F_{v}} (c_{v}) = {\tilde{σ}}_{v}$ . We take $c_{v} = 1$ if ${\tilde{σ}}_{v} = 1$ . By global class field theory, the idele $𝐜$ with $𝐜_{v} = c_{v}$ for each $v$ is the product of the norm of an idele $𝐛$ of $L$ with an element $c \in F^{\times}$ . Recall that

ℂ_{F} ∕ N_{H_{F} ∕ F} ℂ_{H_{F}} ≅ {Cl}_{F},

so we have that

F^{\times} N_{H_{F} ∕ F} 𝕀_{H_{F}} = F^{\times} \prod_{v} 𝒪_{F_{v}}^{\times},

where we take $𝒪_{F_{v}} = F_{v}$ if $v$ is archimedean. Since $L$ contains $H_{F}$ , the idele $𝐛$ may be taken to be a unit at all places. But, as each $c_{v}$ is a local unit at all $v$ and

{(N_{L ∕ F} 𝐛 \cdot c)}_{v} = 𝐜_{v} = c_{v}

for all $v$ , this means that $c$ must be a unit at all places as well. That is, $c \in 𝒪_{F}^{\times}$ . As $N_{L ∕ F} 𝐛$ is a local norm from $E$ everywhere, we have

ρ_{E_{w} ∕ F_{v}} (c) = ρ_{E_{w} ∕ F_{v}} (c_{v}) = σ_{v}

for every $v$ , as desired.

Finally, recall that $I_{E}$ is the free abelian group generated by the prime ideals of $𝒪_{E}$ . For an element of $I_{E}$ to be fixed under $G$ , every prime in its decomposition must appear with the same exponent as its conjugates. That is, $I_{E}^{G}$ is generated by the $\prod_{i = 1}^{g} 𝔓_{i}$ , where $𝔓_{1}, \dots 𝔓_{g}$ are the primes of $E$ lying over a prime $𝔭$ of $F$ . Of course, ${(\prod_{i = 1}^{g} 𝔓_{i})}^{e_{v}} = 𝔭 𝒪_{E}$ , where $e_{v}$ is the ramification index of the place $v$ corresponding to $𝔭$ , so we have

I_{E}^{G} ∕ I_{F} ≅ ⨁_{v} ℤ ∕ e_{v} ℤ ≅ ⨁_{v \in S} I_{v} .

□

Remark 1.3.16. §

Every map but the map between the two rows is canonical in the exact sequence of Theorem 1.3.15. The remaining map depends only upon a choice of generator of $G$ . It can be made canonical by considering instead the map

{Cl}_{E}^{G} ∕ j_{E ∕ F} ({Cl}_{F}) \otimes_{ℤ} G \to 𝒪_{F}^{\times} ∕ N_{E ∕ F} 𝒪_{E}^{\times}

given on the image of a tensor of $[𝔞] \in {Cl}_{E}^{G}$ and $σ \in G$ by writing $𝔞^{σ - 1} = α 𝒪_{E}^{\times}$ and taking the image of $N_{E ∕ F} α \in 𝒪_{F}^{\times}$ in the quotient.

Next, we generalize the situation slightly.

Notation 1.3.17. §

For a set $S$ of places in a number field $F$ , we let $S_{f}$ denote its subset of finite places and $S_{\infty}$ its subset of infinite places.

Definition 1.3.18. §

Let $S$ denote a set of places of $F$ .

a.

The $S$ -class group ${Cl}_{F, S}$ of $F$ is the quotient of the class group by the subgroup generated by the classes of the finite primes in $S$ .

b.

The Hilbert $S$ -class field $H_{F, S}$ of $F$ is the maximal unramified abelian extension of $F$ in which all primes in $S$ split completely.

c.

The ring of $S$ -integers $𝒪_{F, S}$ of $F$ is

𝒪_{F, S} = {a \in F ∣ v_{𝔭} (a) \geq 0 for all 𝔭 \notin S_{f}},

where $𝔭$ is used to denote a finite prime of $F$ and $v_{𝔭}$ its additive valuation.

d.

The $S$ -ideal group $I_{F, S}$ is the group of nonzero fractional ideals in $𝒪_{F, S}$ , and the $S$ -principal ideal group $P_{F, S}$ is the subgroup of principal fractional ideals.

e.

The $S$ -unit group in $F$ is $𝒪_{F, S}^{\times}$ .

Notation 1.3.19. §

If $S$ is a set of primes of $F$ and let $E ∕ F$ is a finite extension, then we let $S_{E}$ denote the set of places of $E$ lying over those in $S$ . For brevity, we denote $𝒪_{E, S_{E}}$ , ${Cl}_{F, S_{E}}$ , and so on more succinctly by $𝒪_{E, S}$ , ${Cl}_{E, S}$ , and so on similarly. That is, we use $S$ in the subscript to denote $S_{E}$ . If $E ∕ F$ is algebraic, we may still speak of its $S$ -integers $𝒪_{E, S}$ as the union of $S$ -integers in the finite subextensions of $F$ in $E$ .

Let us fix a set of places of $S$ for the rest of this section.

Remark 1.3.20. §

The Artin isomorphism $ϕ_{F}$ induces an isomorphism

ϕ_{F, S} : {Cl}_{F, S} \to Gal (H_{F, S} ∕ F) .

We have an analogue of the exact sequence of Theorem 1.3.15 for $S$ -class groups and $S$ -units. The proof is much as before, and is therefore omitted.

Theorem 1.3.21. §

Let $E ∕ F$ be a cyclic extension of number fields, and let $G$ be its Galois group. Let $I_{v}$ (resp., $G_{v}$ ) denote the inertia group (resp., decomposition group) in $G$ at a prime $v$ of $F$ , and let

Σ_{E ∕ F}^{S} : ⨁_{v \notin S} I_{v} \oplus ⨁_{v \in S} G_{v} \to G

denote the map that is the product of the natural inclusions. Then we have an exact sequence

\begin{matrix} 0 \to \ker ({Cl}_{F, S} \to_{}^{j_{E ∕ F}} {Cl}_{E, S}) \to H^{1} (G, 𝒪_{E, S}^{\times}) \to I_{E, S}^{G} ∕ I_{F, S} \to {Cl}_{E, S}^{G} ∕ j_{E ∕ F} ({Cl}_{F, S}) \\ \to 𝒪_{F, S}^{\times} ∕ N_{E ∕ F} 𝒪_{E, S}^{\times} \to \ker Σ_{E ∕ F}^{S} \to {({Cl}_{E, S})}_{G} \to_{}^{N_{E ∕ F}} {Cl}_{F, S} \to coker Σ_{E ∕ F}^{S} \to 0 . \end{matrix}

1.4. Kummer theory

For a set of $S$ primes of $F$ , we let $S_{f}$ denote the set of finite places of $F$ in $S$ , we let $S_{\infty}$ denote the set of archimedean places, and for any $n \geq 1$ , we let $S_{n}$ denote the set of primes of $S$ above $p$ for any prime $p$ dividing $n$ . If $E$ is an extension of $F$ , we generally also use the symbol $S$ to denote the set of primes $S_{E}$ of $E$ above those in $S$ . We will let $V$ denote the set of all primes of $F$ , so we may speak of $V_{\infty}$ and so forth. For brevity, let us set $V_{𝑛∞} = V_{n} \cup V_{\infty}$ .

Definition 1.4.1. §

We say that an extension $E$ of $F$ is $S$ -ramified if it is unramified outside of the places in $S$ .

Lemma 1.4.2. §

There exists a maximal $S$ -ramified extension $F_{S}$ of $F$ , and it is Galois over $F$ .

Proof.

A union of $S$ -ramified extensions is $S$ -ramified, so the existence of $F_{S}$ is clear. If $E$ is an $S$ -ramified finite degree extension of $F$ , then so is any conjugate of $E$ over $F$ in an algebraic closure $\bar{F}$ of $F$ containing $E$ , as the inertia degrees at conjugate primes above $p$ in $E$ and $σ (E)$ are the same (and similarly for real places). The product

\prod_{σ : E ↪ \bar{F}} σ (E)

is Galois (in fact, it is the Galois closure of $E$ in $\bar{F}$ ) and also $S$ -ramified as a compositum of $S$ -ramified extensions. Therefore, $F_{S}$ is a union of finite Galois subextensions, hence itself Galois. □

Definition 1.4.3. §

We use $G_{F, S}$ to denote the Galois group $Gal (F_{S} ∕ F)$ , i.e., the Galois group of the maximal $S$ -ramified extension $F_{S}$ of $F$ .

Kummer theory in $S$ -ramified extensions has as its basis the following proposition.

Proposition 1.4.4. §

Let $S$ be a set of primes of $F$ . We have a canonical isomorphism

{Cl}_{F, S} \to_{}^{\sim} H^{1} (G_{F, S}, 𝒪_{F_{S}, S}^{\times}),

given by taking an ideal class $[𝔞]$ to the cocycle that takes $σ \in G_{F, S}$ to $α^{σ - 1}$ , where $α$ is a generator of $𝔞 𝒪_{F_{S}, S}$ .

Proof.

To reduce clutter in the notation, let us set $𝒢 = G_{F, S}$ and $Ω = F_{S}$ . A similar argument to that of the proof of Proposition 1.3.6 produces an isomorphism

P_{Ω, S}^{𝒢} ∕ P_{F, S} \to_{}^{\sim} H^{1} (𝒢, 𝒪_{E, S}^{\times})

that takes $(α)$ to $σ \mapsto α^{σ - 1}$ . Again similarly to before, we have the commutative diagram

S-ideals in an infinite extension. A full diagram description follows.

Diagram description: S-ideals in an infinite extension

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: P subscript (F,S); column 3: I subscript (F,S); column 4: Cl subscript (F,S); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: P subscript (capital Omega ,S) superscript (script G); column 3: I subscript (capital Omega ,S) superscript (script G); column 4: Cl subscript (capital Omega ,S) superscript (script G).

Arrows and lines:

An arrow from 0 (row 1, column 1) to P subscript (F,S), without a label.
An arrow from P subscript (F,S) to I subscript (F,S), without a label.
An arrow from P subscript (F,S) to P subscript (capital Omega ,S) superscript (script G), without a label.
An arrow from I subscript (F,S) to Cl subscript (F,S), without a label.
An arrow from I subscript (F,S) to I subscript (capital Omega ,S) superscript (script G), without a label.
An arrow from Cl subscript (F,S) to 0 (row 1, column 5), without a label.
An arrow from Cl subscript (F,S) to Cl subscript (capital Omega ,S) superscript (script G), without a label.
An arrow from 0 (row 2, column 1) to P subscript (capital Omega ,S) superscript (script G), without a label.
An arrow from P subscript (capital Omega ,S) superscript (script G) to I subscript (capital Omega ,S) superscript (script G), without a label.
An arrow from I subscript (capital Omega ,S) superscript (script G) to Cl subscript (capital Omega ,S) superscript (script G), without a label.

(1.4.1)

The lower row arises as a direct limit of like sequences for intermediate finite extensions $E$ of $F$ in $Ω$ . However, since $S$ contains the finite primes that are ramified in any such extension $E ∕ F$ , the map $I_{F, S} \to I_{E, S}^{𝒢}$ is not merely an injection, but an isomorphism. Moreover, $j_{Ω ∕ F}$ is the zero map, since $Ω$ contains $H_{F}$ by definition, and every ideal in $I_{F}$ becomes principal in $H_{F}$ . Hence, the snake lemma provides an isomorphism

{Cl}_{F, S} \to P_{Ω, S}^{𝒢} ∕ P_{F, S}

taking $[𝔞]$ to $(α)$ where $(α) = 𝔞 𝒪_{Ω, S}$ . □

Proposition 1.4.5. §

Suppose that $S$ contains $V_{𝑛∞}$ . Then there is a canonical exact sequence

1 \to 𝒪_{F, S}^{\times} ∕ 𝒪_{F, S}^{\times n} \to H^{1} (G_{F, S}, μ_{n}) \to {Cl}_{F, S} [n] \to 0 .

Proof.

For any $α \in 𝒪_{F_{S}, S}^{\times}$ , the extension $F_{S} (α^{1 ∕ n}) ∕ F_{S}$ is unramified outside of $S$ and therefore trivial, as the only primes that can ramify in such a Kummer extension are the real places, those $𝔭$ with $v_{𝔭} (α) \neq 0$ , and those primes dividing $n$ , all of which are contained in $S$ . We then have that

1 \to μ_{n} \to 𝒪_{F_{S}, S}^{\times} \to_{}^{n} 𝒪_{F_{S}, S}^{\times} \to 1

is exact, and the result follows immediately from the exact sequence

H^{0} (G_{F, S}, 𝒪_{F_{S}, S}^{\times}) \to_{}^{n} H^{0} (G_{F, S}, 𝒪_{F_{S}, S}^{\times}) \to H^{1} (G_{F, S}, μ_{n}) \to H^{1} (G_{F, S}, 𝒪_{F_{S}, S}^{\times}) \to_{}^{n} H^{1} (G_{F, S}, 𝒪_{F_{S}, S}^{\times}) .

□

In the case that $S = \emptyset$ , we have the following.

Lemma 1.4.6. §

Fix $n \geq 1$ such that $F$ contains $μ_{n}$ , and let $B_{n} \leq 𝒪_{F}^{\times}$ be the subgroup

B_{n} = {a \in 𝒪_{F}^{\times} ∣ F (a^{1 ∕ n}) ∕ F is unramified} .

There is a canonical exact sequence

1 \to B_{n} ∕ 𝒪_{F}^{\times n} \to H^{1} (G_{F, \emptyset}, μ_{n}) \to {Cl}_{F} [n] .

Proof.

From the short exact sequence

1 \to μ_{n} \to 𝒪_{F_{∅}}^{\times} \to_{}^{n} 𝒪_{F_{∅}}^{\times n} \to 1,

we obtain an exact sequence

𝒪_{F}^{\times} \to_{}^{n} 𝒪_{F_{∅}}^{\times n} \cap 𝒪_{F}^{\times} \to H^{1} (G_{F, \emptyset}, μ_{n}) \to H^{1} (G_{F, \emptyset}, 𝒪_{F_{∅}}^{\times}) \to_{}^{n} H^{1} (G_{F, \emptyset}, 𝒪_{F_{∅}}^{\times n}) .

(1.4.2)

As the $n$ th power map $𝒪_{F_{∅}}^{\times} \to_{}^{n} 𝒪_{F_{∅}}^{\times}$ takes values in $𝒪_{F_{∅}}^{\times n}$ , the kernel of the rightmost map in (1.4.2) is contained in the kernel of

H^{1} (G_{F, \emptyset}, 𝒪_{F_{∅}}^{\times}) \to_{}^{n} H^{1} (G_{F, \emptyset}, 𝒪_{F_{∅}}^{\times}),

which is isomorphic to ${Cl}_{F} [n]$ by Proposition 1.4.4. Noting that

B_{n} = F_{∅}^{\times n} \cap 𝒪_{F}^{\times} = 𝒪_{F_{∅}}^{\times n} \cap 𝒪_{F}^{\times},

equation (1.4.2) yields the result. □

Definition 1.4.7. §

A number field $F$ is said to be abelian if it is an abelian extension of $ℚ$ .

Definition 1.4.8. §

A number field $F$ is said to be totally real if it has no complex places.

Remark 1.4.9. §

There exits a maximal totally real subfield $F^{+}$ of any number field $F$ , as the compositum of any two totally real fields is totally real.

Definition 1.4.10. §

A number field $F$ is CM if it has no real places and is a degree $2$ extension of $F^{+}$ .

Example 1.4.11. §

Let $n \geq 3$ . Then the cyclotomic field $ℚ (μ_{n})$ is CM, and

ℚ {(μ_{n})}^{+} = ℚ (ζ_{n} + ζ_{n}^{- 1}),

where $ζ_{n}$ is a primitive $n$ th root of unity. As a consequence of this and the Kronecker-Weber theorem, every abelian field is either totally real or CM.

Fix a CM field $F$ , and let $τ$ be the nontrivial element of $Gal (F ∕ F^{+})$ . Given a $ℤ [Gal (F ∕ F^{+})]$ -module $A$ , we have submodules

A^{\pm} = {a \in A ∣ τ (a) = \pm a} .

Note that

A^{+} \cap A^{-} = A [2] .

and $A ∕ (A^{+} + A^{-})$ is $2$ -torsion. If multiplication by $2$ is invertible on $A$ , then

𝐴 ≅ A^{+} \oplus A^{-}, a \mapsto \frac{a + τ (a)}{2} + \frac{a - τ (a)}{2} .

Lemma 1.4.12. §

The groups $𝒪_{F}^{\times}$ and ${(𝒪_{F}^{\times})}^{+}$ have the same $ℤ$ -rank, and ${(𝒪_{F}^{\times})}^{-}$ is the group $μ (F)$ of roots of unity in $F$ .

Proof.

The first statement is an immediate consequence of Dirichlet’s unit theorem. Since it holds, ${(𝒪_{F}^{\times})}^{-}$ consists only of elements of finite order, which is to say, roots of unity. Since every root of unity $ξ$ satisfies $τ (ξ) = ξ^{- 1}$ , we have the result. □

We note that for an odd prime $p$ , the map $j_{F ∕ F^{+}}$ provides a canonical identification of $A_{F^{+}}$ with $A_{F}^{+}$ by Lemma 1.3.5.

Lemma 1.4.13. §

The map ${Cl}_{F^{+}} \to {Cl}_{F}^{+}$ induced by $j_{F ∕ F^{+}}$ has kernel of order dividing $2$ .

Proof.

Note that if $τ (x) x = 1$ for $x \in 𝒪_{F}^{\times}$ , then $x$ must be a root of unity. On the other hand, the group of $τ (y) y^{- 1}$ with $y \in 𝒪_{F}^{\times}$ contains $μ {(F)}^{2} = μ {(F)}^{τ - 1}$ . Thus $H^{1} (G, 𝒪_{F}^{\times}) ≅ {\hat{H}}^{- 1} (G, 𝒪_{F}^{\times})$ is isomorphic to a quotient of $μ (F) ∕ μ {(F)}^{2}$ . The result then follows from Proposition 1.3.6. □

Remark 1.4.14. §

If $L$ and $M$ are Galois extensions of a field $K$ with $L$ contained in $M$ , then $Gal (L ∕ K)$ acts on $H^{i} (Gal (M ∕ L), A)$ for any $ℤ_{p} [Gal (M ∕ K)]$ -module $A$ . The action is induced by the following action of $τ \in Gal (M ∕ K)$ on a cochain $f \in C^{i} (Gal (M ∕ L), A)$ :

(τ \cdot f) (σ_{1}, \dots, σ_{i}) = τ \cdot f (τ^{- 1} σ_{1} τ, \dots, τ^{- 1} σ_{i} τ) .

On cohomology, this action factors through an action on $Gal (L ∕ K)$ since, on $Gal (M ∕ L)$ , this action is the conjugation action on cohomology, which is trivial.

For a finitely generated abelian group $A$ , let us use $r (A)$ to denote its rank and

r_{p} (A) = \dim_{𝔽_{p}} A [p]

to denote its $p$ -rank for a prime $p$ .

Theorem 1.4.15. §

Let $F$ be a CM field such that $μ_{p} \subset F$ for an odd prime $p$ . We then have

r_{p} ({Cl}_{F}^{+}) - δ \leq r_{p} ({Cl}_{F}^{-}) \leq r_{p} ({Cl}_{F}^{+}) + r (𝒪_{F}^{\times}),

where $δ = 0$ if $F (μ {(F)}^{1 ∕ p}) ∕ F$ is ramified at $p$ and $1$ otherwise.

Proof.

Note that

H^{1} (G_{F, \emptyset}, μ_{p}) ≅ Hom (Gal (H_{F} ∕ F), μ_{p}),

and

Hom {(Gal (H_{F} ∕ F), μ_{p})}^{\pm} ≅ Hom {({Cl}_{F}, μ_{p})}^{\pm} ≅ Hom ({Cl}_{F}^{\mp}, μ_{p}),

as $τ$ acts on $μ_{p}$ by inversion. Combining this with Lemma 1.4.6, with $B = B_{p}$ as in said lemma, we have

r_{p} ({Cl}_{F}^{\mp}) = r_{p} (H^{1} {(G_{F, \emptyset}, μ_{p})}^{\pm}) \leq r_{p} ({Cl}_{F}^{\pm}) + r_{p} ({(B ∕ 𝒪_{F}^{\times p})}^{\pm}) .

By Lemma 1.4.12, we have that $𝒪_{F}^{\times} = 𝒪_{F}^{+} \cdot μ (F)$ and

r_{p} ({(B ∕ 𝒪_{F}^{\times p})}^{-}) = r_{p} (B \cap μ (F)) = δ,

while

r_{p} ({(B ∕ 𝒪_{F}^{\times p})}^{+}) \leq r_{p} ({(𝒪_{F}^{\times} ∕ 𝒪_{F}^{\times p})}^{+}) = r ({(𝒪_{F}^{\times})}^{+}) = r (𝒪_{F}^{\times}) .

The result follows. □

1.5. Leopoldt’s conjecture

For each place $v$ of $F$ , Let

\hat{F_{v}^{\times}} = \underset{n}{lim_{\leftarrow}} F_{v}^{\times} ∕ F_{v}^{\times p^{n}},

and consider its subgroup

𝒰_{v} = \underset{n}{lim_{\leftarrow}} 𝒪_{F_{v}}^{\times} ∕ 𝒪_{F_{v}}^{\times p^{n}} .

If $v$ is finite, then $\hat{F_{v}^{\times}} ≅ ℤ_{p} \oplus 𝒰_{v}$ , and $𝒰_{v}$ is the group of $p$ -power roots of unity in $F_{v}$ if $v$ does not lie over $p$ while $𝒰_{v}$ is the group of $1$ -units if $v$ lies over $p$ . If $v$ is infinite, then $𝒰_{v} = \hat{F_{v}^{\times}}$ , and both groups are trivial unless $v$ is real and $p = 2$ , in which case they are $ℤ ∕ 2 ℤ$ .

Let us set

E_{F} = 𝒪_{F}^{\times} \otimes_{ℤ} ℤ_{p} ≅ \underset{n}{lim_{\leftarrow}} 𝒪_{F}^{\times} ∕ 𝒪_{F}^{\times p^{n}} .

We may consider the natural map

ι_{F} = {(ι_{v})}_{v \in V_{p}} : E_{F} \to ⨁_{v \in V_{p}} 𝒰_{v} .

Clearly, the kernel of $𝒪_{F}^{\times} \to 𝒪_{F_{v}}^{\times}$ is trivial for $v \in V_{p}$ . Yet, the problem may arise that there exist, for instance, two units $x, y \in 𝒪_{F}^{\times}$ generating a rank two subgroup and $a, b \in ℤ_{p}$ such that $ι_{v} {(x)}^{a} ι_{v} {(y)}^{b} = 1$ . So, in theory, $ι_{F}$ could have a kernel. This brings us to Leopoldt’s conjecture.

Conjecture 1.5.1 (Leopoldt). §

The map $ι_{F} : E_{F} \to \oplus_{v \in V_{p}} 𝒰_{v}$ is injective.

Remark 1.5.2. §

We could, equivalently, consider the map

ι_{F}^{'} : E_{F} \to ⨁_{v \in V_{𝑝∞}} 𝒰_{v},

that includes the archimdean places, setting $𝒰_{v} = \hat{F_{v}^{\times}}$ for such $v$ . The point is that $𝒰_{v} = 1$ for archimedean $v$ unless $p = 2$ and $v$ is real, in which case $𝒰_{v} ≅ ℝ^{\times} ∕ ℝ^{\times 2}$ .

We have that $\ker ι_{F}^{'} \subseteq \ker ι_{F}$ by definition. On the other hand, we have ${(\ker ι_{F})}^{2} \subseteq \ker ι_{F}^{'}$ . In particular, the two kernels have the same $ℤ_{p}$ -rank. Moreover, the $2$ -torsion in $E_{F}$ is $μ_{2}$ , and $- 1$ is not in the kernel of $ι_{v}$ for any $v$ , so $\ker ι_{F} = 0$ if and only if $\ker ι_{F}^{'} = 0$ .

Example 1.5.3. §

For $F = ℚ$ , Leopoldt’s conjecture holds as $E_{ℚ} = 1$ for $p \neq 2$ and $E_{ℚ} = μ_{2}$ for $p = 2$ .

Let $S$ denote a finite set of primes of $F$ containing $V_{𝑝∞}$ . We wish to state several equivalent forms of this conjecture. For this, we set

E_{F, S} = 𝒪_{F, S}^{\times} \otimes_{ℤ} ℤ_{p},

and extend $ι_{F}$ to a map

ι_{F, S} = {(ι_{v})}_{v \in S} : E_{F, S} \to ⨁_{v \in S} \hat{F_{v}^{\times}} .

Let $𝔛_{F, S}$ denote the Galois group of the maximal abelian pro- $p$ unramified outside $S$ extension of $F$ . Let $\hat{G_{v}^{ab}}$ denote the Galois group of the maximal abelian pro- $p$ extension of $F_{v}$ for each $v$ . The following exact sequences will be useful.

Theorem 1.5.4. §

There are two exact sequences fitting into a commutative diagram

Two global reciprocity exact sequences. A full diagram description follows.

Diagram description: Two global reciprocity exact sequences

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: kernel iota subscript (F); column 3: script E subscript (F); column 4: direct sum subscript (v in S) script U subscript (v); column 5: fraktur X subscript (F,S); column 6: A subscript (F); column 7: 0.
Row 2, from left to right: column 1: 0; column 2: kernel iota subscript (F,S); column 3: script E subscript (F,S); column 4: direct sum subscript (v in S) hat of (F subscript (v) superscript (times)); column 5: fraktur X subscript (F,S); column 6: A subscript (F,S); column 7: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to kernel iota subscript (F), without a label.
An arrow from kernel iota subscript (F) to script E subscript (F), without a label.
A hooked arrow from kernel iota subscript (F) to kernel iota subscript (F,S), without a label.
An arrow from script E subscript (F) to direct sum subscript (v in S) script U subscript (v), without a label.
A hooked arrow from script E subscript (F) to script E subscript (F,S), without a label.
An arrow from direct sum subscript (v in S) script U subscript (v) to fraktur X subscript (F,S) (row 1, column 5), without a label.
A hooked arrow from direct sum subscript (v in S) script U subscript (v) to direct sum subscript (v in S) hat of (F subscript (v) superscript (times)), without a label.
An arrow from fraktur X subscript (F,S) (row 1, column 5) to A subscript (F), without a label.
Equality joins fraktur X subscript (F,S) (row 1, column 5) and fraktur X subscript (F,S) (row 2, column 5), without a label.
An arrow from A subscript (F) to 0 (row 1, column 7), without a label.
A double-headed arrow from A subscript (F) to A subscript (F,S), without a label.
An arrow from 0 (row 2, column 1) to kernel iota subscript (F,S), without a label.
An arrow from kernel iota subscript (F,S) to script E subscript (F,S), without a label.
An arrow from script E subscript (F,S) to direct sum subscript (v in S) hat of (F subscript (v) superscript (times)), labelled iota subscript (F,S).
An arrow from direct sum subscript (v in S) hat of (F subscript (v) superscript (times)) to fraktur X subscript (F,S) (row 2, column 5), labelled rho subscript (F,S).
An arrow from fraktur X subscript (F,S) (row 2, column 5) to A subscript (F,S), without a label.
An arrow from A subscript (F,S) to 0 (row 2, column 7), without a label.

(1.5.1)

where $ρ_{F, S}$ is the product over $v \in S$ of the composition of the $p$ -completion of the local reciprocity map $ρ_{v} : \hat{F_{v}^{\times}} \to \hat{G_{v}^{ab}}$ with the natural map $j_{v}$ from $\hat{G_{v}^{ab}}$ onto the decomposition group at $v$ in $𝔛_{F, S}$ , and where the maps $𝔛_{F, S} \to A_{F} \to A_{F, S}$ are the natural quotient maps (under the indentifications given by the Artin map).

Proof.

In the horizontal sequences in the diagram (1.5.1), we note that $im ρ_{F, S}$ (resp., the corresponding map in the upper sequence) is the compositum of the decomposition groups (resp., inertia groups) at all $v \in S$ in $𝔛_{F, S}$ . Being that $𝔛_{F, S}$ already has trivial inertia groups at $v \notin S$ , the quotient $coker ρ_{F, S}$ is therefore the Galois group of the maximal unramified abelian $p$ -extension of $F$ in which all primes in $S$ split completely (resp., maximal unramified abelian $p$ -extension of $F$ ), and is therefore canonically isomorphic to $A_{F, S}$ (resp., $A_{F}$ ) via Artin reciprocity.

For the upper horizontal sequence, the exactness at $\oplus_{v \in S} 𝒰_{v}$ will follow from the exactness at $\oplus_{v \in S} \hat{F_{v}^{\times}}$ in the lower horizontal sequence by noting that $E_{F}$ consists exactly of the elements of $E_{F, S}$ that have image under $ι_{F, S}$ lying in $\oplus_{v \in S} 𝒰_{v}$ . We are therefore reduced to proving the latter exactness.

Recall that $H^{1} (G_{F, S}, μ_{p^{n}})$ is identified via Kummer theory with the quotient $B_{n} ∕ F^{\times p^{n}}$ , where $B_{n}$ is the subgroup of $x \in F^{\times}$ such that $x 𝒪_{F, S} = 𝔞^{p^{n}}$ for some fractional ideal $𝔞$ of $𝒪_{F, S}$ . In other words, we have an exact sequence

1 \to 𝒪_{F, S}^{\times} ∕ 𝒪_{F, S}^{\times p^{n}} \to B_{n} ∕ F^{\times p^{n}} \to A_{F, S} [p^{n}] \to 0 .

It then follows from the finiteness of $A_{F, S}$ that

\underset{n}{lim_{\leftarrow}} B_{n} ∕ F^{\times p^{n}} ≅ E_{F, S},

We claim that there is an exact sequence

B_{n} ∕ F^{\times p^{n}} \to ⨁_{v \in S} F_{v}^{\times} ∕ F_{v}^{\times p^{n}} \to 𝔛_{F, S} ∕ p^{n} 𝔛_{F, S},

where the first map is induced by the localization maps and the second map is $ρ_{F, S}$ taken modulo $p^{n}$ . In that all of the terms of this sequence are finite, we can take the inverse limit as we vary $n$ to obtain an exact sequence

E_{F, S} \to_{}^{ι_{F, S}} ⨁_{v \in S} \hat{F_{v}^{\times}} \to_{}^{ρ_{F, S}} 𝔛_{F, S},

(1.5.2)

finishing the verification of the exactness of the lower sequence.

Let us use $ρ_{n, v}$ and $j_{n, v}$ to denote the modulo $p^{n}$ reductions of $ρ_{v}$ and $j_{v}$ for any $v$ . Any $a \in B_{n}$ has valuation a multiple of $p^{n}$ at $v \notin S$ , so $ρ_{v} (a)$ lies in the compositum of the inertia group and the subgroup of $p^{n}$ th powers in $\hat{G_{v}^{ab}}$ . In particular, we have that $j_{n, v} (ρ_{n, v} (a)) = 1$ for such $v$ . Global class field theory then tells us that

\prod_{v \in S} j_{v} (ρ_{v} (a)) = \prod_{v} j_{v} (ρ_{v} (a)) = 1,

which tells us that (1.5.2) is a complex.

Let $M_{n}$ be the maximal $S$ -ramified abelian extension of $F$ of exponent $p^{n}$ . Its Galois group $Gal (M_{n} ∕ F) ≅ 𝔛_{F, S} ∕ p^{n} 𝔛_{F, S}$ is the quotient of $Gal (F^{ab} ∕ F)$ by the composition of all inertia groups at primes $v \notin S$ and the $p^{n}$ th powers of all of the decomposition groups. By global class field theory, we therefore have an isomorphism

\frac{𝕀_{F}}{F^{\times} \cdot 𝕀_{F}^{p^{n}} \cdot \prod_{v \notin S} 𝒪_{v}^{\times}} \to_{}^{\sim} Gal (M_{n} ∕ F) .

where for simplicity of notation, we have set $𝒪_{v} = 𝒪_{F_{v}}$ . Similarly, if we let $L_{n}^{'}$ be the maximal unramified abelian extension of $F$ of exponent $p^{n}$ in which every prime in $S$ splits completely, so that $Gal (L_{n}^{'} ∕ F) ≅ A_{F, S} ∕ p^{n} A_{F, S}$ , class field theory again provides an isomorphism

\frac{𝕀_{F}}{F^{\times} \cdot 𝕀_{F}^{p^{n}} \cdot (\prod_{v \in S} F_{v}^{\times} \times \prod_{v \notin S} 𝒪_{v}^{\times})} \to_{}^{\sim} Gal (L_{n}^{'} ∕ F) .

We see, then, that we have isomorphisms

\frac{𝕀_{F}^{p^{n}} \cdot (\prod_{v \in S} F_{v}^{\times} \times \prod_{v \notin S} 𝒪_{v}^{\times})}{(F^{\times} \cap 𝕀_{F}^{p^{n}} (\prod_{v \in S} F_{v}^{\times} \times \prod_{v \notin S} 𝒪_{v}^{\times})) \cdot 𝕀_{F}^{p^{n}} \prod_{v \in S} 𝒪_{v}^{\times}} ≅ \frac{F^{\times} \cdot 𝕀_{F}^{p^{n}} \cdot (\prod_{v \in S} F_{v}^{\times} \times \prod_{v \notin S} 𝒪_{v}^{\times})}{F^{\times} \cdot 𝕀_{F}^{p^{n}} \cdot \prod_{v \notin S} 𝒪_{v}^{\times}} ≅ Gal (M_{n} ∕ L_{n}^{'}),

where in the first step, we have used the second isomorphism theorem. Since

⨁_{v \in S} F_{v}^{\times} ∕ F_{v}^{\times p^{n}} ≅ \frac{𝕀_{F}^{p^{n}} \cdot (\prod_{v \in S} F_{v}^{\times} \times \prod_{v \notin S} 𝒪_{v}^{\times})}{𝕀_{F}^{p^{n}} \prod_{v \in S} 𝒪_{v}^{\times}}

and

B_{n} = F^{\times} \cap 𝕀_{F}^{p^{n}} (\prod_{v \in S} F_{v}^{\times} \times \prod_{v \notin S} 𝒪_{v}^{\times}),

we have an exact sequence

B_{n} \to ⨁_{v \in S} F^{\times} ∕ F^{\times p^{n}} \to Gal (M_{n} ∕ L_{n}^{'}),

where the maps agree with the maps in question, hence the result. □

Remark 1.5.5. §

Theorem 1.5.4 can also be derived using Poitou-Tate duality and Kummer theory.

Proposition 1.5.6. §

The kernel of $ι_{F, S}$ is contained in $E_{F}$ . In particular, Leopoldt’s conjecture is equivalent to the injectivity of $ι_{F, S}$ .

Proof.

Let $α \in \ker ι_{F, S}$ . Then $α$ may be written as

α = \sum_{i = 1}^{m} a_{i} \otimes c_{i},

were $a_{i} \in 𝒪_{F, S}^{\times}$ and $c_{i} \in ℤ_{p}$ for each $1 \leq i \leq m$ , for some $m \geq 0$ . For each $v \in S_{f}$ , we then have

\sum_{i = 1}^{m} v (a_{i}) c_{i} = 0,

which means that the $c_{i} \in ℤ_{p}$ are $ℤ$ -linearly dependent if some $v (a_{i}) \neq 0$ . If $v (a_{m}) \neq 0$ , without loss of generality, then $α^{v (a_{m})}$ may be written as a sum of $m - 1$ tensors. Continuing in this way, we obtain that some nonzero integer power of $α$ is a $ℤ_{p}$ -linear combination of units at $v$ . Since there are only finitely many $v \in S$ , we have $α^{c} \in E_{F}$ for some $c \in ℤ$ , which forces $α \in E_{F}$ . □

The following theorem is also a corollary of Theorem 1.5.4 and Proposition 1.5.6, which gives in particular equivalent conditions for Leopoldt’s conjecture to hold (noting that $E_{F}$ is $p$ -torsion free). Let ${rank}_{ℤ_{p}} A$ denote the $ℤ_{p}$ -rank of a finitely generated $ℤ_{p}$ -module $A$ .

Theorem 1.5.7. §

The following are equivalent for a given $δ \geq 0$ :

i.: ${rank}_{ℤ_{p}} \ker ι_{F} = δ$ ,
ii.: ${rank}_{ℤ_{p}} im ι_{F} = r_{1} (F) + r_{2} (F) - 1 - δ$ ,
iii.: ${rank}_{ℤ_{p}} \ker ι_{F, S} = δ$ , and
iv.: ${rank}_{ℤ_{p}} 𝔛_{F, S} = r_{2} (F) + 1 + δ$ .

Proof.

For $v \in S$ , we have that

{rank}_{ℤ_{p}} \hat{𝒰_{v}} = {\begin{matrix} [F_{v} : ℚ_{p}] & if v \in V_{p}, \\ 0 & if v \in S - V_{p} . \end{matrix}

We also have

{rank}_{ℤ_{p}} E_{F} = r_{1} (F) + r_{2} (F) - 1

by Dirichlet’s unit theorem, and $A_{F}$ is finite. Note that

r_{1} (F) + 2 r_{2} (F) = [F : ℚ] = \sum_{v \in V_{p}} [F_{v} : ℚ_{p}] .

Hence, Proposition 1.5.6 and the exactness of the upper exact sequence in (1.5.1) yield the result. □

Corollary 1.5.8. §

The $ℤ_{p}$ -module $𝔛_{F, S}$ is finitely generated of $ℤ_{p}$ -rank independent of $S$ containing $V_{𝑝∞}$ .

The $δ$ in Theorem 1.5.7 is known as the Leopoldt defect of $F$

Definition 1.5.9. §

The Leopoldt defect $δ (F)$ is the $ℤ_{p}$ -rank of $\ker ι_{F}$ .

Leopoldt’s conjecture for $F$ is equivalent to the statement that the Leopoldt defect $δ (F)$ is 0. We may also phrase Leopoldt’s conjecture for $F$ in terms of the nonvanishing of a $p$ -adic regulator of $F$ , which replaces the always nonzero complex regulator.

Definition 1.5.10. §

For a $p$ -adic field $E$ , the $p$ -adic logarithm of $E$ is the unique homomorphism $\log_{p} : E^{\times} \to E$ such that $\log_{p} (p) = 0$ and such that for any $x$ in the maximal ideal of $𝒪_{E}$ , one has

\log_{p} (1 + x) = \sum_{k = 1}^{\infty} {(- 1)}^{k - 1} \frac{x^{k}}{k} .

Remark 1.5.11. §

The kernel of $\log_{p}$ on a $p$ -adic field $E$ is $μ (E)$ .

Notation 1.5.12. §

We use $ℂ_{p}$ to denote the completion of the algebraic closure of $ℚ_{p}$ with respect to the unique extension of the $p$ -adic absolute value on $ℚ_{p}$ . We have a $p$ -adic absolute value

| \cdot |_{p} : ℂ_{p} \to ℝ_{\geq 0}

with $| p |_{p} = p^{- 1}$ .

Remark 1.5.13. §

The $p$ -adic logarithm extends to a continuous homomorphism $\log_{p} : ℂ_{p}^{\times} \to ℂ_{p}$ .

It turns out that $ℂ$ and $ℂ_{p}$ are abstractly isomorphic (being algebraically closed of characteristic $0$ and having the same cardinality), and we can fix an embedding $ι : ℂ \to ℂ_{p}$ . Let $d = [F : ℚ]$ , and let $τ_{i} : F ↪ ℂ_{p}$ for $1 \leq i \leq r + 1$ be the compositions $τ = ι \circ σ_{i}$ of the real and complex embeddings $σ_{i}$ of $F$ previously chosen in Section 1.2.

Definition 1.5.14. §

Let $α_{1}, \dots, α_{r}$ be $r$ independent units in $E_{F}$ . The $p$ -adic regulator $R_{p} (F)$ of $E_{F}$ is the determinant of the $r$ -by- $r$ matrix $R_{p} (α_{1}, \dots, α_{r}) = {(c_{i} \log_{p} τ_{i} (α_{j}))}_{i, j}$ , where $c_{i}$ is $1$ if $σ_{i}$ is real and $2$ if $σ_{i}$ is complex.

Remark 1.5.15. §

The $p$ -adic regulator is well-defined up to sign, so as an element of $ℂ_{p}^{\times} ∕ ⟨ - 1 ⟩$ .

The following is immediate.

Proposition 1.5.16. §

Leopoldt’s conjecture for a number field $F$ is equivalent to the statement the nonvanishing of the $p$ -adic regulator $R_{p} (F)$ .

Baker proved that if $α_{1}, \dots α_{n} \in {\bar{ℚ}}^{\times}$ are such that $2 𝜋𝑖, \log α_{1}, \dots, \log α_{n}$ are $ℚ$ -linearly independent, then they are $\bar{ℚ}$ -linearly independent. Via Baker’s method, Brumer proved a $p$ -adic analogue.

Theorem 1.5.17 (Brumer). §

Let $β_{1}, \dots, β_{n}$ be algebraic numbers that are also $p$ -adic units, and suppose that the $p$ -adic logarithms $\log_{p} β_{1}, \dots, \log_{p} β_{n}$ are $ℚ$ -linearly independent. Then these logarithms are also $\bar{ℚ}$ -linearly independent.

Using this result, Brumer was able to prove Leopoldt’s conjecture for an abelian extensions of number fields with $r = 0$ . Note that the only fields with $r = 0$ are $ℚ$ and the imaginary quadratic fields. We need several preliminary results. We begin with the following result, only the first part of which is needed at the moment.

Proposition 1.5.18. §

Let $G$ be a finite abelian group and $f : G \to ℂ$ be a function. Let $\hat{G}$ denote the group of characters $G \to ℂ^{\times}$ .

a.

We have

\prod_{χ \in \hat{G}} (\sum_{σ \in G} χ (σ) f (σ)) = \det {(f (σ τ^{- 1}))}_{σ, τ \in G} .

In fact, the rank of ${(f (σ τ^{- 1}))}_{σ, τ \in G}$ is the number of $χ \in \hat{G}$ such that $\sum_{σ \in G} χ (σ) f (σ) \neq 0$ .

b.

We have

\prod_{\begin{array}{c} χ \in \hat{G} \\ χ \neq 1 \end{array}} (\sum_{σ \in G} χ (σ) f (σ)) = \det {(f (σ τ^{- 1}) - f (σ))}_{σ, τ \neq 1} .

Proof.

We compare two bases of the complex vector space $V$ of functions $G \to ℂ$ : the set of characters $\hat{G}$ and the set of $δ$ -functions

δ_{σ} (τ) = {\begin{matrix} 1 & τ = σ, \\ 0 & τ \neq σ \end{matrix}

for $σ \in G$ . Consider the linear transformation $T : V \to V$ given by

T (g) (τ) = \sum_{σ \in G} f (σ) g (𝜎𝜏) .

Applied to $g = χ$ , we obtain

T (χ) = \sum_{σ \in G} f (σ) χ (σ) χ,

so $χ$ is an eigenvector with eigenvalue $\sum_{σ \in G} χ (σ) f (σ)$ . It follows that $\det T$ is the product of the latter sums over all $χ$ . On the other hand,

T (δ_{σ}) (ρ) = \sum_{τ \in G} f (τ) δ_{σ} (𝜌𝜏) = \sum_{τ \in G} f (τ) δ_{σ τ^{- 1}} (ρ) = \sum_{τ \in G} f (τ^{- 1} σ) δ_{τ} (ρ)

T (δ_{σ}) = \sum_{τ \in G} f (σ τ^{- 1}) δ_{τ}

so the $(τ, σ)$ -entry of the matrix of $T$ with respect to this basis is $f (σ τ^{- 1})$ . In that the determinant and rank of $T$ are independent of the choice of basis, we have part a.

For part b, we consider the codimension $1$ subspace $W$ of $V$ that consisting of the $g : G \to ℂ$ with $\sum_{σ \in G} g (σ) = 0$ . One basis of these functions is given by $\hat{G} - {1}$ , and another is given by the functions $δ_{σ} - | G |^{- 1}$ for $σ \neq 1$ . Also, we see immediately that $T (W) \subseteq W$ . The determinant of $T |_{W}$ with respect to the character basis is clearly the left-hand side of the desired equality. On the other hand, noting that

\sum_{τ \in G} (δ_{τ} - | G |^{- 1}) = 0,

we have

T (δ_{σ} - | G |^{- 1}) = \sum_{τ \in G} f (σ τ^{- 1}) (δ_{τ} - | G |^{- 1}) = \sum_{\begin{array}{c} τ \in G \\ τ \neq 1 \end{array}} (f (σ τ^{- 1}) - f (σ)) (δ_{τ} - | G |^{- 1}),

which has the desired coefficients. □

We omit a proof of the following.

Lemma 1.5.19. §

For a field $K$ and a finite group $G$ , let $V$ and $W$ be $K [G]$ -modules of finite $K$ -dimension. Suppose that there is a field extension $L$ of $K$ such that $V \otimes_{K} 𝐿 ≅ 𝑊 \otimes_{K} L$ as $L [G]$ -modules. Then $𝑉 ≅ 𝑊$ as $K [G]$ -modules.

Proposition 1.5.20. §

Let $F$ be an abelian extension with Galois group $G$ of either $ℚ$ or an imaginary quadratic field. Then $E_{F} \otimes_{ℤ} ℚ ≅ I_{G} \otimes_{ℤ} ℚ$ as $ℚ [G]$ -modules.

Proof.

By the proof of Dirichlet’s unit theorem, we have $E_{F} \otimes_{ℤ} ℝ ≅ V_{0}$ , where $V_{0}$ is as in Notation 1.2.3. That is $V_{0}$ is a hyperplane in the $ℝ$ -span of the archimedean places of $F$ , in this case consisting of the formal sums with coefficients summing to zero (since $F$ is either totally real or purely imaginary). Since $E$ has just one archimedean place, all of the places of $F$ are conjugate under the action determined by precomposition of a representative by the inverse of an element of $G$ . Fixing an embedding $ϕ : F \to ℂ$ then provides an isomorphism $V_{0} ≅ I_{G} \otimes_{ℤ} ℝ$ , so $E_{F} \otimes_{ℤ} ℝ ≅ I_{G} \otimes_{ℤ} ℝ$ . By Lemma 1.5.19, we then have that $E_{F} \otimes_{ℤ} ℚ ≅ I_{G} \otimes_{ℤ} ℚ$ . □

Theorem 1.5.21 (Brumer). §

Leopoldt’s conjecture holds for all finite abelian extensions of $ℚ$ and all finite abelian extensions of any imaginary quadratic field.

Proof.

By Proposition 1.5.20, we may pick $α \in E_{F}$ be such that ${σ (α) ∣ σ \in G - {1}}$ is an independent set of $r$ units of $F$ . Let $ϕ = ι \circ ϕ : K \to ℂ_{p}$ , and consider the function $f : G \to ℂ_{p}$ defined by $f (σ) = \log_{p} ϕ (σ^{- 1} α)$ . Since

\prod_{σ \in G} σ^{- 1} α = \pm 1,

we have

\sum_{σ \in G} f (σ) = 0 .

If $\sum_{σ \in G} χ (σ) f (σ) = 0$ for some nontrivial character $χ \in \hat{G}$ , then

\sum_{σ \in G - {1}} (1 - χ (σ)) f (σ) = 0 .

Since $1 - χ (σ) \in \bar{ℚ}$ . By Theorem 1.5.17, we then have that the quantities $f (σ)$ for $σ \in G - {1}$ are $ℚ$ -linearly dependent, and hence $ℤ$ -linearly dependent. That is, there exist elements $k_{σ} \in ℤ$ , not all zero, such that

\prod_{σ \in G - {1}} {(𝜎𝛼)}^{k_{σ}} \in μ (F) .

This, however, contradicts our choice of $α$ .

Now choose an ordering of $G$ and form the matrix ${(f (σ τ^{- 1}))}_{σ, τ \in G}$ , the $(σ, τ)$ -entry of which is $\log_{p} (ϕ \circ τ) (σ^{- 1} α)$ . It then follows from Proposition 1.5.18a that this matrix has rank $r = | G | - 1$ , and the $σ = 1$ row and $τ = 1$ column are linearly dependent on the others. If we remove them, the resulting $r$ -by- $r$ minor is the $p$ -adic regulator matrix attached to the basis $σ^{- 1} α$ with $σ \in G - {1}$ and the embeddings $ϕ \circ τ$ for $τ \in G - {1}$ . Thus $R_{p} (F) \neq 0$ , so Leopoldt’s conjecture holds for $F$ . □