Iwasawa Theory

Chapter 4
Cyclotomic fields

4.1. Dirichlet $L$ -functions

In this section, we summarize, largely without proof, various results regarding $L$ -functions of Dirichlet characters.

Definition 4.1.1. §

A multiplicative function $χ : ℤ \to ℂ$ is called a Dirichlet character if it is periodic of some period $n \geq 1$ and $χ (a) \neq 0$ for $a \in ℤ$ if and only if $(a, n) = 1$ . The integer $n$ is called the modulus of $χ$ .

Example 4.1.2. §

There is a unique Dirichlet character $1$ which has value $1$ at every $a \in ℤ$ , and it is known as the trivial character.

Definition 4.1.3. §

a.: The conductor $f_{χ}$ of a Dirichlet character $χ$ is the smallest integer $f$ dividing its period such that there exists a Dirichlet character $ψ$ of modulus $f$ with $χ (a) = ψ (a)$ for all $a \in ℤ$ with $(a, n) = 1$ .
b.: We say that a Dirichlet character is primitive if its conductor equals its modulus.

Definition 4.1.4. §

We say that a Dirichlet character $χ$ is even (resp., odd) if $χ (- 1) = 1$ (resp., $χ (- 1) = - 1$ .)

Every character $ϕ : {(ℤ ∕ 𝑛ℤ)}^{\times} \to ℂ^{\times}$ gives rise to a Dirichlet character $χ : ℤ \to ℂ$ of period $n$ with $χ (a) = ϕ (a (𝑚𝑜𝑑 n))$ for $a \in ℤ$ with $(a, n) = 1$ . The resulting character $χ$ has conductor $f$ , where $f$ is minimal such that $ϕ$ factors through ${(ℤ ∕ 𝑓ℤ)}^{\times}$ .

Definition 4.1.5. §

Let $ϕ : {(ℤ ∕ 𝑛ℤ)}^{\times} \to ℂ^{\times}$ , and suppose that the induced Dirichlet character has conductor $f$ . The primitive Dirichlet character attached to $ϕ$ is the primitive Dirichlet character of conductor $f$ that satisfies $ϕ (a) = χ (a^{'})$ for $a \in ℤ$ , $(a, f) = 1$ , where $a^{'} \in ℤ$ is any integer with $a^{'} \equiv a mod f$ and $(a^{'}, n) = 1$ .

Let $F ∕ ℚ$ be an abelian field, and let $n \geq 1$ be such that $F \subseteq ℚ (μ_{n})$ . The cyclotomic character then allows us to identify $Gal (F ∕ ℚ)$ with a quotient of ${(ℤ ∕ 𝑛ℤ)}^{\times}$ .

Notation 4.1.6. §

The set $X (F)$ of primitive Dirichlet characters of $F \subseteq ℚ (μ_{N})$ consists of the primitive characters of conductor dividing $n$ attached to characters of ${(ℤ ∕ 𝑛ℤ)}^{\times}$ that factor through $Gal (F ∕ ℚ)$ .

Remark 4.1.7. §

A Dirichlet character $χ \in X (F)$ is even if and only if the associated character on $Gal (F ∕ ℚ)$ is even.

To any Dirichlet character, we can attach an $L$ -series.

Definition 4.1.8. §

Let $χ$ be a Dirichlet character. The Dirichlet $L$ -series attached to $χ$ is the complex-valued function on $s \in ℂ$ with $Re s > 1$ defined by

L (χ, s) = \sum_{n = 1}^{\infty} \frac{χ (n)}{n^{s}} .

Example 4.1.9. §

For $χ = 1$ , one has $L (1, s) = ζ (s)$ , the Riemann $ζ$ -function.

We note that Dirichlet $L$ -series have Euler product expansions.

Proposition 4.1.10. §

One has

L (χ, s) = \prod_{p prime} \frac{1}{1 - χ (p) p^{- s}}

for all $s \in ℂ$ with $Re s > 1$ .

Theorem 4.1.11. §

The $L$ -series $L (χ, s)$ has a meromorphic continuation to all of $ℂ$ that is analytic if $f_{χ} > 1$ , while $ζ (s)$ is holomorphic aside from a simple pole at $s = 1$ with residue $1$ .

Definition 4.1.12. §

The Dirichlet $L$ -function $L (χ, s)$ of a Dirichlet character $χ$ is the meromorphic continuation of the $L$ -series $L (χ, s)$ to $ℂ$ .

Definition 4.1.13. §

The $Γ$ -function is the unique meromorphic function on $ℂ$ that satisfies

Γ (s) = \int_{0}^{\infty} t^{s - 1} e^{- t} 𝑑𝑡

for all $s \in ℂ$ with $Re s > 0$ and

Γ (s + 1) = s Γ (s)

for all $s$ for which it is defined.

Remark 4.1.14. §

The $Γ$ -function has poles, which are all simple, at exactly the nonpositive integers. It also satisfies $Γ (n) = (n - 1)!$ for any positive integer $n$ .

Definition 4.1.15. §

The Gauss sum attached to a Dirichlet character $χ$ of modulus $n$ is the value

τ (χ) = \sum_{a = 1}^{n} χ (a) e^{2 𝜋𝑖𝑎 ∕ n} .

Definition 4.1.16. §

For a Dirichlet character $χ$ , we let $\bar{χ}$ denote its complex conjugate, which satisfies $\bar{χ} (a) = \bar{χ (a)}$ for all $a \in ℤ$ .

We mention a couple of basic lemmas regarding Gauss sums that will be of use.

Lemma 4.1.17. §

Let $χ$ be a primitive Dirichlet character. Then we have

χ (b) τ (\bar{χ}) = \sum_{a = 1}^{f_{χ}} \bar{χ} (a) e^{2 𝜋𝑖𝑎𝑏 ∕ f_{χ}}

for all $b \in ℤ$ .

Proof.

If $χ (b) = 0$ , then setting $d = (b, f_{χ})$ and $m = d^{- 1} f_{χ}$ , we have

\sum_{a = 1}^{f_{χ}} \bar{χ} (a) e^{2 𝜋𝑖𝑎𝑏 ∕ f_{χ}} = \sum_{a = 1}^{m} \sum_{c = 1}^{d} \bar{χ} (a + 𝑚𝑐) e^{2 𝜋𝑖𝑎𝑏 ∕ f_{χ}},

and

\sum_{c = 1}^{d} \bar{χ} (a + 𝑚𝑐) = 0

for all $a$ . If $χ (b) \neq 0$ , then

χ (b) τ (\bar{χ}) = \sum_{a = 1}^{f_{χ}} \bar{χ} (a b^{- 1}) e^{2 𝜋𝑖𝑎 ∕ f_{χ}},

which gives the desired equality upon reordering the sum. □

Lemma 4.1.18. §

For a primitive Dirichlet character $χ$ , we have

| τ (χ) | = f_{χ}^{1 ∕ 2} .

Proof.

Note that $\bar{τ (χ)} = χ (- 1) τ (\bar{χ})$ . We then have

| τ (χ) | = χ (- 1) \sum_{a = 1}^{f_{χ}} χ (a) τ (\bar{χ}) e^{2 𝜋𝑖𝑎 ∕ f_{χ}},

and by Lemma 4.1.17, this equals

χ (- 1) \sum_{a = 1}^{f_{χ}} (\sum_{b = 1}^{f_{χ}} \bar{χ} (b) e^{2 𝜋𝑖𝑎𝑏 ∕ f_{χ}}) e^{2 𝜋𝑖𝑎 ∕ f_{χ}} = χ (- 1) \sum_{b = 1}^{f_{χ}} \bar{χ} (b) \sum_{a = 1}^{f_{χ}} e^{2 𝜋𝑖𝑎 (b + 1) ∕ f_{χ}} .

The latter sum of exponentials is zero unless $b = f_{χ} - 1$ , in which case it is $f_{χ}$ . Hence,

| τ (χ) | = | χ (- 1) |^{2} f_{χ} = f_{χ} .

□

Definition 4.1.19. §

For a primitive Dirichlet character $χ$ , we set

\begin{array}{rcl} δ_{χ} = (1 - χ (- 1)) ∕ 2, & 𝜖_{χ} = \frac{τ (χ)}{i^{δ_{χ}} \sqrt{f_{χ}}}, and & Λ (χ, s) = {(\frac{f_{χ}}{π})}^{s ∕ 2} Γ (\frac{s + δ_{χ}}{2}) L (χ, s), \end{array}

Theorem 4.1.20. §

Let $χ$ be a primitive Dirichlet character. Then the $L$ -functions of $χ$ and $\bar{χ}$ satisfy the functional equation

Λ (χ, s) = 𝜖_{χ} Λ (\bar{χ}, 1 - s)

for all $s \in ℂ$ .

We give the relationship between Dirichlet $L$ -functions and the Dedekind zeta function of an abelian field.

Proposition 4.1.21. §

Let $F$ be an abelian field. Then

ζ_{F} (s) = \prod_{χ \in X (F)} L (χ, s) .

Proof.

It suffices to check this on $s$ with $Re s > 1$ by uniqueness of the meromorphic continuations. In turn, it suffices to check that for each prime $p$ , we have

\prod_{𝔭 \in V_{p} (F)} (1 - {(𝑁𝔭)}^{- s}) = \prod_{χ \in X (F)} (1 - χ (p) p^{- s}) .

(4.1.1)

As $F ∕ ℚ$ is Galois, we have $𝑁𝔭 = p^{- 𝑓𝑠}$ , where $f$ is the common residue degree of the primes over $p$ in $F$ , so the lefthand side is just ${(1 - p^{- 𝑓𝑠})}^{g}$ , where $g = | V_{p} (F) |$ . Note that $χ (p) = 0$ if $p$ ramified in the fixed field of the kernel of $χ$ . Thus, the product reduces to $χ \in X (E)$ , where $E$ is the maximal subextension of $F ∕ ℚ$ that is unramified at $p$ . Viewing $χ \in X (E)$ as a Galois character, so $χ (p)$ is the value of $χ$ on the Frobenius at $p$ , which is a generator of a cyclic subgroup of order $f$ in $Gal (E ∕ ℚ)$ . Since $𝑓𝑔 = [E : ℚ]$ , there are $g$ characters $χ$ such that $χ (f) = ζ_{f}^{i}$ for a fixed primitive $f$ th root of unity $ζ_{f}$ and given integer $i$ with $0 \leq i \leq f - 1$ . The righthand side of (4.1.1) is then simply

\prod_{i = 0}^{f - 1} {(1 - ζ_{f} p^{- s})}^{g} = {(1 - p^{- 𝑓𝑠})}^{g},

as required. □

Corollary 4.1.22. §

Let $χ$ be a Dirichlet character with associated primitive character nontrivial. Then $L (χ, 1) \neq 0$ .

Proof.

Since $ζ_{F} (s)$ has a simple pole at $s = 1$ , as does $L (χ_{0}, s)$ , for $χ_{0}$ the trivial character of modulus $[F : ℚ]$ , while $L (χ, s)$ is analytic for $χ \neq χ_{0}$ , this is a direct result of Proposition 4.1.21. □

4.2. Bernoulli numbers

Definition 4.2.1. §

For $n \geq 0$ , the $n$ th Bernoulli number $B_{n}$ is the value of the $n$ th derivative of $\frac{t}{e^{t} - 1}$ at $0$ .

In other words, $B_{n}$ is the rational number appearing in the Taylor expansion

\frac{t}{e^{t} - 1} = \sum_{n = 0}^{\infty} B_{n} \frac{t^{n}}{n!}

Example 4.2.2. §

We have

\frac{e^{t} - 1}{t} = \sum_{n = 0}^{\infty} \frac{t^{n}}{(n + 1)!} = 1 + \frac{1}{2} t + \frac{1}{6} t^{2} + \dots,

so $B_{0} = 1$ , $B_{1} = - \frac{1}{2}$ , and $B_{2} = \frac{1}{6}$ after inverting the series.

Remark 4.2.3. §

Note that

\frac{- t}{e^{- t} - 1} = \frac{t e^{t}}{e^{t} - 1} = \frac{t}{e^{t} - 1} + t,

\frac{t}{e^{t} - 1} + \frac{1}{2} t

is an even function, and therefore we have $B_{n} = 0$ for all odd $n \geq 2$ .

We shall require generalizations of these numbers attached to Dirichlet characters.

Definition 4.2.4. §

Let $χ$ be a primitive Dirichlet character, and let $m$ be any multiple of $f_{χ}$ . Then the generalized Bernoulli number $B_{n, χ}$ is the algebraic number appearing in the series expansions

\sum_{a = 1}^{m} χ (a) \frac{t e^{𝑎𝑡}}{e^{𝑚𝑡} - 1} = \sum_{n = 0}^{\infty} B_{n, χ} \frac{t^{n}}{n!} .

Remark 4.2.5. §

The independence from $m$ in the definition of $B_{n, χ}$ is easily seen to boil down to the fact that

\sum_{i = 0}^{r - 1} \frac{x^{i}}{x^{r} - 1} = \frac{1}{x - 1},

taking $r = m ∕ f_{χ}$ and $x = e^{f_{χ} t}$ .

Remark 4.2.6. §

We have $B_{n, 1} = B_{n}$ for all $n \geq 2$ , but $B_{1, 1} = \frac{1}{2} = - B_{1}$ .

Remark 4.2.7. §

We have that $B_{n, χ} = 0$ for $n ≢ δ_{χ} mod 2$ , aside from $B_{1, 1}$ .

We also have Bernoulli polynomials.

Definition 4.2.8. §

The $n$ th Bernoulli polynomial $B_{n} (X) \in ℚ [X]$ is the polynomial appearing in the series expansion

\frac{t e^{𝑋𝑡}}{e^{t} - 1} = \sum_{n = 0}^{\infty} B_{n} (X) \frac{t^{n}}{n!} .

Example 4.2.9. §

We have $B_{0} (X) = 1$ and $B_{1} (X) = X - \frac{1}{2}$ .

Lemma 4.2.10. §

Let $χ$ be a primitive Dirichlet character, and let $m$ be a multiple of $f_{χ}$ . We have

B_{n, χ} = m^{n - 1} \sum_{a = 1}^{m} χ (a) B_{n} (\frac{a}{m})

for $n \geq 1$ .

Proof.

We have

\sum_{n = 0}^{\infty} m^{n - 1} \sum_{a = 1}^{m} χ (a) B_{n} (\frac{a}{m}) \frac{t^{n}}{n!} = \sum_{a = 1}^{m} χ (a) m^{- 1} \sum_{n = 0}^{\infty} B_{n} (\frac{a}{m}) \frac{{(𝑚𝑡)}^{n}}{n!} = \sum_{a = 1}^{m} χ (a) \frac{t e^{𝑎𝑡}}{e^{𝑚𝑡} - 1} .

□

Corollary 4.2.11. §

Let $χ$ be a primitive, nontrivial Dirichlet character of conductor dividing $m$ . Then we have

B_{1, χ} = \frac{1}{m} \sum_{a = 1}^{m} χ (a) a .

Proof.

We compute easily that $B_{1} (x) = x - 1 ∕ 2$ . The result then follows from Lemma 4.2.10 and the fact that the sum over all $χ (a)$ for $1 \leq a \leq m$ is zero, since $χ$ is nontrivial. □

Definition 4.2.12. §

A value of $L (χ, s)$ at $s \in ℤ$ is known as an $L$ -value, or as a special value of the $L$ -function $L (χ, s)$ .

The following proposition gives a relationship between $L$ -values and generalized Bernoulli numbers.

Proposition 4.2.13. §

Let $χ$ be a primitive Dirichlet character. Then we have

L (χ, 1 - n) = - \frac{B_{n, χ}}{n}

for all positive integers $n$ .

Proof.

Let $x \in ℝ$ with $0 < x \leq 1$ , and consider the complex function

f (t) = \frac{t e^{(1 - x) t}}{e^{t} - 1} = \sum_{n = 0}^{\infty} B_{n} (1 - x) \frac{t^{n}}{n!} .

For $s \in ℂ$ , set

g (s) = \lim_{𝜖 \to 0^{+}} \int_{γ_{𝜖}} f (t) t^{s - 2} 𝑑𝑡,

where the path $γ_{𝜖}$ consists of the horizontal infinite path along the real axis to $𝜖$ , following by a counterclockwise traversal around the circle $C_{𝜖}$ of radius $𝜖$ , followed by the horizontal infinite path from $𝜖$ along the positive real axis. Here, $t^{s - 2} = e^{(s - 2) \log t}$ , where we take the branch of the logarithm given by the positive real axis. Then

g (s) = \lim_{𝜖 \to 0^{+}} ((e^{2 𝜋𝑖𝑠} - 1) \int_{𝜖}^{\infty} f (t) t^{s - 2} 𝑑𝑡 + \int_{C_{𝜖}} f (t) t^{s - 2} 𝑑𝑡) .

If $Re s > 1$ , the second term vanishes in the limit, and this simplifies to

{(e^{2 𝜋𝑖𝑠} - 1)}^{- 1} g (s) = \int_{0}^{\infty} f (t) t^{s - 2} 𝑑𝑡 = \sum_{k = 0}^{\infty} \int_{0}^{\infty} t^{s - 1} e^{- (x + k) t} 𝑑𝑡 = \sum_{k = 0}^{\infty} {(x + k)}^{- s} Γ (s) = Γ (s) ζ (s, x),

where we set $ζ (s, x) = \sum_{k = 0}^{\infty} {(x + k)}^{- s}$ . The latter function can be meromorphically continued to all of $ℂ$ which is again analytic away from $s = 1$ . We therefore have

g (s) = (e^{2 𝜋𝑖𝑠} - 1) Γ (s) ζ (s, x)

for all $s \in ℂ - {1}$ .

For $s = 1 - n$ , we obtain

\lim_{s \to 1 - n} (e^{2 𝜋𝑖𝑠} - 1) Γ (s) ζ (s, x) = \lim_{𝜖 \to 0^{+}} \int_{C_{𝜖}} f (t) t^{- 1 - n} 𝑑𝑡 = 2 𝜋𝑖 \cdot \frac{B_{n} (1 - x)}{n!}

by Cauchy’s integral formula. We have

\lim_{s \to 1 - n} (e^{2 𝜋𝑖𝑠} - 1) Γ (s) = 2 𝜋𝑖 \lim_{s \to 1 - n} s Γ (s) = 2 𝜋𝑖 \frac{{(- 1)}^{n - 1}}{(n - 1)!},

so we obtain

ζ (1 - n, x) = {(- 1)}^{n - 1} \frac{B_{n} (1 - x)}{n} = - \frac{B_{n} (x)}{n} .

Finally, setting $f = f_{χ}$ , we need only note that

L (χ, 1 - n) = \sum_{a = 1}^{f} χ (a) f^{n - 1} ζ (1 - n, \frac{a}{f}) = - \frac{1}{n} \sum_{a = 1}^{f} χ (a) f^{n - 1} B_{n} (\frac{a}{f}) = - \frac{B_{n, χ}}{n} .

□

Theorem 4.2.14. §

Let $χ$ be a nontrivial primitive Dirichlet character. We have

L (χ, 1) = {\begin{matrix} \frac{𝜋𝑖𝜏 (χ)}{f_{χ}} B_{1, \bar{χ}} & if χ is odd, \\ - \frac{τ (χ)}{f_{χ}} \sum_{a = 1}^{f_{χ}} \bar{χ} (a) \log | 1 - e^{2 𝜋𝑖𝑎 ∕ f_{χ}} | & if χ is even . \end{matrix}

Proof.

If $χ$ is odd, then the functional equation and the fact that $Γ (1 ∕ 2) = π^{1 ∕ 2}$ imply that

L (χ, 1) = - \frac{𝜋𝑖𝜏 (χ)}{f_{χ}} L (\bar{χ}, 0) = \frac{𝜋𝑖𝜏 (χ)}{f_{χ}} B_{1, \bar{χ}} .

Now let $χ$ be even, and set $f = f_{χ}$ . By Lemma 4.1.17, we then have

L (χ, 1) = \sum_{n = 1}^{\infty} \frac{χ (n)}{n} = \sum_{n = 1}^{\infty} \frac{1}{τ (\bar{χ})} \sum_{a = 1}^{f} \frac{\bar{χ} (a) e^{2 𝜋𝑖𝑎𝑛 ∕ f}}{n} = - \frac{1}{τ (\bar{χ})} \sum_{a = 1}^{f} \bar{χ} (a) \log (1 - e^{2 𝜋𝑖𝑎 ∕ f}) .

By Lemma 4.1.18 (and Lemma 4.1.17), we have that $τ (\bar{χ}) τ (χ) = f$ , and the evenness of $\bar{χ}$ plus the fact that the sum is taken over all $a$ mod $f$ tell us that we may replace $\log (1 - e^{2 𝜋𝑖𝑎 ∕ f})$ with

\log | 1 - e^{2 𝜋𝑖𝑎 ∕ f} | = \frac{1}{2} (\log (1 - e^{2 𝜋𝑖𝑎 ∕ f}) + \log (1 - e^{2 𝜋𝑖 (f - a) ∕ f})) .

□

Combining the analytic class number formula with Proposition 4.1.21 and Theorem 4.1.11, we obtain the following, which we will at times also refer to as the analytic class number formula.

Corollary 4.2.15. §

Let $F$ be an abelian field. Then we have

\prod_{\begin{array}{c} χ \in X (F) \\ χ \neq 1 \end{array}} L (χ, 1) = \frac{2^{r_{1} (F)} {(2 π)}^{r_{2} (F)} h_{F} R_{F}}{w_{F} | d_{F} |^{1 ∕ 2}} .

We note the following.

Lemma 4.2.16. §

Let $F$ be a CM field. Set $Q_{F} = [E_{F} : μ (F) E_{F}^{+}]$ . Then $Q_{F} \in {1, 2}$ and

[E_{F} : E_{F}^{+}] = \frac{Q_{F}}{2} w_{F} .

Proof.

Let $τ$ be the generator of $Gal (F ∕ F^{+})$ . For $α \in E_{F}$ , we have $| α^{1 - τ} | = 1$ under any complex embedding of $F$ , so $α^{1 - τ} \in μ (F)$ . Consider the commutative diagram

Units and roots of unity in a CM field. A full diagram description follows.

Diagram description: Units and roots of unity in a CM field

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 2: 1; column 3: 1; column 4: 1.
Row 2, from left to right: column 1: 1; column 2: left angle bracket minus 1 right angle bracket; column 3: mu (F); column 4: mu (F) superscript (2); column 5: 1.
Row 3, from left to right: column 1: 1; column 2: E subscript (F) superscript (plus); column 3: E subscript (F); column 4: mu (F).
Row 4, from left to right: column 1: 1; column 2: E subscript (F) superscript (plus) / left angle bracket minus 1 right angle bracket; column 3: E subscript (F) / mu (F); column 4: mu (F) / mu (F) superscript (2).
Row 5, from left to right: column 2: 1; column 3: 1; column 4: 1.

Arrows and lines:

An arrow from 1 (row 1, column 2) to left angle bracket minus 1 right angle bracket, without a label.
An arrow from 1 (row 1, column 3) to mu (F) (row 2, column 3), without a label.
An arrow from 1 (row 1, column 4) to mu (F) superscript (2), without a label.
An arrow from 1 (row 2, column 1) to left angle bracket minus 1 right angle bracket, without a label.
An arrow from left angle bracket minus 1 right angle bracket to mu (F) (row 2, column 3), without a label.
An arrow from left angle bracket minus 1 right angle bracket to E subscript (F) superscript (plus), without a label.
An arrow from mu (F) (row 2, column 3) to mu (F) superscript (2), without a label.
An arrow from mu (F) (row 2, column 3) to E subscript (F), without a label.
An arrow from mu (F) superscript (2) to 1 (row 2, column 5), without a label.
An arrow from mu (F) superscript (2) to mu (F) (row 3, column 4), without a label.
An arrow from 1 (row 3, column 1) to E subscript (F) superscript (plus), without a label.
An arrow from E subscript (F) superscript (plus) to E subscript (F), without a label.
An arrow from E subscript (F) superscript (plus) to E subscript (F) superscript (plus) / left angle bracket minus 1 right angle bracket, without a label.
An arrow from E subscript (F) to mu (F) (row 3, column 4), labelled 1 minus tau.
An arrow from E subscript (F) to E subscript (F) / mu (F), without a label.
An arrow from mu (F) (row 3, column 4) to mu (F) / mu (F) superscript (2), without a label.
An arrow from 1 (row 4, column 1) to E subscript (F) superscript (plus) / left angle bracket minus 1 right angle bracket, without a label.
An arrow from E subscript (F) superscript (plus) / left angle bracket minus 1 right angle bracket to E subscript (F) / mu (F), without a label.
An arrow from E subscript (F) superscript (plus) / left angle bracket minus 1 right angle bracket to 1 (row 5, column 2), without a label.
An arrow from E subscript (F) / mu (F) to mu (F) / mu (F) superscript (2), labelled 1 minus tau.
An arrow from E subscript (F) / mu (F) to 1 (row 5, column 3), without a label.
An arrow from mu (F) / mu (F) superscript (2) to 1 (row 5, column 4), without a label.

The snake lemma tells us that the cokernels $K$ of the two maps $τ - 1$ are isomorphic. The lower two rows yield

\begin{array}{rcl} [E_{F} : E_{F}^{+}] = \frac{w_{F}}{| K |} & and & [E_{F} : μ (F) E_{F}^{+}] = \frac{2}{| K |}, \end{array}

and the result follows. □

We remark that for cyclotomic fields, $Q_{F}$ is computable.

Lemma 4.2.17. §

Let $F = ℚ (μ_{m})$ for some $m \geq 1$ with $m ≢ 2 mod 4$ . Then

Q_{F} = {\begin{matrix} 1 & m is a prime power \\ 2 & otherwise . \end{matrix}

Proof.

Let $τ$ be the generator of $Gal (F ∕ F^{+})$ . Note that

Q_{F} = 2 | coker (E_{F} \to_{}^{1 - τ} μ (F)) |^{- 1}

by the proof of Lemma 4.2.16. If $m$ is not a prime power, then $1 - ζ_{m}$ is a unit, and ${(1 - ζ_{m})}^{1 - τ} = - ζ_{m}$ , which generates $μ (F)$ . Thus $Q_{F} = 2$ in this case. Conversely, if $α^{1 - τ} = - ζ_{m}$ generates $μ (F)$ for some $α \in E_{F}$ , we would have $α^{- 1} (1 - ζ_{m}) \in F^{+}$ . If $m$ were a power of a prime $p$ , then $α^{- 1} (1 - ζ_{m})$ would generate the unique prime over $p$ in $F$ . Since this prime is ramified in $F ∕ F^{+}$ , its generator cannot lie in $F^{+}$ . This forces $Q_{F}$ to be $1$ if $m$ is a prime power. □

Notation 4.2.18. §

For a CM field $F$ , we set $R_{F}^{+} = R_{F^{+}}$

Lemma 4.2.19. §

Let $F$ be a CM field. Then

R_{F} = 2^{r_{2} (F) - 2} \frac{w_{F}}{[E_{F} : E_{F}^{+}]} R_{F}^{+} .

Proof.

Let

r = r_{2} (F) - 1 = rank E_{F} = rank E_{F}^{+} .

Suppose that $α_{1}, α_{2}, \dots, α_{r} \in E_{F}^{+}$ satsify

⟨ - 1, α_{1}, α_{2}, \dots, α_{r} ⟩ = E_{F}^{+} .

Then

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

which has index $2 [E_{F} : E_{F}^{+}] ∕ w_{F}$ in $E_{F}$ , so Lemma 1.2.10 tells us that

R_{F} = \frac{w_{F}}{2 [E_{F} : E_{F}^{+}]} R_{F} (α_{1}, α_{2}, \dots, α_{r}) .

On the other hand, note that each $c_{i}$ in Definition 1.2.5 is $2$ for $F$ but $1$ for $F^{+}$ , so

R_{F} (α_{1}, α_{2}, \dots, α_{r}) = 2^{r} R_{F}^{+},

as desired. □

Corollary 4.2.15 implies the following.

Theorem 4.2.20. §

Suppose that $F$ is a CM abelian field. Then

\begin{array}{rcl} h_{F}^{-} = 2 [E_{F} : E_{F}^{+}] \prod_{\begin{array}{c} χ \in X (F) \\ χ odd \end{array}} \frac{- B_{1, χ}}{2} & and & h_{F}^{+} = \frac{1}{R_{F}^{+}} \prod_{\begin{array}{c} χ \in X (F) \\ χ \neq 1 even \end{array}} (\frac{- 1}{2} \sum_{a = 1}^{f_{χ}} χ (a) \log | 1 - e^{2 𝜋𝑖𝑎 ∕ f_{χ}} |) . \end{array}

Proof.

Let $E$ be an arbitrary abelian field. We remark that for $χ \in X (E)$ , the quantity $f_{χ}$ is the conductor of the corresponding character ${(ℤ ∕ f_{χ} ℤ)}^{\times} \to ℂ^{\times}$ . Therefore, the conductor-discriminant fomula tells us that

| d_{E} | = \prod_{χ \in X (E)} f_{χ} .

(4.2.1)

Moreover, a comparison of the functional equations of the Dirichlet $L$ -functions and the Artin $L$ -functions yields that

\prod_{χ \in X (E)} 𝜖_{χ} = 1,

\prod_{χ \in X (E)} τ (χ) = i^{r_{2} (E)} | d_{E} |^{1 ∕ 2} .

(4.2.2)

Taking the quotient of the analytic class number formula for $F$ by that for $F^{+}$ and applying Theorem 4.2.14, we obtain

\prod_{\begin{array}{c} χ \in X (F) \\ χ odd \end{array}} \frac{𝜋𝑖𝜏 (χ)}{f_{χ}} B_{1, \bar{χ}} = \frac{π^{r_{2} (F)}}{| d_{F} ∕ d_{F^{+}} |^{1 ∕ 2}} \frac{R_{F} ∕ R_{F}^{+}}{w_{F} ∕ w_{F^{+}}} h_{F}^{-} .

(4.2.3)

Applying (4.2.1) and (4.2.2) for $E = F$ and $E = F^{+}$ , we see that

\prod_{\begin{array}{c} χ \in X (F) \\ χ odd \end{array}} \frac{𝜋𝑖𝜏 (χ)}{f_{χ}} = \frac{{(- π)}^{r_{2} (F)}}{| d_{F} ∕ d_{F^{+}} |^{1 ∕ 2}},

and Lemma 4.2.19 tells us that

\frac{R_{F} ∕ R_{F}^{+}}{w_{F} ∕ w_{F^{+}}} = 2^{r_{2} (F) - 1} [E_{F} : E_{F}^{+}],

since $w_{F^{+}} = 2$ . Equation (4.2.3) is then immediately reduced to the desired form.

On the other hand, the analytic class number formula for $F^{+}$ and Theorem 4.2.14,

\prod_{\begin{array}{c} χ \in X (F) \\ χ \neq 1 even \end{array}} (\frac{- τ (χ)}{f_{χ}} \sum_{a = 1}^{f_{χ}} \bar{χ} (a) \log | 1 - e^{2 𝜋𝑖𝑎 ∕ f_{χ}} |) = \frac{2^{r_{1} (F^{+})} h_{F}^{+} R_{F}^{+}}{2 | d_{F^{+}} |^{1 ∕ 2}},

Applying (4.2.1) and (4.2.2) and noting that replacing $\bar{χ} (a)$ by $χ (a)$ in the resulting sum makes no difference in the result, we obtain the formula for $h_{F}^{+}$ . □

4.3. Cyclotomic units

The product appearing in the formula for $h_{F}^{+}$ in Theorem 4.2.20 may appear itself something like a regulator. This is essentially the case.

Definition 4.3.1. §

If $F$ is an abelian field contained in $ℚ (μ_{m})$ for $m \geq 1$ , we let $S = V_{𝑚∞}$ and define the group of cyclotomic $S$ -units $C_{F, S}$ of $F$ to be the subgroup

C_{F, S} = ⟨ 1 - ζ_{m}^{a} ∣ 1 \leq a < m ⟩ \cap F^{\times}

of $𝒪_{F, S}^{\times}$ , where $ζ_{m}$ is a primitive $m$ th root of unity. The group of cyclotomic units of $F$ is then defined as the intersection $C_{F} = E_{F} \cap C_{F, S}$ .

Remark 4.3.2. §

The definition of $C_{F}$ is independent of the multiple $m$ of the conductor of $F^{+}$ .

We have the following result of Hasse, which is due to Kummer in the case of $ℚ (μ_{p})$ for a prime $p$ . We will prove a generalization of this result to arbitrary cyclotomic fields in Theorem 4.7.1.

Theorem 4.3.3 (Hasse). §

Let $F = ℚ (μ_{p^{n}})$ for an odd prime $p$ and $n \geq 1$ . Then we have

h_{F}^{+} = [E_{F}^{+} : C_{F}^{+}] .

Proof.

The set

{ξ_{a} = \frac{ζ_{p^{n}}^{a ∕ 2} - ζ_{p^{n}}^{- a ∕ 2}}{ζ_{p^{n}}^{1 ∕ 2} - ζ_{p^{n}}^{- 1 ∕ 2}} | 1 < a < p^{n} ∕ 2, (a, p) = 1}

forms an independent set of generators of $C_{F}^{+}$ . Let us let $R_{cyc}$ denote the regulator of the latter set. Then $R_{cyc}$ is the absolute value of the determinant of the matrix with rows and columns indexed by the integers $a$ prime to $p$ with $1 < a < p^{n} ∕ 2$ with entries in the row and column corresponding to $(a, b)$ given by $\log | σ_{a} (ξ_{b}) |$ , where $σ_{a} (ζ_{p^{n}}) = ζ_{p^{n}}^{a}$ . Now

\log | σ_{a} (ξ_{b}) | = \log | 1 - ζ_{p^{n}}^{𝑎𝑏} | - \log | 1 - ζ_{p^{n}}^{a} | .

Proposition 1.5.18 applied to the group $Gal (F^{+} ∕ ℚ)$ yields

R_{cyc} = | \prod_{\begin{array}{c} χ \in X (F^{+}) \\ χ \neq 1 \end{array}} (\sum_{\begin{array}{c} b = 1 \\ (b, p) = 1 \end{array}}^{p^{n} ∕ 2 - 1} χ (b) \log | 1 - ζ_{p^{n}}^{b} |) | = | \prod_{\begin{array}{c} χ \in X (F^{+}) \\ χ \neq 1 \end{array}} \frac{1}{2} \sum_{\begin{array}{c} c = 1 \\ (c, p) = 1 \end{array}}^{p^{n} - 1} χ (c) \log | 1 - ζ_{p^{n}}^{c} | | .

As $χ$ has conductor dividing $p^{n}$ and

1 - ζ_{n}^{c} = \prod_{j = 0}^{k - 1} (1 - ζ_{𝑛𝑘}^{c + 𝑗𝑘})

for $n, k \geq 1$ and $c ≢ 0 mod n$ , we have

\sum_{\begin{array}{c} c = 1 \\ (c, p) = 1 \end{array}}^{p^{n} - 1} χ (c) \log | 1 - ζ_{p^{n}}^{c} | = \sum_{\begin{array}{c} c = 1 \\ (c, p) = 1 \end{array}}^{f_{χ} - 1} χ (c) \log | 1 - ζ_{f_{χ}}^{c} |,

the middle step by Theorem 4.2.14. By Theorem 4.2.20, it then follows that $R_{cyc} = h_{F}^{+} R_{F}^{+}$ . On the other hand, we have $R_{cyc} = R_{F}^{+} [E_{F}^{+} : C_{F}^{+}]$ by Lemma 1.2.10. □

A standard choice of primitive $m$ th roots of unity for $m \geq 1$ , viewing $\bar{ℚ}$ as a subset of $ℂ$ , is to take $ζ_{m} = e^{2 𝜋𝑖 ∕ m}$ for $m \geq 1$ . This choice has the advantage that $ζ_{n}^{n ∕ m} = ζ_{m}$ for $m$ dividing $n$ . Let us make such a choice. We first remark that the elements $1 - ζ_{m}$ for $m$ divisible by two distinct primes are in fact units.

Lemma 4.3.4. §

If $m$ is divisible by two distinct primes, then $1 - ζ_{m} \in C_{ℚ (μ_{m})}$ .

Proof.

For a positive integer $d$ , let $Φ_{d}$ denote the $d$ th cyclotomic polynomial. We have

Φ_{m} (1) = \prod_{\begin{array}{c} i = 1 \\ (i, m) = 1 \end{array}}^{m} (1 - ζ_{m}^{i}),

so it suffices to show that $Φ_{m} (1) = \pm 1$ . We have

\frac{x^{m} - 1}{x - 1} = \prod_{\begin{array}{c} d ∣ m \\ d > 1 \end{array}} Φ_{d} (x) .

Plugging in $x = 1$ , we obtain

m = \prod_{\begin{array}{c} d ∣ m \\ d > 1 \end{array}} Φ_{d} (1) .

Note $Φ_{p^{k}} (1) = p^{k}$ for any power $p^{k}$ of a prime $p$ . Expressing $m = \prod_{i = 1}^{g} p_{i}^{k_{i}}$ as a product of powers of distinct primes $p_{i}$ , we then also have

m = \prod_{i = 1}^{g} Φ_{p_{i}^{k_{i}}} (1) .

Since each $Φ_{d} (1)$ is an integer, it follows that $Φ_{m} (1) = \pm 1$ , as desired. □

Next, we note the following the compatibility of the elements $1 - ζ_{m}$ under norms.

Lemma 4.3.5. §

For $m \geq 1$ and a prime $ℓ$ , we have

N_{ℚ (μ_{𝑚ℓ}) ∕ ℚ (μ_{m})} (1 - ζ_{𝑚ℓ}) = {\begin{matrix} 1 - ζ_{m} & if ℓ ∣ m, \\ - ζ_{m}^{ℓ^{- 1}} \frac{1 - ζ_{m}}{1 - ζ_{m}^{ℓ^{- 1}}} & if ℓ ∤ m . \end{matrix}

Proof.

Note that

\prod_{i = 1}^{ℓ} (1 - ζ_{𝑚ℓ} ζ_{ℓ}^{i}) = 1 - ζ_{m} .

If $ℓ$ divides $m$ , then the left-hand side runs over the conjugates of $1 - ζ_{𝑚ℓ}$ under $Gal (ℚ (μ_{𝑚ℓ}) ∕ ℚ (μ_{m}))$ , so the product equals the norm.

If $ℓ$ does not divide $m$ , then let $a, b \in ℤ$ with $𝑎ℓ + 𝑏𝑚 = 1$ . We then have $ζ_{m}^{a} ζ_{ℓ}^{b} = ζ_{𝑚ℓ}$ , so the conjugates of $ζ_{𝑚ℓ}$ have the form $ζ_{𝑚ℓ} ζ_{ℓ}^{i}$ with $i ≢ - b mod ℓ$ . Note that $b \equiv m^{- 1} mod ℓ$ , so moving this term from the product to the other side, we have

N_{ℚ (μ_{m} ℓ) ∕ ℚ (μ_{m})} (1 - ζ_{𝑚ℓ}) = \frac{1 - ζ_{m}}{1 - ζ_{m}^{- ℓ^{- 1}}} = - ζ_{m}^{ℓ^{- 1}} \frac{1 - ζ_{m}}{1 - ζ_{m}^{ℓ^{- 1}}} .

□

4.4. Reflection theorems

We now refine Theorem 1.4.15 by working with eigenspaces. Start with a totally real field $F$ . Let

χ : G_{F} \to {\bar{ℚ}}^{\times}

be a character with finite image. Any embedding $φ$ of $\bar{F}$ in $ℂ$ fixes an element $c_{φ} \in G_{F}$ that is the restriction of complex conjugation in $Gal (ℂ ∕ ℝ)$ , since $F$ is taken to a subfield of $ℝ$ under the embedding. All such complex conjugations in $G_{F}$ arise in this way, and they form $[F : ℚ]$ distinct conjugacy classes in $G_{ℚ}$ for the real embeddings of $F$ in $\bar{F} = \bar{ℚ}$ . In $G_{F}^{ab}$ , these complex conjugations restrict to exactly $[F : ℚ]$ distinct elements, with the elements of the same class restricting to the same element.

Definition 4.4.1. §

We say that a character $χ : G_{F} \to {\bar{ℚ}}^{\times}$ of a totally real field $F$ is totally even if $χ$ is trivial on all complex conjugations and totally odd if $χ$ is nontrivial on all complex conjugations. If $F = ℚ$ , we say more simply that $χ$ is even or odd in the respective cases.

We let $F_{χ}$ denote the extension of $F$ that is the fixed field of the kernel of $χ$ , which will itself be totally real if $χ$ is totally even and CM if $χ$ is totally odd. If $F = ℚ$ , these are the only cases.

We now suppose that $χ$ has order prime to a given odd prime $p$ . We fix an embedding $ι_{p} : \bar{ℚ} \to \bar{ℚ_{p}}$ , which allows us to view $χ$ as a character with values in ${\bar{ℚ_{p}}}^{\times}$ , and hence in ${\bar{ℤ_{p}}}^{\times}$ .

One key character of interest to us is the Teichmüller character

ω : G_{F} \to ℚ_{p}^{\times}

which has image contained in $μ_{p - 1} (ℤ_{p})$ and is defined by the equality

σ (ζ) = ζ^{ω (σ)}

for any $σ \in G_{F}$ and $ζ \in μ_{p}$ . Note that the Teichmüller character is an odd character on $G_{F}$ .

Theorem 4.4.2 (Leopoldt’s Spiegelungsatz). §

Let $F$ be a totally real field, and let $χ : G_{F} \to {\bar{ℚ_{p}}}^{\times}$ be a totally odd character of finite order prime to $p$ . Let $E$ be an abelian extension of $F$ of degree prime to $p$ that contains $F_{χ} (μ_{p})$ . Then we have

r_{p} (A_{E}^{(ω χ^{- 1})}) - δ_{χ} \leq r_{p} (A_{E}^{(χ)}) \leq r_{p} (A_{E}^{(ω χ^{- 1})}) + r_{p} ({(𝒪_{E}^{\times} ∕ 𝒪_{E}^{\times p})}^{(ω χ^{- 1})}),

where $δ_{χ}$ is $0$ unless $χ = ω$ and the extension $E (μ {(F)}^{1 ∕ p}) ∕ E$ is unramified, in which case it is $1$ .

Proof.

Let $Δ = Gal (E ∕ F)$ . Let $𝒪$ be the ring generated over $ℤ_{p}$ by the character values of $Δ$ . Let $k$ denote the residue field of $𝒪$ , and let $k_{ψ}$ denote the residue field of $𝒪_{ψ}$ , the ring of values of $ψ$ , for any $ψ \in Δ^{*}$ . As $𝒪$ is unramified over $ℤ_{p}$ , we have $[𝒪 : 𝒪_{ψ}] = [k : k_{ψ}]$ .

For a $ℤ_{p} [Δ]$ -module $B$ , we let $B_{𝒪} = B \otimes_{ℤ_{p}} 𝒪$ . We remark that Lemma 2.8.7 implies that

r_{p} (B_{𝒪}^{ψ}) = [k : k_{ψ}] r_{p} (B^{(ψ)}) .

Note also that we have

r_{p} (B_{𝒪}^{ψ}) = [k : 𝔽_{p}] \dim_{k} ({(B ∕ 𝑝𝐵)}_{𝒪}^{ψ}),

r_{p} (B^{(ψ)}) = {[k_{ψ} : 𝔽_{p}]}^{- 1} \dim_{k} ({(B ∕ 𝑝𝐵)}_{𝒪}^{ψ}) .

(4.4.1)

Since $𝒪_{χ} = 𝒪_{ω χ^{- 1}}$ and since $δ_{χ} = 0$ unless $χ = ω$ , in which case $k_{χ} = 𝔽_{p}$ , equation (4.4.1) tells us that the desired inequalities are equivalent to

\dim_{k} (A_{𝒪}^{ω χ^{- 1}}) - δ_{χ} \leq \dim_{k} (A_{𝒪}^{χ}) \leq \dim_{k} (A_{𝒪}^{ω χ^{- 1}}) + \dim_{k} ({(𝒪_{E}^{\times} ∕ 𝒪_{E}^{\times p})}_{𝒪}^{ω χ^{- 1}}),

where we have set $A = A_{E} ∕ p A_{E}$ to shorten notation.

Note that we have the following isomorphisms of groups

{Hom}_{ℤ_{p}} {(A_{E}, μ_{p})}_{𝒪} ≅ {Hom}_{ℤ_{p}} (A_{E}, {(μ_{p})}_{𝒪}) ≅ {Hom}_{𝒪} ({(A_{E})}_{𝒪}, {(μ_{p})}_{𝒪})

the first step following from the freeness of $𝒪$ over $ℤ_{p}$ and the second from the adjointness of $Hom$ and $\otimes$ . Moreover, Lemma 2.8.7 implies that

{Hom}_{𝒪} {({(A_{E})}_{𝒪}, {(μ_{p})}_{𝒪})}^{ψ} ≅ {Hom}_{𝒪} ({(A_{E})}_{𝒪}^{ω ψ^{- 1}}, {(μ_{p})}_{𝒪})

for any $ψ \in Δ^{*}$ . Recalling Lemma 1.4.6, we then have an exact sequence

0 \to {((B \cap 𝒪_{E}^{\times}) ∕ 𝒪_{E}^{\times p})}_{𝒪}^{ψ} \to {Hom}_{𝒪} ({(A_{E})}_{𝒪}^{ω ψ^{- 1}}, {(μ_{p})}_{𝒪}) \to {(A_{E})}_{𝒪}^{ψ} [p],

where $B$ is the set of elements of $E^{\times}$ that have $p$ th roots that generate unramified extensions of $E$ . Since ${(μ_{p})}_{𝒪}$ is a one-dimensional $k$ -vector space, we have

{Hom}_{𝒪} ({(A_{E})}_{𝒪}^{ω ψ^{- 1}}, {(μ_{p})}_{𝒪}) ≅ {Hom}_{k} (A_{𝒪}^{ω ψ^{- 1}}, k),

which as the $k$ -dual of a $k$ -vector space has dimension equal to $\dim_{k} (A_{𝒪}^{ω ψ^{- 1}})$ .

In the case that $ψ = χ$ , we then have that

\dim_{k} (A_{𝒪}^{ω χ^{- 1}}) \leq \dim_{k} ({((B \cap μ (E)) ∕ μ {(E)}^{p})}_{𝒪}^{χ}) + \dim_{k} (A_{𝒪}^{(χ)}) = δ_{χ} + \dim_{k} (A_{E}^{(χ)}),

since the $p$ -power roots of unity in $E$ have trivial $χ$ -eigenspace unless $[χ] = [ω]$ , which happens if and only if $χ = ω$ , as $ω$ takes its values in $ℤ_{p}$ . On the other hand, if we take $ψ = ω χ^{- 1}$ , then we have

\dim_{k} (A_{𝒪}^{χ}) \leq \dim_{k} ({(𝒪_{E}^{\times} ∕ 𝒪_{E}^{\times p})}_{𝒪}^{ω χ^{- 1}}) + \dim_{k} (A_{𝒪}^{ω χ^{- 1}}),

finishing the proof. □

In the special case that $F = ℚ$ and $E = ℚ (μ_{p})$ , we remark that $δ_{ω} = 0$ , as $ℚ (μ_{p^{2}}) ∕ F$ is ramified at the unique prime over $p$ . Moreover, we have the following.

Lemma 4.4.3. §

Let $k$ be an even integer. Then

{(𝒪_{ℚ (μ_{p})}^{\times} \otimes_{ℤ} ℤ_{p})}^{(ω^{k})} ≅ {\begin{matrix} ℤ_{p} & k ≢ 0 mod (p - 1) \\ 0 & k \equiv 0 mod (p - 1) \end{matrix} .

Corollary 4.4.4. §

For any even integer $k$ , we have

r_{p} (A_{ℚ (μ_{p})}^{(ω^{k})}) \leq r_{p} (A_{ℚ (μ_{p})}^{(ω^{1 - k})}) \leq r_{p} (A_{ℚ (μ_{p})}^{(ω^{k})}) + 1 .

Corollary 4.4.5. §

We have $A_{ℚ (μ_{p})}^{(ω)} = A_{ℚ (μ_{p})}^{(1)} = 0$ .

Proof.

We know that

A_{ℚ (μ_{p})}^{(1)} = A_{ℚ (μ_{p})}^{Gal (ℚ (μ_{p}) ∕ ℚ)} ≅ A_{ℚ} = 0 .

As Theorem 4.4.2 and Lemma 4.4.3 tell us that $r_{p} (A_{ℚ (μ_{p})}^{(1)}) = r_{p} (A_{ℚ (μ_{p})}^{(ω)})$ , so we are done. □

4.5. Stickelberger theory

Let us fix an integer $m \geq 1$ and a primitive $m$ th root of unity $ζ_{m}$ throughout this section.

Definition 4.5.1. §

Let $F = ℚ (μ_{m})$ , and let $G = Gal (F ∕ ℚ)$ .

a.: For $a \in ℤ$ with $(a, m) = 1$ , let $σ_{a} \in G$ be such that $σ_{a} (ζ_{m}) = ζ_{m}^{a}$ . The Stickelberger element $𝜃_{F}$ is the element of $ℚ [G]$ given by
$𝜃_{F} = \frac{1}{m} \sum_{\begin{array}{c} a = 1 \\ (a, m) = 1 \end{array}}^{m} a σ_{a}^{- 1} .$
b.: The Stickelberger ideal of $F$ is the ideal $I_{F} = ℤ [G] 𝜃_{F} \cap ℤ [G]$ of $ℤ [G]$ .

Lemma 4.5.2. §

Let $J$ denote the ideal of $ℤ [G]$ generated by elements of the form $σ_{b} - b$ for $b \in ℤ$ with $(b, m) = 1$ . Then $J = {x \in ℤ [G] ∣ x 𝜃_{F} \in ℤ [G]}$ .

Proof.

Let us use $⟨ α ⟩$ to denote the fractional part of $α \in ℚ$ . We note

σ_{b} 𝜃_{F} = \sum_{\begin{array}{c} a = 1 \\ (a, m) = 1 \end{array}}^{m} \frac{a}{m} σ_{b} σ_{a}^{- 1} = \sum_{\begin{array}{c} a = 1 \\ (a, m) = 1 \end{array}}^{m} ⟨ \frac{𝑎𝑏}{m} ⟩ σ_{a}^{- 1} .

Since $⟨ \frac{𝑎𝑏}{m} ⟩ - ⟨ \frac{a}{m} ⟩ b \in ℤ$ , we have $(σ_{b} - b) 𝜃_{F} \in I_{F}$ for all $b \in ℤ$ prime to $m$ , and hence $J 𝜃_{F} \subseteq ℤ [G]$ .

Now take $x = \sum_{b} e_{b} σ_{b}$ with $x 𝜃_{F} \in ℤ [G]$ . Writing this out, we have

\sum_{\begin{array}{c} a = 1 \\ (a, m) = 1 \end{array}}^{m} (\sum_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} e_{b} ⟨ \frac{𝑎𝑏}{m} ⟩) σ_{a}^{- 1} \in ℤ [G],

which implies that

\sum_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} e_{b} b \in 𝑚ℤ .

But note that $m = (m + 1) - σ_{1} \in J$ , so $𝑚ℤ \subset J$ . We then have

x = \sum_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} e_{b} (σ_{b} - b) + \sum_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} e_{b} b \in J,

finishing the proof. □

Definition 4.5.3. §

Let $q$ be a power of a prime $ℓ$ and $χ : 𝔽_{q}^{\times} \to ℂ^{\times}$ be a character, which we extend to a function $χ : 𝔽_{q} \to ℂ$ by $χ (0) = 0$ . The Gauss sum attached to $χ$ is

g (χ) = - \sum_{α \in 𝔽_{q}^{\times}} χ (α) e^{2 𝜋𝑖 Tr (α) ∕ ℓ}

where $Tr = {Tr}_{𝔽_{q} ∕ 𝔽_{ℓ}}$ is the trace map.

Lemma 4.5.4. §

Let $q$ be a power of a prime $ℓ$ prime to $m$ . Let $χ : 𝔽_{q}^{\times} \to μ_{m}$ be a character, so $g (χ) \in ℚ (μ_{ℓ𝑚})$ . Let $b \in ℤ$ be relatively prime to $m$ , and let $σ_{b} \in Gal (ℚ (μ_{ℓ𝑚}) ∕ ℚ (μ_{ℓ}))$ be the unique lift of $σ_{b} \in G$ . Then

g {(χ)}^{σ_{b} - b} \in ℚ (μ_{m}) .

In particular, we have $g {(χ)}^{m} \in ℚ (μ_{m})$ .

Proof.

For $τ \in Gal (ℚ (μ_{ℓ𝑚}) ∕ ℚ (μ_{m}))$ with $τ (ζ_{ℓ}) = ζ_{ℓ}^{c}$ , we have

g {(χ)}^{τ} = - \sum_{α \in 𝔽_{q}} χ (α) e^{2 𝜋𝑖 Tr (𝑐𝛼) ∕ ℓ} = χ {(c)}^{- 1} g (χ) .

On the other hand, we have $g {(χ)}^{σ_{b}} = g (χ^{b})$ as $σ_{b}$ fixes $μ_{ℓ}$ , so we see that

{(g {(χ)}^{σ_{b} - b})}^{τ} = g {(χ^{b})}^{τ} g {(χ)}^{- 𝑏𝜏} = χ^{b} {(c)}^{- 1} χ {(c)}^{- b} g {(χ)}^{σ_{b} - b} = g {(χ)}^{σ_{b} - b},

as desired. □

Lemma 4.5.5. §

Let $q$ be a power of a prime $ℓ$ and $χ : 𝔽_{q}^{\times} \to ℂ^{\times}$ be a character. Then

g (χ) g (\bar{χ}) = χ (- 1) ℓ .

We state Stickelberger’s theorem for $F = ℚ (μ_{m})$ . A similar result holds for abelian fields in general.

Theorem 4.5.6 (Stickelberger). §

Let $F = ℚ (μ_{m})$ , set $G = Gal (F ∕ ℚ)$ . Then the Stickelberger ideal of $F$ annihilates the class group: $I_{F} \cdot {Cl}_{F} = 0$ .

Proof.

Fix $C \in {Cl}_{F}$ , and let $𝔩$ be a prime ideal representing $C$ in $𝒪_{F}$ that lies above a completely split prime $ℓ$ of $ℚ$ . Note that $ℓ \equiv 1 mod m$ , and let $c \in ℤ$ be a primitive root modulo $ℓ$ . Let $χ : 𝔽_{ℓ}^{\times} \to ℂ^{\times}$ denote the character with $χ (c) = e^{2 𝜋𝑖 ∕ m}$ . There is unique prime $𝔏$ of $ℚ (μ_{ℓ𝑚})$ lying above $𝔩$ , and $𝔏^{ℓ - 1} = 𝔩 \cdot ℤ [μ_{ℓ𝑚}]$ . We use $v_{𝔏}$ to denote the additive valuation attached to $𝔏$ . For $b \in ℤ$ prime to $m$ , and $σ_{b} \in Gal (ℚ (μ_{ℓ𝑚}) ∕ ℚ (μ_{ℓ}))$ the unique lift of $σ_{b} \in Gal (F ∕ ℚ)$ , we set

t_{b} = v_{σ_{b}^{- 1} 𝔏} (g (χ)) .

By Lemma 4.5.5, we have that $g (χ) ∣ (ℓ)$ , so $t_{b} \leq ℓ - 1$ , and by Lemma 4.5.4, we have in the smaller field $F$ that

v_{σ_{b}^{- 1} 𝔩} (g {(χ)}^{ℓ - 1}) = t_{b} .

In other words, we have the factorization

g {(χ)}^{ℓ - 1} 𝒪_{F} = \prod_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} {(σ_{b}^{- 1} 𝔩)}^{t_{b}},

\sum_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} t_{b} σ_{b}^{- 1}

annihilates the class of $𝔩$ .

Now take $τ \in Gal (F (μ_{ℓ}) ∕ F)$ given by $τ (ζ_{ℓ}) = ζ_{ℓ}^{c}$ . Then since every prime over $ℓ$ is totally ramified $F (μ_{ℓ}) ∕ F$ , we have that $τ$ is in the inertia group of all such primes. Note that

v_{σ_{b}^{- 1} 𝔏} (ζ_{ℓ} - 1) = 1

for all $b$ . We calculate

\frac{g (χ)}{{(ζ_{ℓ} - 1)}^{t_{b}}} \equiv \frac{g {(χ)}^{τ}}{{(ζ_{ℓ}^{c} - 1)}^{t_{b}}} \equiv \frac{χ {(c)}^{- 1} g (χ)}{c^{t_{b}} {(ζ_{ℓ} - 1)}^{t_{b}}} mod σ_{b}^{- 1} 𝔏 .

This forces $e^{2 𝜋𝑖 ∕ m} \equiv c^{- t_{b}} mod σ_{b}^{- 1} 𝔏$ and therefore modulo $σ_{b}^{- 1} 𝔩$ , since both sides of the latter congruence lie in $F$ . In other words, we have

e^{2 𝜋𝑖𝑏 ∕ m} \equiv c^{- t_{b}} mod 𝔩 .

On the other hand, there exists some $a$ prime to $m$ such that

e^{2 𝜋𝑖 ∕ m} \equiv c^{- (ℓ - 1) a ∕ m} mod 𝔩 .

We therefore have that

t_{b} \equiv \frac{(ℓ - 1) 𝑎𝑏}{m} mod (ℓ - 1),

forcing

t_{b} = (ℓ - 1) ⟨ \frac{𝑎𝑏}{m} ⟩ .

It follows that

(ℓ - 1) \sum_{\begin{array}{c} b = 1 \\ (n, m) = 1 \end{array}}^{m} ⟨ \frac{𝑎𝑏}{m} ⟩ σ_{b}^{- 1} = (ℓ - 1) σ_{a} 𝜃_{F}

annihilates $C$ .

Now suppose $x \in ℤ [G]$ is such that $x 𝜃_{F} \in I_{F}$ . We then have

{(g {(χ)}^{σ_{a}^{- 1} x})}^{ℓ - 1} 𝒪_{F} = 𝔩^{(ℓ - 1) x 𝜃_{F}} .

By Lemmas 4.5.2 and 4.5.4, we have $g {(χ)}^{σ_{a}^{- 1} x} \in F$ . Therefore, the identity

(g {(χ)}^{σ_{a}^{- 1} x}) 𝒪_{F} = 𝔩^{x 𝜃_{F}}

actually holds, and so we see that $x 𝜃_{F}$ annihilates $C$ . So, $I_{F}$ annihilates $C$ , and we are done. □

This has an interesting application for the field $ℚ (μ_{p})$ .

Theorem 4.5.7 (Herbrand). §

Let $p$ be an odd prime, and set $F = ℚ (μ_{p})$ . Let $j ≢ 1 mod p - 1$ be an odd integer, and suppose that $A_{F}^{(ω^{j})} \neq 0$ . Then $B_{1, ω^{- j}} \in p ℤ_{p}$ . Moreover, we have $A_{F}^{(ω)} = 0$ .

Proof.

By Stickelberger’s theorem, we have that $I_{F} \cdot A_{F} = 0$ . In particular, we have that $I_{j} = e_{ω^{j}} I_{F}$ annihilates $A_{F}^{(ω^{j})}$ , where $e_{ω^{j}} \in ℤ_{p} [G]$ is the idempotent attached to $ω^{j}$ . Note that for, $b$ prime to $p$ , we have

e_{ω^{j}} (σ_{b} - b) 𝜃_{F} = (ω^{j} (b) - b) \frac{1}{p} \sum_{a = 1}^{p - 1} a ω^{- j} (a) e_{ω^{j}} = (ω^{j} (b) - b) B_{1, ω^{- j}} e_{ω^{j}},

where we have applied Corollary 4.2.11 in the last step. It follows that $(ω^{j} (b) - b) B_{1, ω^{- j}}$ annihilates $A_{F}^{(ω^{j})}$ for all $b$ prime to $p$ . Choosing $b$ to be a primitive root of $1$ , we have that $ω^{j} (b) ≢ b mod p$ , so if $A_{F}^{(ω^{j})}$ is nontrivial, then $B_{1, ω^{- j}}$ must be divisible by $p$ . For $j = 1$ , we note that

(ω^{j} (1 + p) - (1 + p)) B_{1, ω^{- 1}} = - p B_{1, ω^{- 1}} = - \sum_{i = 1}^{p} ω {(a)}^{- 1} a \equiv 1 mod p,

so we get that $1$ annihliates $A_{F}^{(ω)}$ , hence the result. □

As with the plus part, the minus part of the class number of a cyclotomic field of prime power roots of unit can be interpreted as an index, as in the following result of Iwasawa. The proof is deferred to its generalization to arbitrary cyclotomic fields in Theorem 4.7.1.

Theorem 4.5.8 (Iwasawa). §

Let $F = ℚ (μ_{p^{n}})$ for a prime $p$ and $n \geq 1$ . Then

h_{F}^{-} = [ℤ {[G]}^{-} : I_{F}^{-}] .

4.6. Distributions

Definition 4.6.1. §

Let ${X_{i} ∣ i \in I}$ be a collection of finite sets, were $I$ is a directed set under $\leq$ , and let $π_{𝑖𝑗} : X_{i} \to X_{j}$ for $i \geq j$ be a collection surjective maps. Let $A$ be an abelian group. An $A$ -valued distribution on the collection $(X_{i}, π_{𝑖𝑗})$ is a set of maps $ψ_{i} : X_{i} \to A$ for $i \in I$ that satisfy the distribution relation

ψ_{j} (x) = \sum_{y \in π_{𝑖𝑗}^{- 1} (x)} ψ_{i} (y)

for all $j \leq i$ and $x \in X_{i}$ .

Remark 4.6.2. §

Given a collection $(X_{i}, π_{𝑖𝑗})$ as above, we may consider the inverse limit

X = \underset{i \in I}{lim_{\leftarrow}} X_{i} .

Let $π_{i} : X \to X_{i}$ be the map induced by the system. Let $Step (X, A)$ denote the set of $A$ -valued step functions on $X$ . Supposing now that $A$ is a ring, a distribution ${ψ_{i} : X_{i} \to A ∣ i \in I}$ on the collection $(X_{i}, π_{𝑖𝑗})$ (or more simply, on $X$ ) gives rise to an $A$ -module homomorphism

\tilde{ψ} : Step (X, A) \to A

as follows. If $χ_{Y}$ denotes the characteristic function of a compact-open subset $Y$ of $X$ , then we let

\tilde{ψ} (χ_{π_{i}^{- 1} (x)}) = ψ_{i} (x)

for any $i \in I$ and $x \in X_{i}$ . We take $\tilde{ψ}$ as the $A$ -linear extension of this map to the group of all step functions. The distribution relation insures that it is well-defined. Conversely, given an $A$ -module homomorphism $\tilde{ψ} : Step (X, A) \to A$ , we may define $ψ_{i} (x)$ to be $\tilde{ψ} (χ_{π_{i}^{- 1}} (x))$ , and the $ψ_{i}$ provide a distribution on $X$ .

Example 4.6.3. §

Let $I$ be the set of positive integers, ordered in the usual manner. Let $X_{i} = ℤ ∕ p^{i} ℤ$ , and let $π_{𝑖𝑗}$ for $j \leq i$ be the reduction modulo $p^{j}$ map. Let $a \in ℤ_{p}$ . Define

ψ_{i} (x) = {\begin{matrix} 1 & if x \equiv a mod p^{i}, \\ 0 & otherwise . \end{matrix}

Then ${ψ_{i} ∣ i \geq 0}$ is an $R$ -valued distribution for any ring $R$ , called the $δ$ -distribution at $a$ . The corresponding functional $δ_{a}$ satisfies $δ_{a} (f) = f (a)$ , where $f \in Step (ℤ_{p}, R)$ is any congruence function.

Let us focus on a specific case of interest.

Definition 4.6.4. §

Let $A$ be an abelian group, and let $D$ be a divisible abelian group with finitely topologically generated Pontryagin dual.

a.

By an $A$ -valued distribution on $D$ , we mean a function $ψ : D \to A$ with the property that

ψ (d) = \sum_{\begin{array}{c} c \in D \\ 𝑛𝑐 = d \end{array}} ψ (c)

(4.6.1)

for all positive integers $n$ and $d \in D$ .

b.

By an $A$ -valued punctured distribution on $D$ , we mean a function $ψ : D - {0} \to A$ satisfying the distribution relation (4.6.1) for all positive integers $n$ and $d \in D - {0}$ .

Remark 4.6.5. §

For an abelian group $A$ and a torsion divisible abelian group $D$ , the $A$ -valued distributions on $D$ are in one-to-one correspondence with the $A$ -valued distributions ${ψ_{n} ∣ n \geq 1}$ on the collection of $n$ -torsion subgroups $D [n]$ in $D$ for $n \geq 1$ , together with the transition maps $π_{n, m} : D [n] \to D [m]$ for $m$ dividing $n$ given by multiplication by $\frac{n}{m}$ . That is, $ψ$ and the maps $ψ_{n}$ take the same values on the elements of $\frac{1}{n} ℤ ∕ ℤ$ . If $A$ is a ring, then the maps $ψ$ also give rise to a functional $\tilde{ψ} : Step ({lim_{\leftarrow}}_{n} D [n], A) \to A$ , as noted above.

Remark 4.6.6. §

Punctured distributions on $D$ do not quite give rise to distributions on the sets $D [n] - {0}$ , since multiplication by $\frac{n}{m}$ does not preserve these sets.

Example 4.6.7. §

Let $I$ be the set of positive integers, ordered by divisibility. Fix $k \geq 0$ , and for $0 \leq a < n$ with $n \geq 1$ , set

ψ_{n}^{(k)} (\frac{a}{n}) = n^{k - 1} B_{k} (\frac{a}{n}) .

For $m$ dividing $n$ , we have

ψ_{n}^{(k)} (\frac{a}{m}) = m^{k - 1} B_{k} (\frac{a}{m}) = \sum_{j = 0}^{n ∕ m - 1} n^{k - 1} B_{k} (\frac{a + 𝑗𝑚}{n}) = \sum_{\begin{array}{c} b = 0 \\ b \equiv a mod m \end{array}}^{n - 1} ψ_{n}^{(k)} (\frac{b}{n}) .

Thus, we can safely make the following definition.

Definition 4.6.8. §

For $k \geq 0$ , the $k$ th Bernoulli distribution $ψ^{(k)}$ is the $ℚ$ -valued distribution on $ℚ ∕ ℤ$ defined by

ψ^{(k)} (\frac{a}{n}) = n^{k - 1} B_{k} (⟨ \frac{a}{n} ⟩),

where $⟨ α ⟩$ denotes the smallest nonegative rational number representing $α \in ℚ ∕ ℤ$ .

We also mention the following example of something close to a distribution.

Example 4.6.9. §

Define $ψ : ℚ ∕ ℤ - {0} \to ℚ {(μ_{\infty})}^{\times}$ by $ψ (\frac{i}{n}) = 1 - ζ_{n}^{i}$ . If $m ∣ n$ and $i ≢ 0 mod m$ , we have

ψ (\frac{i}{m}) = 1 - ζ_{m}^{i} = \prod_{k = 0}^{n ∕ m - 1} (1 - ζ_{n}^{i + 𝑘𝑚}) = \prod_{\begin{array}{c} j = 0 \\ j \equiv i mod m \end{array}}^{n - 1} ψ (\frac{j}{n}),

so $ψ$ satisfies the distribution relations under multiplication. Thus $ψ$ is a punctured distribution on $ℚ ∕ ℤ$ .

We will be interested in the following resulting distribution.

Notation 4.6.10. §

Let $ψ_{cyc}$ be the $ℝ$ -valued punctured distribution on $ℚ ∕ ℤ$ given by

ψ_{cyc} (α) = - \frac{1}{2} \log | 1 - e^{2 𝜋𝑖 ⟨ α ⟩} |

for $α \in ℚ ∕ ℤ$ .

Remark 4.6.11. §

Note that an $A$ -valued (punctured) distribution $ψ$ on $ℚ ∕ ℤ$ gives rise to an $A$ -valued map $\tilde{ψ}$ on (nontrivial) Dirichlet characters $χ$ in that Dirichlet characters are step functions on $\hat{ℤ}$ (that are zero at zero). In particular, if $χ$ has modulus dividing $m$ , then

\tilde{ψ} (χ) = \sum_{a = 0}^{m - 1} χ (a) ψ (\frac{a}{m}) .

Example 4.6.12. §

By Lemma 4.2.10, we have

ψ^{(n)} (χ) = B_{n, χ}

for a primitive Dirichlet character $χ$ . In particular, $ψ^{(n)} (χ) = 0$ if $n ≢ χ (- 1) mod 2$ , unless $n = 1$ and $χ = 1$ . Similarly, $ψ_{cyc} (χ) = 0$ unless $χ$ is even.

4.7. Sinnott’s theorem

In this section, we fix $m > 1$ with $m ≢ 2 mod 4$ . We set $F = ℚ (μ_{m})$ and $G = Gal (F ∕ ℚ)$ . The goal of this section is to prove the following generalization of the results of Hasse and Iwasawa for $F$ , which is due to Sinnott.1

Theorem 4.7.1 (Sinnott). §

Let $F = ℚ (μ_{m})$ for $m > 1$ with $m ≢ 2 mod 4$ . Then we have

\begin{array}{rcl} [E_{F}^{+} : C_{F}^{+}] = 2^{a} h_{F}^{+} & and & [ℤ {[G]}^{-} : I_{F}^{-}] = 2^{b} h_{F}^{-}, \end{array}

where

\begin{array}{rcl} a = {\begin{matrix} 0 & if g = 1 \\ 2^{g - 2} + 1 - g & if g \geq 2 \end{matrix} & and & b = {\begin{matrix} 0 & if g = 1 \\ 2^{g - 2} - 1 & if g \geq 2, \end{matrix} \end{array}

for $g$ the number of primes dividing $m$ .

Notation 4.7.2. §

For $χ \in \hat{G}$ , we have the idempotent

e_{χ} = \frac{1}{φ (m)} \sum_{\begin{array}{c} a = 1 \\ (a, m) = 1 \end{array}}^{m} χ (a) σ_{a}^{- 1} \in ℂ [G] .

We also have idempotents

e^{\pm} = \frac{1 \pm σ_{- 1}}{2} \in ℚ [G] .

The following is essentially immediate from the definitions.

Lemma 4.7.3. §

We have $e^{\pm} A = \frac{1}{2} ℤ {[G]}^{\pm}$ inside $ℚ [G]$ . In particular, we see that

[e^{\pm} ℤ [G] : ℤ {[G]}^{\pm}] = 2^{φ (m) ∕ 2} .

Notation 4.7.4. §

For any $ℤ [G]$ -module $A$ , set $A_{0} = \ker (N_{G} : A \to A)$ .

Remark 4.7.5. §

For a $ℤ [G]$ -module $A$ , we note that $e^{-} (1 - e_{1}) A = e^{-} A$ .

Notation 4.7.6. §

For each prime $p$ dividing $m$ , set

λ_{p} = \sum_{χ \in \hat{G}} (1 - \bar{χ} (p)) e_{χ} \in ℚ [G] .

For each positive integer $f$ dividing $m$ , set $G_{f} = Gal (ℚ (μ_{m}) ∕ ℚ (μ_{f}))$ . Let $U$ denote the $ℤ [G]$ -module generated by the elements

u_{f} = N_{G_{f}} \prod_{p ∣ f} λ_{p} \in ℚ [G]

for positive integers $f$ dividing $m$ , where the product is taken over primes dividing $f$ .

We briefly sketch a proof of the following proposition.

Proposition 4.7.7. §

Let $g$ be the number of primes dividing $m$ . Then we have the following equalities:

(e^{\pm} ℤ [G] : e^{\pm} U) = {\begin{matrix} 1 & if g = 1 \\ 2^{2^{g - 2}} & if g \geq 2 . \end{matrix}

Proof.

If $g = 1$ , then $U$ is generated by $N_{G}$ and $λ_{p}$ for the unique prime $p$ dividing $m$ . We have $u_{1} = N_{G} = | G | e_{1}$ and $u_{p} = λ_{p} = 1 - e_{1}$ . Then

[e_{1} ℤ [G] + ℤ [G] : U] = | G | = [e_{1} ℤ [G] + ℤ [G] : ℤ [G]],

so $[ℤ [G] : U] = 1$ . Moreover, note that $e^{-} e_{1} = 0$ , and from this it is easily seen that $e^{-} ℤ [G] = e^{-} U$ , and as a result, $[e^{+} ℤ [G] : e^{+} U] = 1$ as well.

For $g \geq 2$ , we indicate only a few details of the proof. One uses the fact that $U$ is the product over primes $p$ dividing $m$ of the modules $U_{p}$ generated by $N_{I_{p}}$ and $λ_{p}$ , where $I_{p} < Gal (F ∕ ℚ)$ is the inertia group at $p$ , to see that $(ℤ [G] : U) = 1$ . On the other hand,

(ℤ [G] : U) = (e^{+} ℤ [G] : e^{+} U) (ℤ {[G]}^{-} : U^{-}) .

One checks that the order of

{\hat{H}}^{- 1} (Gal (F ∕ F^{+}), U) = U^{-} ∕ (σ_{- 1} - 1) U ≅ e^{-} ℤ [G] ∕ e^{-} U

is $2^{2^{g - 1}}$ . We then have

(e^{+} ℤ [G] : e^{+} U) (e^{-} ℤ [G] : e^{-} U) = 2^{2^{g - 1}},

and the proof is finished upon showing that $(e^{-} ℤ [G] : e^{-} U) = 2^{2^{g - 2}}$ , which we omit. □

Recall that $I_{G}$ denotes the augmentation ideal in $ℤ [G]$ .

Corollary 4.7.8. §

Let $g$ be the number of primes dividing $m$ . Then

(e^{+} I_{G} : e^{+} U_{0}) = {\begin{matrix} φ {(m)}^{- 1} & if g = 1 \\ 2^{2^{g - 2}} φ {(m)}^{- 1} & if g \geq 2, \end{matrix}

Proof.

The quotient $e^{+} ℤ [G] ∕ e^{+} I_{G}$ is isomorphic to $ℤ$ via the augmentation map, while $e^{+} U ∕ e^{+} U_{0}$ is generated by the class of $u_{1} = N_{G}$ , and the image of $N_{G} \in e^{+} ℤ [G]$ under the augmentation map is $| G | = φ (M)$ . It follows that

(e^{\pm} ℤ [G] : e^{\pm} U) = φ (M) (e^{+} I_{G} : e^{+} U_{0}),

and we apply Proposition 4.7.7. □

Notation 4.7.9. §

For a punctured $ℂ$ -valued distribution $ψ$ on $ℚ ∕ ℤ$ , let $T_{ψ}$ be the subgroup of $ℂ [G]$ generated by the elements

η_{ψ} (c) = \sum_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} ψ (\frac{𝑏𝑐}{m}) σ_{b}^{- 1}

for positive integers $c$ with $c ≢ 0 mod m$ .

Remark 4.7.10. §

The group $T_{ψ}$ is a $ℤ [G]$ -module, as $σ_{a} η_{ψ} (c) = η_{ψ} (𝑎𝑐)$ for $a$ prime to $m$ . As a $ℤ [G]$ -module, it is then generated by the elements $η_{ψ} (d)$ for $d$ positive dividing $m$ .

At times, we will view the elements of $\hat{G}$ also as primitive Dirichlet characters.

Proposition 4.7.11. §

Let $ψ$ be a punctured $ℂ$ -valued distribution on $ℚ ∕ ℤ$ . Then

(1 - e_{1}) T_{ψ} = ω_{ψ} U,

where

ω_{ψ} = \sum_{χ \in \hat{G} - {1}} ψ (\bar{χ}) e_{χ} \in ℂ [G] .

Proof.

For $d \geq 1$ dividing $m$ , set $f = \frac{m}{d}$ . Let $χ$ be a nontrivial character of $\hat{G}$ . Then $e_{χ} η_{ψ} (d)$ vanishes if $f$ does not divide the conductor $f_{χ}$ of $χ$ , and if $f ∣ f_{χ}$ , then

e_{χ} η_{ψ} (d) = e_{χ} \sum_{\begin{array}{c} b = 1 \\ (b, m) = 1 \end{array}}^{m} ψ (\frac{b}{f}) \bar{χ} (b) = e_{χ} \frac{φ (m)}{φ (f)} (\prod_{p ∣ f} (1 - \bar{χ} (p))) ψ (\bar{χ}) .

Noting that $e_{χ} ω_{ψ} = e_{χ} ψ (\bar{χ})$ , that $e_{χ} λ_{p} = e_{χ} (1 - \bar{χ} (p))$ , and that

e_{χ} N_{G_{f}} = {\begin{matrix} e_{χ} \frac{φ (m)}{φ (f)} & if f_{χ} ∣ f \\ 0 & otherwise, \end{matrix}

we conclude that

e_{χ} η_{ψ} (d) = e_{χ} ω_{ψ} u_{f} .

This holds for all $χ \neq 1$ , and we also ahve that $e_{1} ω_{ψ} = 0$ , so we obtain $(1 - e_{1}) η_{ψ} (d) = ω_{ψ} u_{f}$ . In that this holds for all $d$ , the result follows. □

Lemma 4.7.12. §

Let $ψ$ be a punctured $ℂ$ -valued distribution on $ℚ ∕ ℤ$ . In the notation of Proposition 4.7.11, if $ψ (χ) = 0$ for all nontrivial $χ \in \hat{G}$ with $χ (- 1) = \mp 1$ , then

(e^{\pm} U_{0} : (1 - e_{1}) T_{ψ}) = | \prod_{\begin{array}{c} χ \in \hat{G} - {1} \\ χ (- 1) = \pm 1 \end{array}} ψ (χ) | .

Proof.

By our condition on $χ$ , the element $ω_{ψ}$ of Proposition 4.7.11 is

ω_{ψ} = \sum_{\begin{array}{c} χ \in \hat{G} - {1} \\ χ (- 1) = \pm 1 \end{array}} ψ (χ) e_{χ} .

Then $ω_{ψ} \in (1 - e_{1}) e^{\pm} ℂ [G]$ by assumption on $ψ$ , and Proposition 4.7.11 implies that

(1 - e_{1}) T_{ψ} = ω_{ψ} U = (1 - e_{1}) e^{\pm} ω_{ψ} U_{0} .

Note that $e_{1} λ_{p} = 0$ for any prime $p$ dividing $m$ , so $e_{1} u_{f} = 0$ if $f$ is a positive divisor of $m$ other than $1$ . Since the $u_{f}$ generate $U$ as a $ℤ [G]$ -module and $u_{1} = N_{G}$ , we therefore have $U = U_{0} + N_{G} ℤ$ . It follows that $(1 - e_{1}) U = U_{0}$ . Multiplication by $ω_{ψ}$ determines an $ℂ$ -linear endomorphism of $(1 - e_{1}) e^{\pm} ℂ [G]$ that takes $e^{\pm} U_{0}$ onto $(1 - e_{1}) T_{ψ}$ . The idempotent $e_{χ}$ for nontrivial $χ \in \hat{G}$ with $χ (- 1) = \pm 1$ is an eigenvector of this endomorphism with eigenvalue $ψ (χ)$ . The determinant is of course the product of these eigenvalues. The result then follows by Lemma 1.2.9. □

Remark 4.7.13. §

For any $ℤ [G]$ -module $A$ that is free over $ℤ$ , we have $A_{0} = A \cap (1 - e_{1}) A$ , since $e_{1} A_{0} = 0$ and the kernel of $N_{G}$ is the image of $e_{1}$ on $A \otimes_{ℤ} ℚ$ .

Example 4.7.14. §

The $ℝ$ -vector space $V$ spanned by the elements of $G$ has $V_{0}$ equal to the elements with coefficients summing to $0$ . For $S$ the set of primes above $m$ in $F$ , we have $T = T_{ψ_{cyc}} = ρ (C_{F, S})$ is contained in $V$ , and note that $T_{0} = ρ (C_{F})$ by the product formula.

Lemma 4.7.15. §

For $ψ = ψ_{cyc}$ and $T = T_{ψ_{cyc}}$ , we have

[(1 - e_{1}) T : T_{0}] = 2^{- g} φ (m) .

Proof.

Note that

(1 - e_{1}) T ∕ T_{0} ≅ ((1 - e_{1}) T + T) ∕ T ≅ (e_{1} T + T) ∕ T ≅ e_{1} T ∕ T^{G} .

We have

e_{1} T = \frac{1}{φ (m)} N_{G} ρ (C_{F, S}) = \frac{1}{φ (m)} ρ (C_{F, S}^{N_{G}}) .

Note that $| {(1 - ζ_{f})}^{N_{G}} | = 1$ if $f$ is not a prime power, and ${(1 - ζ_{p^{k}})}^{N_{G}} = p^{φ (m) ∕ φ (p^{k})}$ . It follows that

e_{1} T = \frac{1}{2} \sum_{p ∣ m} \frac{1}{φ (p^{k_{p}})} \log p \cdot N_{G} ℤ,

where $k_{p} \geq 1$ is the additive $p$ -adic valuation of $m$ .

Next, note that $α \in C_{F, S}$ satisfies $j (α) \in T^{G}$ if and only if $j (α^{σ - 1}) = 0$ for all $σ \in G$ , which is equivalent to $α^{σ - 1} \in μ (F)$ , which is in turn equivalent to $α^{1 + τ} \in ℚ^{\times}$ , with $τ$ complex conjugation. Let

P = {α \in C_{F, S} ∣ α^{1 + j} \in ℚ^{\times}},

and note that $T^{G} = ρ (P) = \frac{1}{2} ρ (P^{1 + τ})$ . For an odd prime $p$ dividing $m$ , set

α_{p} = \prod_{a = 1}^{(p - 1) ∕ 2} (1 - ζ_{p}^{a}),

and set $α_{2} = 1 - ζ_{4}$ if $m$ is even. Then each $α_{p}$ for $p$ dividing $m$ lies in $P$ , so $P^{1 + τ}$ contains the group $H$ generated by all primes dividing $m$ . Since $P^{1 + τ}$ is a subgroup of the positive rationals, the quotient $P^{1 + τ} ∕ H$ is torsion-free, and on the other hand ${(P^{1 + j})}^{φ (m)} = {(P^{1 + j})}^{N_{G}} \subseteq H$ , which forces $P^{1 + τ} = H$ . Thus

T^{G} = \frac{1}{4} \sum_{p ∣ m} \log p \cdot N_{G} ℤ .

It follows that

[(1 - e_{1}) T : T_{0}] = [e_{1} T : T^{G}] = \prod_{p ∣ m} \frac{φ (p^{k})}{2} = \frac{φ (m)}{2^{g}} .

□

Lemma 4.7.16. §

Let $ρ : E_{F, S} \to V^{+}$ denote the $ℤ [G]$ -module homomorphism

ρ (α) = - \frac{1}{2} \sum_{σ \in G} \log | σ (α) | σ^{- 1} .

Then

(e^{+} I_{G} : ρ (E_{F})) = \frac{R_{F}^{+}}{Q_{F}} .

Proof.

Let $X = (1 - e_{1}) e^{+} V$ , in which $ρ (E_{F})$ forms a lattice of full rank $r = \frac{φ (m)}{2} - 1$ . The lattice $e^{+} I_{G}$ has a basis $e^{+} (1 - σ_{a}^{- 1})$ for $1 < a < \frac{m}{2}$ with $(a, m) = 1$ . Fix a complex embedding of $F$ , hence an absolute value. For an independent system of units $α_{1}, \dots, α_{r} \in E_{F}^{+}$ generating $E_{F} ∕ μ_{F}$ , we have

ρ (α_{i}) = - \sum_{\begin{array}{c} a = 1 \\ (a, m) = 1 \end{array}}^{⌈ \frac{m}{2} ⌉ - 1} \log | σ_{a} (α_{i}) | σ_{a}^{- 1} = \sum_{\begin{array}{c} a = 2 \\ (a, m) = 1 \end{array}}^{⌈ \frac{m}{2} ⌉ - 1} \log | σ_{a} (α_{i}) | e^{+} (1 - σ_{a}^{- 1}) .

Since the matrix with entries $\log | η_{i}^{σ_{a}} |$ has determinant $2^{- r} R_{F}$ by definition and $R_{F} = \frac{2^{r}}{Q_{F}} R_{F}^{+}$ , we are done. □

Lemma 4.7.17. §

We have

[e^{-} ℤ [G] 𝜃_{F} : I_{F}^{-}] = w_{F} .

Proof.

Let $Θ_{F} = ℤ [G] 𝜃_{F}$ for brevity. Since $(σ_{a} - a) 𝜃_{F} \in I_{F}$ for all $a \in ℤ$ , we have that

Θ_{F} = I_{F} + 𝜃_{F} ℤ,

and therefore $Θ_{F} ∕ I_{F} ≅ℤ ∕ 𝑚ℤ$ as $m$ is minimal with $m 𝜃_{F}$ integral. From the fact that $⟨ α ⟩ + ⟩ 1 - α ⟩ = 1$ for $α \notin ℤ$ , one see that $e^{+} 𝜃_{F} = \frac{1}{2} N_{G} ℤ$ . Since $(σ_{2} - 2) 𝜃_{F} \in I_{F}$ and $e^{+} (σ_{2} - 2) = - \frac{1}{2} N_{G}$ , we then have that $e^{+} Θ_{F} = e^{+} I_{F}$ and therefore ${(ℤ [G] 𝜃)}_{F}^{+} = I_{F}^{+}$ , which in turn implies that

Θ_{F}^{-} ∕ I_{F}^{-} ≅ Θ_{F} ∕ I_{F} ≅ℤ ∕ 𝑚ℤ .

If $m$ is even, then $σ_{m ∕ 2} 𝜃_{F} = \frac{1}{2} N_{G} = e^{+} 𝜃_{F}$ . Therefore, we have $e^{+} Θ_{F} \subseteq Θ_{F}$ , and in turn this implies that $e^{-} Θ_{F} \subseteq Θ_{F}$ . In other words, we have $[e^{-} Θ_{F} : Θ_{F}^{-}] = 1$ .

If $m$ is odd, then $e^{-} σ_{a} 𝜃_{F} = σ_{a} 𝜃_{F} - \frac{1}{2} N_{G}$ , so

e^{-} Θ_{F} + Θ_{F} = \frac{1}{2} N_{G} ℤ + 𝜃_{F},

and therefore

e^{-} Θ_{F} ∕ Θ_{F}^{-} ≅ \frac{1}{2} N_{G} ℤ ∕ (Θ_{F} \cap \frac{1}{2} N_{G} ℤ) .

Note that $N_{G} = (1 + j) 𝜃_{F} \in Θ_{F}$ but $\frac{1}{2} N_{G} \notin Θ_{F}$ since $m Θ_{F} \subset ℤ [G]$ and $m$ is odd. Therefore, we have $[e^{-} Θ_{F} : Θ_{F}^{-}] = 2$ .

For arbitrary $m$ , we conclude that

[e^{-} ℤ [G] 𝜃_{F} : I_{F}^{-}] = [e^{-} Θ_{F} : Θ_{F}^{-}] [Θ_{F}^{-} : I_{F}^{-}] = \frac{w_{F}}{m} \cdot m = w_{F} .

□

We are now ready to prove Sinnott’s theorem.

Proof

Proof of Theorem 4.7.1. First consider $ψ = ψ_{cyc}$ , and set $T = T_{ψ}$ . Since $T_{0} = ρ (C_{F})$ , we may write our index as a product

[E_{F}^{+} : C_{F}^{+}] = [ρ (E_{F}) : ρ (C_{F})] = (ρ (E_{F}) : e^{+} I_{G}) (e^{+} I_{G} : e^{+} U_{0}) (e^{+} U_{0} : (1 - e_{1}) T) ((1 - e_{1}) T : T_{0}) .

The latter four relative indices are computed by Lemma 4.7.16, Lemma 4.7.12, Corollary 4.7.8, and Lemma 4.7.15, respectively. Plugging in, we obtain

[E_{F}^{+} : C_{F}^{+}] = \frac{Q_{F}}{R_{F}^{+}} \cdot \frac{{(2^{2^{g - 1}})}^{1 ∕ 2}}{φ (m)} \cdot | \prod_{\begin{array}{c} χ \in \hat{G} - {1} \\ χ even \end{array}} ψ_{cyc} (χ) | \cdot \frac{φ (m)}{2^{g}} = 2^{a} \frac{1}{R_{F}^{+}} \prod_{\begin{array}{c} χ \in \hat{G} - {1} \\ χ even \end{array}} ψ_{cyc} (χ) = 2^{a} h_{F}^{+},

where the last equality follows from Theorem 4.2.20.

Next, consider $ψ = ψ^{(1)}$ , the first Bernoulli distribution, which by definition has $T_{ψ^{(1)}} = e^{-} ℤ [G] 𝜃_{F}$ . We write the index in question as a product as follows:

[ℤ {[G]}^{-} : I_{F}^{-}] = (ℤ {[G]}^{-} : e^{-} ℤ [G]) (e^{-} ℤ [G] : e^{-} U) (e^{-} U : e^{-} ℤ [G] 𝜃_{F}) (e^{-} ℤ [G] 𝜃_{F} : I_{F}^{-}) .

The latter four relative indices are computed by Lemmas 4.7.3, Proposition 4.7.7, 4.7.12, and 4.7.17, respectively. Noting also that $2^{b} Q_{F} = 2^{2^{g - 2}}$ if $g \geq 2$ and $2^{b} Q_{F} = 1$ if $g = 1$ , we obtain

[ℤ {[G]}^{-} : I_{F}^{-}] = 2^{- φ (m) ∕ 2} \cdot 2^{b} Q_{F} \cdot | \prod_{\begin{array}{c} χ \in \hat{G} \\ χ odd \end{array}} ψ^{(1)} (χ) | \cdot w_{F} = 2^{b} \cdot 2 [E_{F} : E_{F}^{+}] \prod_{\begin{array}{c} χ \in \hat{G} \\ χ odd \end{array}} \frac{- B_{1, χ}}{2} = 2^{b} h_{F}^{-}

where the second equality uses that $Q_{F} w_{F} = 2 [E_{F} : E_{F}^{+}]$ by Lemma 4.2.16, and the final equality follows from Theorem 4.2.20. □

Chapter 4 Cyclotomic fields

4.1. Dirichlet L-functions

Definition 4.1.1. §

Example 4.1.2. §

Definition 4.1.3. §

Definition 4.1.4. §

Definition 4.1.5. §

Notation 4.1.6. §

Remark 4.1.7. §

Definition 4.1.8. §

Example 4.1.9. §

Proposition 4.1.10. §

Theorem 4.1.11. §

Definition 4.1.12. §

Definition 4.1.13. §

Remark 4.1.14. §

Definition 4.1.15. §

Definition 4.1.16. §

Lemma 4.1.17. §

Lemma 4.1.18. §

Definition 4.1.19. §

Theorem 4.1.20. §

Proposition 4.1.21. §

Corollary 4.1.22. §

4.2. Bernoulli numbers

Definition 4.2.1. §

Example 4.2.2. §

Remark 4.2.3. §

Definition 4.2.4. §

Remark 4.2.5. §

Remark 4.2.6. §

Remark 4.2.7. §

Definition 4.2.8. §

Example 4.2.9. §

Lemma 4.2.10. §

Corollary 4.2.11. §

Definition 4.2.12. §

Proposition 4.2.13. §

Theorem 4.2.14. §

Corollary 4.2.15. §

Lemma 4.2.16. §

Lemma 4.2.17. §

Notation 4.2.18. §

Lemma 4.2.19. §

Theorem 4.2.20. §

4.3. Cyclotomic units

Definition 4.3.1. §

Remark 4.3.2. §

Theorem 4.3.3 (Hasse). §

Lemma 4.3.4. §

Lemma 4.3.5. §

4.4. Reflection theorems

Definition 4.4.1. §

Theorem 4.4.2 (Leopoldt’s Spiegelungsatz). §

Lemma 4.4.3. §

Corollary 4.4.4. §

Corollary 4.4.5. §

4.5. Stickelberger theory

Definition 4.5.1. §

Lemma 4.5.2. §

Definition 4.5.3. §

Lemma 4.5.4. §

Lemma 4.5.5. §

Theorem 4.5.6 (Stickelberger). §

Theorem 4.5.7 (Herbrand). §

Theorem 4.5.8 (Iwasawa). §

4.6. Distributions

Definition 4.6.1. §

Remark 4.6.2. §

Example 4.6.3. §

Definition 4.6.4. §

Remark 4.6.5. §

Remark 4.6.6. §

Example 4.6.7. §

Definition 4.6.8. §

Example 4.6.9. §

Notation 4.6.10. §

Remark 4.6.11. §

Example 4.6.12. §

Chapter 4
Cyclotomic fields

4.1. Dirichlet $L$ -functions