Iwasawa Theory

Chapter 5
Kubota-Leopoldt $p$ -adic $L$ -functions

5.1. $p$ -adic measures

In this section, we study $ℂ_{p}$ -valued distributions.

Definition 5.1.1. §

We say that a $ℂ_{p}$ -valued distribution ${ψ_{i}}_{i \in I}$ on an inverse system of finite sets $X_{i}$ is bounded if there exists a constant $B \in ℝ_{\geq 0}$ such that $| ψ_{i} (x) | \leq B$ for all $x \in X_{i}$ for all $i \in I$ , where $| \cdot |$ is the unique extension of the $p$ -adic valuation on $ℚ_{p}$ to $ℂ_{p}$ .

Remark 5.1.2. §

To say that ${ψ_{i}}$ is bounded is the same as saying the corresponding functional $ψ$ on step functions on the profinite space $X = {lim_{\leftarrow}}_{i} X_{i}$ satisfies

| ψ (χ) | \leq B ∥ χ ∥,

where $∥ χ ∥ = \sup_{x \in X} | χ (x) |$ (which is actually a maximum, as $X$ is compact).

Notation 5.1.3. §

For a topological subring $𝒪$ of $ℂ_{p}$ , let $C (X, 𝒪)$ denote the space of continuous functions from $X$ to $𝒪$ , endowed with the compact-open topology.

Remark 5.1.4. §

The set $Step (X, ℂ_{p})$ is dense in $C (X, ℂ_{p})$ .

Definition 5.1.5. §

For a topological subring $𝒪$ of $ℂ_{p}$ and a profinite space $X$ , an $𝒪$ -valued measure on $X$ is a bounded linear functional

μ : C (X, 𝒪) \to 𝒪 .

We write

\int_{X} 𝑔𝑑𝜇

for the value $μ (g)$ .

Remark 5.1.6. §

Measures on $X$ are in one-to-one correspondence with bounded distributions, since $Step (X, ℂ_{p})$ is dense in $C (X, ℂ_{p})$ .

Example 5.1.7. §

The $δ$ -distribution at $x \in X$ gives rise to the Dirac measure

\int_{X} 𝑔𝑑 δ_{x} = g (x) .

We briefly discuss measures on $ℤ_{p}$ .

Remark 5.1.8. §

Let $g : ℤ_{p} \to ℂ_{p}$ be a continuous function, and let $μ$ be a $ℂ_{p}$ -valued measure on $ℤ_{p}$ with corresponding distribution ${μ_{n}}$ . Then

\int_{ℤ_{p}} 𝑔𝑑𝜇 = \lim_{n \to \infty} \sum_{a = 0}^{p^{n} - 1} g_{n} (a) μ_{n} (a) .

Let $𝒪$ denote the valuation ring of a finite extension of $ℚ_{p}$ .

Proposition 5.1.9. §

There is a canonical bijection between $𝒪$ -valued measures $μ$ on $ℤ_{p}$ and elements $λ$ of $𝒪 ⟦ ℤ_{p} ⟧$ , seen explicitly as follows. Write $λ \in 𝒪 ⟦ ℤ_{p} ⟧$ as $λ = {(λ_{n})}_{n}$ with $λ_{n} \in 𝒪 [ℤ ∕ p^{n} ℤ]$ . Then $μ$ is the measure associated to the distribution ${μ_{n}}_{n \geq 1}$ with $μ_{n} : ℤ ∕ p^{n} ℤ \to 𝒪$ corresponds to $λ$ if and only if

λ_{n} = \sum_{a = 0}^{p^{n} - 1} μ_{n} (a) {[a]}_{n}

where ${[a]}_{n} \in 𝒪 [ℤ ∕ p^{n} ℤ]$ is the group element attached to $a$ .

Proof.

Clearly, the data of the $λ_{n}$ determine the $μ_{n}$ and conversely. One need only see that $f$ is well-defined if and only if the $μ_{n}$ satisfy the distribution relations. But, from the definitions, the element $λ_{n + 1}$ maps to $λ_{n}$ if and only if

μ_{n} (a) = \sum_{b = 0}^{p^{n} - 1} μ_{n + 1} (a + p^{n} b),

as required. □

Remark 5.1.10. §

We have $μ (χ_{a + p^{n} ℤ_{p}}) = μ_{n} (a)$ , so knowing each $μ_{n}$ determines $μ$ on step functions explicitly.

Remark 5.1.11. §

Since $𝒪 ⟦ T ⟧ ≅ 𝒪 ⟦ ℤ_{p} ⟧$ via the continuous $𝒪$ -linear isomorphism taking $T + 1$ to the group element of $1$ , we have a canonical bijection between $𝒪$ -valued measures on $ℤ_{p}$ and power series in $𝒪 ⟦ T ⟧$ .

Corollary 5.1.12. §

The power series $f$ attached to an $𝒪$ -valued measure $μ$ on $ℤ_{p}$ is given by

f (T) = \sum_{i = 0}^{\infty} (\int_{ℤ_{p}} (\binom{x}{i}) 𝑑𝜇 (x)) T^{i} \in 𝒪 ⟦ T ⟧ .

Proof.

Let $f_{n} \in 𝒪 ⟦ T ⟧ ∕ (ω_{n})$ be the image of $f$ . By Proposition 5.1.9, the measure $μ$ attached to $f$ given by the distribution ${μ_{n}}_{n \geq 1}$ is related to $f_{n}$ through the formula

f_{n} (T) = \sum_{a = 0}^{p^{n} - 1} μ_{n} (a) {(T + 1)}^{a} = \sum_{i = 0}^{\infty} \sum_{a = 0}^{p^{n} - 1} (\binom{a}{i}) μ_{n} (a) T^{i},

so the inverse limit $f$ of the $f_{n}$ satisfies the desired equation. □

Corollary 5.1.13. §

Let $f$ be the power series attached to an $𝒪$ -valued measure $μ$ on $ℤ_{p}$ . If $t \in 𝔪$ , where $𝔪$ is the maximal ideal of $𝒪$ , the value $f (t)$ may be calculated by

f (t) = \int_{ℤ_{p}} {(1 + t)}^{x} 𝑑𝜇 (x),

where $μ$ is the measure corresponding to $f$ .

Theorem 5.1.14 (Mahler). §

We have

C (ℤ_{p}, 𝒪) = {\sum_{i = 0}^{\infty} c_{i} (\binom{x}{i}) ∣ c_{i} \in 𝒪, c_{i} \to 0},

and the representation of $g \in C (ℤ_{p}, 𝒪)$ as a sum as in the latter set is unique.

Proof.

Suppose that there is a sequence ${(c_{i})}_{i \geq 1}$ of elements of $𝒪$ that converges to $0$ . Since each $| (\binom{x}{i}) |$ is bounded by $1$ on $ℤ_{p}$ , any $g = \sum_{i = 0}^{\infty} c_{i} (\binom{x}{i})$ with $c_{i} \to 0$ is the uniform limit of its continuous partial sums, hence continuous.

Consider the difference operator $\nabla$ on $g \in C (ℤ_{p}, 𝒪)$ defined by $\nabla (g) (x) = g (x + 1) - g (x)$ . Then

\nabla (\binom{x}{i}) = (\binom{x + 1}{i}) - (\binom{x}{i}) = (\binom{x}{i - 1}),

so if $g$ has the form in the theorem, then $\nabla^{i} (g) (0) = c_{i}$ . In other words, the representation of $g$ as a sum is unique if it exists.

We now show existence. For this, it suffices to consider $ℤ_{p}$ -valued functions by choice of a basis and projection. We have a $ℤ_{p}$ -linear map from the set of sequences in $ℤ_{p}$ that converge to $0$ to $C (ℤ_{p}, ℤ_{p})$ given by ${(c_{i})}_{i \geq 0} \mapsto \sum_{i = 0}^{\infty} c_{i} (\binom{x}{i})$ . It suffices to show that this map is surjective. This can be derived via recursion from the claim that the set of eventually zero sequences in $𝔽_{p}$ surjects onto $C (ℤ_{p}, 𝔽_{p})$ via the reduction modulo $p$ of this map.

Note that

C (ℤ_{p}, 𝔽_{p}) = \underset{n}{lim_{\to}} Maps (ℤ ∕ p^{n} ℤ, 𝔽_{p}),

For $0 \leq i \leq p^{n} - 1$ , the map $x \mapsto (\binom{x}{i}) mod p$ lies in $Maps (ℤ ∕ p^{n} ℤ, 𝔽_{p})$ , since

{(1 + T)}^{x + p^{n}} \equiv {(1 + T)}^{x} (1 + T^{p^{n}}) \equiv {(1 + T)}^{x} mod (p, T^{p^{n}}) ℤ_{p} ⟦ T ⟧ .

Thus, our map restricts to a map

{{(c_{i})}_{0 \leq i < p^{n}} ∣ c_{i} \in 𝔽_{p}} \to Maps (ℤ ∕ p^{n} ℤ, 𝔽_{p})

that is injective by our earlier uniqueness argument using $\nabla$ and surjective by equality of $𝔽_{p}$ -dimensions. This proves the desired surjectivity. □

The following is a matter of switching the order of a sum and an integral.

Corollary 5.1.15. §

For $g = \sum_{i = 0}^{\infty} c_{i} (\binom{x}{i}) \in C (ℤ_{p}, 𝒪)$ and $μ$ the measure attached to

f = \sum_{i = 0}^{\infty} a_{i} T^{i} \in 𝒪 ⟦ T ⟧,

we have

\int_{ℤ_{p}} 𝑔𝑑𝜇 = \sum_{i = 0}^{\infty} a_{i} c_{i} .

Typically, we are more interested in measures on $ℤ_{p}^{\times}$ , or the units in a slightly larger ring. Let us recall that $1 + q ℤ_{p}$ , where $q = p$ if $p$ is odd and $q = 4$ for $p = 2$ , is isomorphic to $ℤ_{p}$ via the map that takes $u^{a}$ to $a$ for any $a \in ℤ_{p}$ , where $u$ is a fixed topological generator of $1 + q ℤ_{p}$ , such as $1 + q$ . In this way, measures on $1 + q ℤ_{p}$ are made to correspond to measures on $ℤ_{p}$ .

Definition 5.1.16. §

For an $𝒪$ -valued measure $ν$ on $1 + q ℤ_{p}$ , let $μ$ be the $𝒪$ -valued measure on $ℤ_{p}$ defined by

\int_{ℤ_{p}} g (u^{x}) 𝑑𝜇 (x) = \int_{1 + q ℤ_{p}} 𝑔𝑑𝜈 .

The power series in $𝒪 ⟦ T ⟧$ attached to $ν$ is the power series corresponding to $μ$ by Proposition 5.1.9.

Lemma 5.1.17. §

The power series $f$ attached to an $𝒪$ -valued measure $ν$ on $1 + q ℤ_{p}$ satisfies

f (u^{s} - 1) = \int_{1 + q ℤ_{p}} x^{s} 𝑑𝜈 (x)

for $s \in ℤ_{p}$ , and $f$ is uniquely determined by this formula.

Proof.

Let $μ$ be the measure on $ℤ_{p}$ corresponding to $ν$ and $f$ . Set $t = u^{s} - 1$ for some $s \in ℤ_{p}$ . Then Corollary 5.1.13 tells us that

f (u^{s} - 1) = \int_{ℤ_{p}} u^{𝑠𝑥} 𝑑𝜇 (x) = \int_{1 + q ℤ_{p}} x^{s} 𝑑𝜈 (x) .

We leave the last simple statement to the reader. □

Remark 5.1.18. §

We can also attach a measure on $ℤ_{p}$ to a measure on $ℤ_{p}^{\times}$ , by extension by zero. Similarly, we can restrict measures on $ℤ_{p}$ to the latter multiplicative subgroups.

5.2. $p$ -adic $L$ -functions

Definition 5.2.1. §

Let $p$ be a prime number, and let $m \geq 1$ be prime to $p$ . Set

ℤ_{p, m} = \underset{n}{lim_{\leftarrow}} (ℤ ∕ m p^{n} ℤ) .

Then

ℤ_{p, m} \to_{}^{\sim} ℤ_{p} \times ℤ ∕ 𝑚ℤ,

and we let $c_{p}$ denote the first coordinate of the image of $c \in ℤ_{p, m}$ .

Note that

ℤ_{p, m}^{\times} = \underset{n}{lim_{\leftarrow}} {(ℤ ∕ m p^{n} ℤ)}^{\times} ≅ ℤ_{p}^{\times} \times {(ℤ ∕ 𝑚ℤ)}^{\times},

and, setting $q = p$ for $p$ odd and $q = 4$ for $p = 2$ , we also have

ℤ_{p, m}^{\times} \to_{}^{\sim} (1 + q ℤ_{p}) \times {(ℤ ∕ 𝑞𝑚ℤ)}^{\times} .

Definition 5.2.2. §

For $c \in ℤ_{p, m}^{\times}$ , we let $c_{p}$ denote its image in $ℤ_{p}^{\times}$ and ${⟨ c ⟩}_{p}$ denote its image in $1 + q ℤ_{p}$ .

Note also that $ℤ_{p, m}^{\times}$ is canonically isomorphic to the Galois group of $ℚ (μ_{m p^{\infty}}) ∕ ℚ$ . We will typically be interested in measures on $ℤ_{p, m}^{\times}$ .

Let us set $Δ = {(ℤ ∕ 𝑞𝑚ℤ)}^{\times}$ . We have

ℤ_{p} ⟦ ℤ_{p, m}^{\times} ⟧ ≅ ℤ_{p} [Δ] ⟦ 1 + q ℤ_{p} ⟧ ≅ ℤ_{p} [Δ] ⟦ T ⟧,

the latter isomorphism taking the group element $u$ to $T + 1$ .

Let $𝒪$ be the valuation ring of a finite extension of $ℚ_{p}$ . By the same discussion as before, replacing $𝒪$ by the group ring $𝒪 [Δ]$ , we have the following.

Lemma 5.2.3. §

There is a canonical bijection between $𝒪$ -valued measures on $ℤ_{p, m}^{\times}$ and elements of $𝒪 ⟦ T ⟧ [Δ]$ .

Explicitly, the power series $f \in 𝒪 ⟦ T ⟧ [Δ]$ attached to an $𝒪$ -valued measure $ν$ on $ℤ_{p, m}^{\times}$ satisfies

f (u^{s} - 1) = \sum_{σ \in Δ} \int_{1 + q ℤ_{p}} x^{s} 𝑑𝜈 (𝜎𝑥) \cdot σ

If $ν$ arises from a distribution $ψ = (ψ_{n})$ on the groups ${(ℤ ∕ m p^{n} ℤ)}^{\times}$ , then $f$ is then given by the compatible system of elements

\sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{p^{n} m} ψ_{n} (a) {[a]}_{n} \in 𝒪 [{(ℤ ∕ p^{n} 𝑚ℤ)}^{\times}],

where ${[a]}_{n}$ denotes the group element of $a$ .

The Bernoulli distribution will be the key to our definition of $p$ -adic $L$ -functions, but it is not necessarily integral. Therefore, we introduce the following modification.

Definition 5.2.4. §

Set $N = p^{n} m$ , let $c \in ℤ_{p, m}^{\times}$ , and take $k \geq 1$ . For $x \in ℤ ∕ 𝑁ℤ$ , we view $\frac{x}{N}$ as an element of $\frac{1}{N} ℤ ∕ ℤ$ and define

E_{n}^{(k)} (x) = \frac{1}{k} N^{k - 1} B_{k} (⟨ \frac{x}{N} ⟩)

and

E_{n, c}^{(k)} (x) = E_{n}^{(k)} (x) - c_{p}^{k} E_{n}^{(k)} (c^{- 1} x) .

Note that $E_{n}^{(k)} = \frac{1}{k} ψ_{k}^{(N)}$ , so the $E_{n}^{(k)}$ and $E_{n, c}^{(k)}$ form distributions on $ℚ ∕ ℤ$ .

Proposition 5.2.5. §

For $N = p^{n} m$ with $n \geq 1$ and $k \geq 1$ , we have $E_{n, c}^{(k)} (x) \in ℤ_{p}$ and

E_{n, c}^{(k)} (x) \equiv x^{k - 1} E_{n, c}^{(1)} (x) mod p^{n} ℤ_{p}

for all $x \in ℤ ∕ 𝑁ℤ$ .

Proof.

We have

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

from which we see that the $k$ th Bernoulli polynomial has the form

B_{k} (X) = X^{k} - \frac{k}{2} X^{k - 1} + 𝑘𝑓 (X)

with $f \in ℚ [X]$ of degree $k - 2$ (or $0$ if $k = 1$ ), the leading term of $f (X)$ being $\frac{k - 1}{6} X^{k - 2}$ . Let $e_{k} \geq 0$ be minimal such that $p^{e_{k}} f (X) \in ℤ_{p} ⟦ X ⟧$

Let $a \in ℤ$ with $0 \leq a < N$ lift $x \in ℤ ∕ 𝑁ℤ$ , and let $b \in ℤ$ with $0 \leq b < N$ and $y \in ℤ_{p}$ be such that $\frac{c_{p}^{- 1} a}{N} = \frac{b}{N} + y$ . We then have

b^{j} \equiv c_{p}^{- j} a^{j} - 𝑗𝑁 c_{p}^{- j + 1} a^{j - 1} y mod N^{2}

for $j \geq 1$ , which yields

E_{n, c}^{(k)} (x) \equiv \frac{1}{k} (\frac{1}{N} a^{k} - \frac{k}{2} a^{k - 1} - c_{p}^{k} (\frac{1}{N} b^{k} - \frac{k}{2} b^{k - 1})) \equiv a^{k - 1} (c_{p} y + \frac{1}{2} (c_{p} - 1)) mod p^{n - e_{k}} ℤ_{p} .

Since this holds for all $k$ , we have in particular that $E_{n, c}^{(1)} (x)$ is in $ℤ_{p}$ for all $n$ , noting that $c_{p} \equiv 1 mod 2 ℤ_{p}$ , and that

E_{n, c}^{(k)} (x) \equiv x^{k - 1} E_{n, c}^{(1)} (x) mod p^{n - e_{k}} ℤ_{p}

is integral for sufficiently large $n$ as well. By the distribution relation for the $E_{n, c}^{(k)}$ , this integrality then holds for all $n$ . If we choose $n \geq e_{k}^{'}$ , where $e_{k}^{'}$ is minimal such that $p^{e_{k}^{'}} (f (X) - \frac{k - 1}{6} X^{k - 2}) \in ℤ_{p} [X]$ , then we can refine the above to to

E_{n, c}^{(k)} (x) \equiv x^{k - 1} E_{n, c}^{(1)} (x) + N x^{k - 2} \frac{k - 1}{6} (1 - c_{p}^{2}) mod p^{n} ℤ_{p}

Since $c_{p}^{2} \equiv 1 mod 6 ℤ_{p}$ , this reduces to

E_{n, c}^{(k)} (x) \equiv x^{k - 1} E_{n, c}^{(1)} (x) mod p^{n} ℤ_{p},

and the congruence then follows for arbitrary $n$ by the distribution relation. □

Remark 5.2.6. §

Together the $E_{n, c}^{(k)}$ form a $ℤ_{p}$ -valued measure $E_{c}^{(k)}$ on $ℤ_{p, m}$ , hence on $ℤ_{p, m}^{\times}$ as well by restriction. We can integrate the resulting measure against functions on $ℤ_{p, m}^{\times}$ that arise as limits of Dirichlet characters of conductor dividing $p^{n} m$ for some $n$ .

Definition 5.2.7. §

We let $E_{c}^{(k)}$ denote the measure defined by the $E_{n, c}^{(k)}$ .

Remark 5.2.8. §

Given $g \in C (ℤ_{p, m}^{\times}, 𝒪)$ , we have

\int_{ℤ_{p, m}^{\times}} g (x) d E_{c}^{(k)} (x) = \int_{ℤ_{p, m}^{\times}} g (x) x_{p}^{k - 1} d E_{c}^{(1)} (x)

for every $k \geq 1$ .

Remark 5.2.9. §

When $χ : ℤ_{p, m} \to 𝒪^{\times}$ is a continuous multiplicative function, we have

\int_{ℤ_{p, m}} χ (x) d E_{c}^{(k)} (x) = (1 - χ (c) c_{p}^{k}) \frac{B_{k, χ}}{k} .

Note that if $χ$ has finite order, so is the map attached to a primitive Dirichlet character of conductor dividing $m p^{n}$ for some $n$ , then $χ E_{c}^{(k)}$ defines an $𝒪$ -valued measure on $ℤ_{p, m}$ , with volume given by the above formula. Here, the $χ E_{n, c}^{(k)}$ are really only defined for $p^{n} m$ a multiple of the conductor of $χ$ , but the $i$ th terms of the distribution for $i$ less than the minimal such $n$ can be defined by the distribution relations.

Definition 5.2.10. §

Let $ν$ be an $𝒪$ -valued measure on $ℤ_{p, m}^{\times}$ . We define its $p$ -adic Mellin transform to be the $𝒪$ -valued function $M_{p} (ν)$ on $ℤ_{p}$ given by

M_{p} (ν) (s) = \int_{ℤ_{p, m}^{\times}} {⟨ x ⟩}_{p}^{s} x_{p}^{- 1} 𝑑𝜈 (x) .

Remark 5.2.11. §

When $p$ is odd, $x_{p} = {⟨ x ⟩}_{p} ω (x)$ for any $x \in ℤ_{p, m}^{\times}$ , where $ω$ is the Teichmüller character, which factors through ${(ℤ ∕ 𝑝ℤ)}^{\times}$ . For $p = 2$ , we simply define $ω : ℤ_{2, m}^{\times} \to μ_{2} (ℤ_{2})$ by the above formula.

Remark 5.2.12. §

If $ν$ is an $𝒪$ -valued measure on $ℤ_{p, m}^{\times}$ , then so is $𝜓𝜈$ for any Dirichlet character $ψ$ of conductor dividing $p^{n} m$ for some $n$ . In particular, we have

M_{p} (ν) (s) = \int_{ℤ_{p, m}^{\times}} {⟨ x ⟩}^{s - 1} d (ω^{- 1} ν) .

Definition 5.2.13. §

Let $χ : ℤ_{p, m}^{\times} \to ℂ_{p}^{\times}$ be a finite-order character. We define the Kubota-Leopoldt $p$ -adic $L$ -function of $χ$ to be the $ℂ_{p}$ -valued function on $ℤ_{p}$ given by

L_{p} (χ, s) = - {(1 - χ (c) {⟨ c ⟩}_{p}^{1 - s})}^{- 1} M_{p} (χ E_{c}^{(1)}) (1 - s)

for $s \in ℤ_{p}$ and $c \in ℤ_{p, m}^{\times}$ such that $χ (c) \neq 1$ if $χ \neq 1$ .

Rewriting this, we have

\begin{matrix} \begin{aligned} - (1 - χ (c) {⟨ c ⟩}_{p}^{1 - s}) L_{p} (χ, s) & = \int_{ℤ_{p, m}^{\times}} χ (x) {⟨ x ⟩}_{p}^{1 - s} x_{p}^{- 1} d E_{c}^{(1)} (x) \\ = \int_{ℤ_{p, m}^{\times}} {⟨ x ⟩}_{p}^{- s} d (χ ω^{- 1} E_{c}^{(1)}) (x) . \end{aligned} \end{matrix}

(5.2.1)

Remark 5.2.14. §

The factor ${(1 - χ (c) {⟨ c ⟩}_{p}^{1 - s})}^{- 1}$ in the definition of $L_{p} (χ, s)$ removes the dependence of the definition of the $p$ -adic $L$ -function on the value $c$ . Note that such a factor (without the inverse) was used in defining $E_{c}^{(1)}$ in the first place.

A finite order character $χ : ℤ_{p, m}^{\times} \to ℂ^{\times}$ takes values in $\bar{ℚ}$ and may be viewed as a $p$ -adic character through a choice of embedding of $\bar{ℚ}$ in $\bar{ℚ_{p}}$ , we have the following.

Proposition 5.2.15. §

Let $χ$ be a primitive Dirichlet character of conductor $p^{n} m$ for some $n \geq 0$ , and let $χ$ also denote the resulting character $χ : ℤ_{p, m}^{\times} \to ℂ_{p}^{\times}$ , fixing a place over $p$ in $\bar{ℚ}$ . For $k \geq 1$ , we have

L_{p} (χ, 1 - k) = - (1 - χ ω^{- k} (p) p^{k - 1}) \frac{B_{k, χ ω^{- k}}}{k} = (1 - χ ω^{- k} (p) p^{k - 1}) L (χ ω^{- k}, 1 - k) .

Proof.

Set $χ_{k} = χ ω^{- k}$ . We note that

(1 - χ_{k} (c) c_{p}^{k}) L_{p} (χ, 1 - k) = \int_{ℤ_{p, m}^{\times}} χ_{k} (x) x_{p}^{k - 1} d E_{c}^{(1)} (x) = \int_{ℤ_{p, m}^{\times}} χ_{k} (x) d E_{c}^{(k)} (x),

and we split the latter integral into a difference of an integral over $ℤ_{p, m}$ by an integral over $p ℤ_{p, m}$ , given that $χ_{k}$ is trivial on elements of $ℤ_{p, m}$ not prime to $m$ . By Remark 5.2.9, the former is

\int_{ℤ_{p, m}} χ_{k} (x) d E_{c}^{(k)} (x) = (1 - χ_{k} (c) c_{p}^{k}) \frac{B_{k, χ_{k}}}{k} .

Since $E_{n, c}^{(k)} (𝑝𝑏) = E_{n - 1, c}^{(k)} (b)$ for $b \in ℤ ∕ m p^{n} ℤ$ , the latter is

\begin{array}{l} \int_{p ℤ_{p, m}} χ_{k} (x) d E_{c}^{(k)} (x) & = \sum_{a = 1}^{m p^{n - 1}} χ_{k} (𝑝𝑎) E_{n, c}^{(k)} (𝑝𝑎) \\ = χ_{k} (p) p^{k - 1} \sum_{a = 1}^{m p^{n - 1}} χ_{k} (a) E_{n - 1, c}^{(k)} (a) \\ = χ_{k} (p) p^{k - 1} \int_{ℤ_{p, m}} χ_{k} (x) d E_{c}^{(k)} (x) . \end{array}

Taking the difference of the two terms, we have the result. □

Corollary 5.2.16. §

The $p$ -adic $L$ -function of $χ$ is independent of the choice of $c$ in its definition.

Proof.

The function $L_{p} (χ, s)$ is continuous, and its values at the dense subset of $ℤ_{p}$ consisting of the nonnegative integers are independent of $c$ by Proposition 5.2.15. □

5.3. Iwasawa power series

Definition 5.3.1. §

A finite order $p$ -adic character $χ$ on $ℤ_{p, m}^{\times}$ is a of the first kind if $χ$ is trivial on $1 + q ℤ_{p}$ and of the second kind if it is trivial on $Δ$ .

In general, a finite order $p$ -adic character $χ$ on $ℤ_{p, m}^{\times}$ is a unique product $χ = χ_{t} χ_{w}$ of a $p$ -adic character $χ_{t}$ of the first kind and a $p$ -adic character $χ_{w}$ of the second kind. We use the subscripts “t” and “w” to indicate “tame” and “wild”, respectively, though the terminology is technically incorrect if $p = 2$ . If we view $χ$ as corresponding to a primitive Dirichlet character of conductor $m p^{n}$ for $n \geq 0$ , then $χ$ is of the first kind if and only if it has conductor dividing $𝑚𝑞$ , and $χ$ is of the second kind if and only if it has $p$ -power order and conductor $p^{n}$ for some $n \geq 1$ (and then necessarily at least $2$ if $p = 2$ ). If $χ$ is of the second kind, it is necessarily even.

Notation 5.3.2. §

For any Dirichlet character $χ$ , let $Λ_{χ} = 𝒪_{χ} ⟦ T ⟧$ , where $𝒪_{χ}$ is the $ℤ_{p}$ -algebra generated by the values of $χ$ , fixing a choice of a embedding $\bar{ℚ} ↪ \bar{ℚ_{p}}$ . Let $K_{χ}$ denote the quotient field of $𝒪_{χ}$ , and let $Q (Λ_{χ})$ denote the quotient field of $Λ_{χ}$ , which contains $K_{χ} ⟦ T ⟧$ .

Proposition 5.3.3. §

Let $χ$ be a primitive even Dirichlet character of conductor $m$ or $𝑚𝑞$ . There exists a unique element $F_{χ} \in Q (Λ_{χ})$ such that

F_{χ} (ξ u^{s} - 1) = L_{p} (𝜒𝜌, s)

for all $s \in ℤ_{p}$ and $ξ$ of $p$ -power order, where $ρ$ is of the second kind satisfying $ρ (u) = ξ^{- 1}$ .

Proof.

It follows from (5.2.1) that

\begin{array}{l} - (1 - 𝜒𝜌 (c) {⟨ c ⟩}_{p}^{1 - s}) L_{p} (𝜒𝜌, s) & = \sum_{σ \in Δ} (\int_{1 + q ℤ_{p}} {⟨ x ⟩}_{p}^{- s} ρ (x) d E_{c}^{(1)} (x)) χ ω^{- 1} (σ) . \end{array}

Let $a \in ℤ_{p}$ be such that ${⟨ c ⟩}_{p} = u^{a}$ , and set

h_{χ, c} (T) = χ (c) {⟨ c ⟩}_{p} {(1 + T)}^{- a} - 1 .

Then

h_{χ, c} (ξ u^{s} - 1) = χ (c) {⟨ c ⟩}_{p} ξ^{- a} u^{- 𝑠𝑎} - 1 = - (1 - 𝜒𝜌 (c) {⟨ c ⟩}_{p}^{1 - s}) .

Similarly, if we let $f_{χ, c} \in 𝒪_{χ} ⟦ T ⟧$ be such that

f_{χ, c} (u^{s} - 1) = - (1 - χ (c) {⟨ c ⟩}_{p}^{1 - s}) L_{p} (χ, s)

for all $s \in ℤ_{p}$ , then

f_{χ, c} (ξ u^{s} - 1) = - (1 - 𝜒𝜌 (c) {⟨ c ⟩}_{p}^{1 - s}) L_{p} (𝜒𝜌, s) .

Thus $F_{χ} = \frac{f_{χ, c}}{h_{χ, c}}$ has the desired property. In that the integral power series $f_{χ, c}$ satisfies $f_{χ, c} ({(1 + q)}^{s} - 1) = h_{χ, c} ({(1 + q)}^{s} - 1) L_{p} (χ, s)$ for all $s \in ℤ_{p}$ , it is unique, and therefore so is $F_{χ}$ . □

Remark 5.3.4. §

In the notation of Proposition 5.3.3, we have If $χ \neq 1$ , then we may take $c \in Δ \subset ℤ_{p, m}^{\times}$ to be such that $χ (c) \neq 1$ , so $h_{χ, c} = χ (c) - 1$ , and $(χ (c) - 1) F_{χ} \in Λ_{χ}$ . If $χ = 1$ , then we may take $c = u \in 1 + q ℤ_{p}$ , so $h_{χ, u} = u {(T + 1)}^{- 1} - 1$ .

Definition 5.3.5. §

Let $F_{n} = ℚ (μ_{m p^{n}})$ , and let $G_{n} = Gal (F_{n} ∕ ℚ)$ . For any $b \in ℤ_{p, m}^{\times}$ , set

Θ_{n}^{(k)} (b) = \sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{m p^{n}} E_{n}^{(k)} (𝑎𝑏) σ_{a}^{- 1} .

We set $Θ_{n}^{(k)} = Θ_{n}^{(k)} (1)$ and refer to $Θ_{n}^{(k)}$ as the $k$ th higher Stickelberger element for $F_{n}$ .

Since the $E_{n}^{(k)}$ form a distribution, the $Θ_{n}^{(k)}$ give a compatible system in the inverse limit. Set $G_{\infty} = Gal (F_{\infty} ∕ ℚ)$ . We have a continuous isomorphism $ℤ_{p, m}^{\times} \to_{}^{\sim} G_{\infty}$ via $a \mapsto σ_{a}$ , under which $1 + q ℤ_{p}$ is idenitifed with $Γ = Gal (F_{\infty} ∕ F)$ for $F = ℚ (μ_{𝑚𝑞})$ , and $Δ = {(ℤ ∕ 𝑚𝑞ℤ)}^{\times}$ is identified with the torsion subgroup of $G_{\infty}$ , which we also denote by $Δ$ . For locally compact $ℤ_{p}$ -algebra $R$ , we then have an identification

R ⟦ G_{\infty} ⟧ = R [Δ] ⟦ Γ ⟧ ≅𝑅 [Δ] ⟦ T ⟧

of topological rings, where $T = γ - 1$ for $γ = σ_{u}$ .

Notation 5.3.6. §

Let

Θ_{\infty}^{(k)} (b) = {(Θ_{n}^{(k)} (b))}_{n} \in ℚ_{p} [Δ] ⟦ T ⟧,

and set $Θ_{\infty}^{(k)} = Θ_{\infty}^{(k)} (1)$ .

Remark 5.3.7. §

Since

(1 - c_{p}^{k} σ_{c}^{- 1}) \sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{m p^{n}} E_{n}^{(k)} (a) σ_{a}^{- 1} = \sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{m p^{n}} E_{n, c}^{(k)} (a) σ_{a}^{- 1} \in ℤ_{p} [G_{n}],

we have $(1 - c_{p}^{k} σ_{c}^{- 1}) Θ_{\infty}^{(k)} \in ℤ_{p} [Δ] ⟦ T ⟧$ . Aside from the use of $σ_{a}^{- 1}$ in place of $σ_{a}$ , the latter is the power series corresponding to the measure given by the $E_{n, c}^{(k)}$ on $ℤ_{p, m}^{\times}$ .

Notation 5.3.8. §

For any nontrivial primitive even Dirichlet character $χ$ of conductor $m$ or $𝑚𝑞$ , let

f_{χ} = - \tilde{ω χ^{- 1}} (Θ_{\infty}^{(1)}),

where $\tilde{ω χ^{- 1}} : ℚ_{p} [Δ] ⟦ T ⟧ \to K_{χ} ⟦ T ⟧$ is the unique continuous $ℚ_{p} ⟦ T ⟧$ -linear map that restricts to $χ_{1}^{- 1} = ω χ^{- 1}$ on $Δ$ . Set

f_{1} = (1 - u {(1 + T)}^{- 1}) \tilde{ω} (Θ_{\infty}^{(1)}) .

Definition 5.3.9. §

For any primitive even Dirichlet character of conductor $m$ or $𝑚𝑞$ , the power series $f_{χ}$ is called the Iwasawa power series of $χ$ .

Remark 5.3.10. §

For any nontrivial $χ$ , the image of $f_{χ}$ in $Gal (F_{n} ∕ ℚ)$ is

- \sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{m p^{n}} (\frac{a}{m p^{n}} - \frac{1}{2}) χ ω^{- 1} (a) σ_{{⟨ a ⟩}_{p}}^{- 1} = - \frac{1}{m p^{n}} \sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{m p^{n}} 𝑎𝜒 ω^{- 1} (a) σ_{{⟨ a ⟩}_{p}}^{- 1},

where the second equality is by the nontriviality of $χ_{1}$ . Then $- f_{χ}$ becomes identified with

{(\tilde{ω χ^{- 1}} (𝜃_{F_{n}}))}_{n} \in ℚ_{p} ⟦ Γ ⟧,

where $ω χ^{- 1}$ is defined in the obvious fashion. In other words, $f_{χ}$ is the negative of the $ω χ^{- 1}$ -specialization of the inverse limit of Stickelberger elements of the fields $F_{n}$ .

For $χ$ nontrivial, the power series $f_{χ}$ agrees with $F_{χ}$ defined above.

Lemma 5.3.11. §

For any primitive even Dirichlet character $χ$ of conductor $m$ or $𝑚𝑞$ , we have $f_{χ} = F_{χ}$ . For $χ = 1$ , we have $f_{1} = h_{1} F_{1}$ , where $h_{1} = 1 - u {(1 + T)}^{- 1}$ .

Proof.

We have

\begin{array}{l} - (1 - χ (c)) f_{χ} (u^{s} - 1) & = \tilde{ω χ^{- 1}} (\sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{𝑚𝑝} \int_{1 + q ℤ_{p}} x^{- s} d E_{c}^{(1)} (x) σ_{a}^{- 1}) \\ = \int_{ℤ_{p, m}^{\times}} χ_{1} (x) {⟨ x ⟩}_{p}^{- s} d E_{c}^{(1)} (x) \\ = \int_{ℤ_{p, m}^{\times}} χ (x) {⟨ x ⟩}_{p}^{1 - s} x_{p}^{- 1} d E_{c}^{(1)} (x) \\ = - (1 - χ (c)) L_{p} (χ, s) . \end{array}

It follows that $f_{χ} = F_{χ}$ . The case that $χ = 1$ is similar and left to the reader. □

We now prove the integrality of the Iwasawa power series $f_{χ}$ for odd $p$ .

Proposition 5.3.12. §

Let $χ$ be a primitive even Dirichlet character of conductor $m$ or $𝑚𝑞$ . Then $\frac{1}{2} f_{χ} \in Λ_{χ}$ .

Proof.

We prove this in the case that $p$ is odd. For $χ = 1$ , this is immediate from Lemma 5.3.11 and Remark 5.3.4. For $χ$ nontrivial, we are already done if $χ$ is not of $p$ -power order, as $χ (c) - 1$ can be chosen to be a unit. In particular, we may suppose that $m \neq 1$ , so is divisible by a prime $ℓ \neq p$ . We claim that

Θ_{\infty}^{(1)} - ℓ^{- 1} Θ_{\infty}^{(1)} (ℓ) \in ℤ_{p} ⟦ G_{\infty} ⟧ .

To see this, note that

Θ_{n}^{(1)} - ℓ^{- 1} Θ_{n}^{(1)} (ℓ) = \sum_{\begin{array}{c} a = 1 \\ (a, 𝑚𝑝) = 1 \end{array}}^{m p^{n}} (\frac{a}{m p^{n}} - ℓ^{- 1} ⟨ \frac{ℓ𝑎}{m p^{n}} ⟩) σ_{a}^{- 1} \in ℤ_{p} [G_{n}]

for all $n$ , since $ℓ \in ℤ_{p}^{\times}$ and $ℓ \frac{a}{m p^{n}} - ⟨ \frac{ℓ𝑎}{m p^{n}} ⟩ \in ℤ$ by definition. Setting $m^{'} = \frac{m}{ℓ}$ , for $\tilde{Δ} \subset {1, \dots, m p^{n}}$ a set of representatives of $Δ$ viewed inside $ℤ ∕ m p^{n} ℤ$ and a fixed $b \in ℤ_{p}$ , we have that the coefficient of $γ^{- b} \in 𝒪_{χ} [G_{n}]$ of $\tilde{ω χ^{- 1}} Θ_{n}^{(1)} (ℓ)$ is

\sum_{a \in \tilde{Δ}} ⟨ \frac{ℓ𝑎𝑏}{m p^{n}} ⟩ ω χ^{- 1} (a) = \sum_{\begin{array}{c} a = 1 \\ a \in \tilde{Δ} \end{array}}^{m^{'} p^{n}} \frac{𝑎𝑏}{m^{'} p^{n}} \sum_{i = 0}^{ℓ - 1} ω χ^{- 1} (a + i m^{'} p^{n}),

and the latter sum is $0$ since $ℓ$ divides the conductor of $ω χ^{- 1}$ . (Note that one value of $a + i m^{'} p^{n}$ in the sum will not be prime to $m$ if $ℓ ∤ m^{'}$ , but $ω χ^{- 1} (a + i m^{'} p^{n}) = 0$ for this value.) Thus, $\tilde{ω χ^{- 1}} (Θ_{\infty}^{(1)} (ℓ)) = 0$ , so

f_{χ} = \tilde{ω χ^{- 1}} (Θ_{\infty}^{(1)} - ℓ^{- 1} Θ_{\infty}^{(1)} (ℓ)) \in Λ_{χ} .

□

Putting this all together, we have the following.

Theorem 5.3.13. §

Let $χ$ be a primitive even $p$ -adic Dirichlet character of the first kind. There exists a unique element $f_{χ} \in Λ_{χ}$ such that if $χ$ is nontrivial, we have

f_{χ} (ξ u^{s} - 1) = L_{p} (𝜒𝜌, s),

and if $χ = 1$ , then for $h_{1} = u {(1 + T)}^{- 1} - 1$ , we have

f_{1} (ξ u^{s} - 1) = h_{1} (ξ u^{s} - 1) L_{p} (ρ, s)

for all $s \in ℤ_{p}$ and $ξ$ of $p$ -power order, where $ρ$ is of the second kind satisfying $ρ (u) = ξ^{- 1}$ .

Recall that $X_{\infty}$ denotes the unramified Iwasawa module over $F_{\infty}$ . The interpretation of $f_{χ}$ in terms of Stickelberger elements also gives the following.

Proposition 5.3.14. §

For any primitive Dirichlet character $χ$ of conductor $m$ or $𝑚𝑞$ , the Iwasawa power series $\frac{1}{2} f_{χ} \in Λ_{χ}$ annihilates $X_{\infty}^{(ω χ^{- 1})}$ .

Proof.

We again suppose that $p$ is odd. Recall that $Θ_{\infty}^{(1)} - ℓ^{- 1} Θ_{\infty}^{(1)} (ℓ)$ is integral, and

\tilde{ω ψ^{- 1}} (Θ_{\infty}^{(1)} (ℓ)) = 0

for every nontrivial even character $ψ$ of conductor $m$ or $𝑚𝑝$ . Write $χ = 𝜈𝜌$ where $ν$ has order prime to $p$ and $ρ$ has $p$ -power order. By varying $ρ$ over its $G_{ℚ_{p}}$ -conjugates, this implies that $e_{ω ν^{- 1}} Θ_{\infty}^{(1)} (ℓ) = 0$ , where $e_{ω ν^{- 1}}$ is the idempotent for $ω ν^{- 1}$ on the prime-to- $p$ part of $Δ$ . Then $e_{ω ν^{- 1}} Θ_{\infty}^{(1)} \in 𝒪_{ν} ⟦ G_{\infty} ⟧$ . By Remark 5.3.10, it annihilates $e_{ω ν^{- 1}} X_{\infty}$ . By projection, we then have that $f_{χ}$ annihilates the quotient $X_{\infty}^{(ω χ^{- 1})}$ . □

Corollary 5.3.15. §

Suppose that $p$ is odd. For any even $k \geq 2$ not divisible by $p - 1$ and every $j \geq 1$ , we have

B_{1, ω^{k - 1}} \equiv \frac{B_{j, ω^{k - j}}}{j} mod p .

In particular, we have

B_{1, ω^{k - 1}} \equiv \frac{B_{k}}{k} mod p .

Proof.

We have $L_{p} (0, ω^{k}) = - B_{1, ω^{k - 1}}$ and $L_{p} (1 - j, ω^{k}) = - \frac{B_{j}, ω^{k - j}}{j}$ , so this follows from the fact that $ω^{k} \neq 1$ , then $L_{p} (χ, s) = f_{χ} (u^{s} - 1)$ , and $u^{1 - j} - 1 \equiv 0 mod p$ . □

Corollary 5.3.16. §

Suppose that $p$ is odd and $j \equiv k mod p^{n - 1} (p - 1)$ are even positive integers not divisible by $p - 1$ . Then

(1 - p^{j - 1}) \frac{B_{j}}{j} \equiv (1 - p^{k - 1}) \frac{B_{k}}{k} mod p^{n} .

Proof.

We have $L_{p} (1 - j, ω^{j}) = - (1 - p^{j - 1}) \frac{B_{j}}{j}$ . As

ω^{j} (x) {⟨ x ⟩}_{p}^{j - 1} = x_{p}^{j} {⟨ x ⟩}_{p}^{- 1}

and $x_{p}^{j} \equiv x_{p}^{k} mod p^{n}$ , we have the result so long as

1 - ω^{j} (c) {⟨ c ⟩}_{p}^{j} = 1 - c_{p}^{j}

can be taken to be a unit, which occurs if $j ≢ 0 mod p - 1$ . □

5.4. Coleman theory

Let $E$ be an unramified extension of $ℚ_{p}$ with valuation ring $𝒪$ . Let $q$ denote the order of the residue field of $𝒪$ . Let $E_{n} = E (μ_{p^{n + 1}})$ , and let $𝒪_{n}$ denote its valuation ring, for $n \geq 0$ . Fix a sequence ${(ζ_{p^{n}})}_{n}$ of primitive $p^{n}$ th roots of unity in $E_{n}$ such that $ζ_{p^{n + 1}}^{p} = ζ_{p^{n}}$ for each $n \geq 1$ . Let $Λ = 𝒪 ⟦ T ⟧$ .

Notation 5.4.1. §

Let $[p]$ denote the continuous $ℤ_{p}$ -linear endomorphism of $Λ$ given on $f \in Λ$ by

[p] (f) (T) = f ({(1 + T)}^{p} - 1) .

Lemma 5.4.2. §

The image of $[p]$ is equal to the set of all $f \in Λ$ such that

f (ζ_{p}^{i} (1 + T) - 1) = f (T)

for all $i \in ℤ$ .

Proof.

We need only show that every $f$ with the above property is in the image of $[p]$ , which is to say that it can be expanded in a power series in $P = [p] (T) = {(1 + T)}^{p} - 1$ . For this, suppose inductively that we have written $f$ as

f = \sum_{i = 0}^{n - 1} a_{i} P^{i} + P^{n} f_{n}

with $a_{i} \in 𝒪$ for some $n \geq 0$ . Then $f_{n}$ also has the property that $f_{n} (ζ_{p}^{i} (1 + T) - 1) = f_{n} (T)$ for all $i$ . Taking $T = 0$ , we see that $f_{n} (ζ_{p}^{i} - 1) = f_{n} (0)$ for all $i$ , and therefore

f_{n} - f_{n} (0) = P f_{n + 1}

for some $f_{n + 1} \in Λ$ having the desired property, and we set $a_{n} = f_{n} (0)$ . We then have $f = \sum_{i = 0}^{\infty} a_{i} P^{i}$ in the limit. □

Proposition 5.4.3. §

There exist unique maps $𝒩 : Λ \to Λ$ and $𝒮 : Λ \to p Λ$ satisfying

\begin{array}{rcl} ([p] \circ 𝒩) (f) (T) = \prod_{i = 0}^{p - 1} f (ζ_{p}^{i} (1 + T) - 1) & and & ([p] \circ 𝒮) (f) (T) = \sum_{i = 0}^{p - 1} f (ζ_{p}^{i} (1 + T) - 1) \end{array}

for all $f \in Λ$ .

Proof.

For $f \in Λ$ , consider

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

which is clearly in $Λ$ as its coefficients are fixed by $Gal (ℚ (μ_{p}) ∕ ℚ)$ . We have $g (T) = g (ζ_{p}^{i} (1 + T) - 1)$ for all $i \in ℤ$ , so by Lemma 5.4.2, we have $g = [p] (𝒩 (f))$ for some $𝒩 (f) \in Λ$ , which is unique by the injectivity of $[p]$ .

If we take

h = \sum_{i = 0}^{p - 1} f (ζ_{p}^{i} (1 + T) - 1) \in Λ

then as

f (ζ_{p}^{i} (1 + T) - 1) \equiv f (T) mod (1 - ζ_{p}),

for each $i$ , we have $h (T) \in p Λ$ . As in the case of $𝒩$ , we have $h (T) = h (ζ_{p}^{i} (1 + T) - 1)$ for all $i$ , so $h = [p] (𝒮 (f))$ for a unique $𝒮 (f) \in p Λ$ . □

Definition 5.4.4. §

Coleman’s norm operator $𝒩 : Λ \to Λ$ and Coleman’s trace operator $𝒮 : Λ \to Λ$ are the maps characterized by Proposition 5.4.3.

Lemma 5.4.5. §

If $f \in Λ$ and $n \geq 1$ , then $f \equiv 1 mod p^{n}$ if and only if $[p] (f) \equiv 1 mod p^{n}$ .

Proof.

We consider the nontrivial direction. Let $m \geq 0$ be maximal with $f \equiv 1 mod p^{m}$ , and let $k \geq 0$ be maximal such that

f \equiv 1 + a p^{m} T^{k} mod (p^{m + 1}, T^{k + 1})

for some nonzero $a \in 𝒪^{\times}$ . Since $[p] (T) \equiv T^{p} mod p$ , we have

[p] (f) \equiv 1 + p^{m} a T^{𝑝𝑘} mod (p^{m + 1}, T^{k + 1}) .

So, if $[p] (f) \equiv 1 mod p^{n}$ , then $n \leq m$ . □

Let $φ$ denote the unique Frobenius element in $Gal (E_{\infty} ∕ ℚ_{p})$ , where $E_{\infty} = ⋃_{n} E_{n}$ , which we also let act on $Λ$ through its action on coefficients.

Proposition 5.4.6. §

If $f \in Λ^{\times}$ , then $𝒩 (f) \equiv φ (f) mod p$ . If $f \equiv 1 mod p^{n}$ for some positive integer $n$ , then $𝒩 (f) \equiv 1 mod p^{n + 1}$ .

Proof.

Take $f \in Λ^{\times}$ , and suppose that $f \equiv 1 mod p^{k}$ for some $k \geq 0$ . We then have

f (ζ_{p}^{i} (1 + T) - 1) \equiv f (T) mod p^{k} (1 - ζ_{p})

for each $i \in ℤ$ , and our assumption on $f$ implies that

([p] \circ 𝒩) (f) = \prod_{i = 1}^{p - 1} f (ζ_{p}^{i} (1 + T) - 1) \equiv f {(T)}^{p} mod p^{k + 1} .

If $k \geq 1$ , then $f^{p} \equiv 1 mod p^{k + 1}$ , so Lemma 5.4.5 tells us that $𝒩 (f) \equiv 1 mod p^{k + 1}$ as well.

If $k = 0$ , then we can at least say that $f {(T)}^{p} \equiv φ (f) (T^{p}) \equiv [p] (f) (T) mod p$ , so

[p] (\frac{𝒩 (f)}{φ (f)}) \equiv 1 mod p,

and therefore Lemma 5.4.5 tells us that $𝒩 (f) \equiv φ (f) mod p$ . □

Corollary 5.4.7. §

Suppose that $f \in Λ^{\times}$ . For $n \geq m$ , we have

𝒩^{n} (φ^{- n} (f)) \equiv 𝒩^{m} (φ^{- m} (f)) mod p^{m + 1} .

Proof.

By repeated application of Proposition 5.4.6 with $k = 0$ , we have

𝒩^{n - m} (f) \equiv φ^{n - m} (f) mod p,

and again by Proposition 5.4.6, the congruence follows by applying $𝒩^{m} \circ φ^{- n}$ to $\frac{𝒩^{n - m} (f)}{φ^{n - m} (f)}$ . □

Corollary 5.4.8. §

Suppose that $f \in Λ^{\times}$ . Then $g = \lim_{i \to \infty} 𝒩^{i} (φ^{- i} (f))$ exists, and $𝒩 (g) = φ (g)$ .

Proof.

By Corollary 5.4.7, the limit $g$ in question exists, and we have

𝒩 (g) = \lim_{i \to \infty} 𝒩^{i + 1} (φ^{- i} (f)) = φ \lim_{i \to \infty} 𝒩^{i + 1} (φ^{- (i + 1)} (f)) = φ (g) .

□

Theorem 5.4.9 (Coleman). §

Suppose that $u = {(u_{n})}_{n \geq 0}$ forms a norm compatible sequence of units with $u_{n} \in 𝒪_{n}^{\times}$ . Then there exists a unique $f \in Λ^{\times}$ such that $f (ζ_{p^{n}} - 1) = φ^{n} (u_{n})$ for all $n \geq 0$ , and it has the property that $𝒩 (f) = φ (f)$ .

Proof.

We choose arbitrary $f_{n} \in Λ^{\times}$ that satisfy $f_{n} (ζ_{p^{n}} - 1) = φ^{n} (u_{n})$ for each $n$ , and we set $g_{n} = 𝒩^{n} (φ^{- n} (f_{2 n}))$ . As ${(g_{n})}_{n}$ is a sequence in a compact set $Λ$ , it has a limit point, which we call $f$ . We claim that this $f$ has the desired property.

For any $n \geq m$ , we have

\begin{array}{l} φ^{m} (u_{m}) = φ^{m} (N_{E_{n} ∕ E_{m}} (u_{n})) & = φ^{m - n} \prod_{i = 0}^{p^{n - m} - 1} f_{n} (ζ_{p^{n - m}}^{i} ζ_{p^{n}} - 1) \\ = (𝒩^{n - m} φ^{m - n} f_{n}) ({[p]}^{n - m} (ζ_{p^{n}} - 1)) \\ = (𝒩^{n - m} φ^{m - n} f_{n}) (ζ_{p^{m}} - 1) . \end{array}

Since $2 n - m \geq n$ , Corollary 5.4.7, tells us that

𝒩^{2 n - m} φ^{m - 2 n} f_{2 n} \equiv 𝒩^{n} φ^{- n} f_{2 n} mod p^{n + 1},

φ^{m} (u_{m}) = 𝒩^{2 n - m} φ^{m - 2 n} f_{2 n} (ζ_{p^{m}} - 1) \equiv g_{n} (ζ_{p^{m}} - 1) mod p^{n + 1} .

This forces $f (ζ_{p^{m}} - 1) = φ^{m} (u_{m})$ by taking the limit over the subsequence of ${(g_{n})}_{n}$ converging to $f$ .

The power series $f$ is unique, as its difference with any other such power series would have infinitely many zeros in the maximal ideal of $𝒪_{\infty}$ . Note that

\begin{matrix} 𝒩 (f) (ζ_{p^{n}} - 1) = 𝒩 (f) ([p] (ζ_{p^{n + 1}} - 1)) = ([p] \circ 𝒩) (f) (ζ_{p^{n + 1}} - 1) \\ = \prod_{i = 0}^{p - 1} f (ζ_{p^{n + 1}}^{i} - 1) = N_{F_{n + 1} ∕ F_{n}} φ^{n + 1} (u_{n + 1}) = φ^{n + 1} (u_{n}) = φ (f) (ζ_{p^{n}} - 1) \end{matrix}

for all $n$ , which similarly forces $𝒩 (f) = φ (f)$ . □

Notation 5.4.10. §

Let $\tilde{Γ} = Gal (E_{\infty} ∕ E)$ . Let $U_{\infty} = {lim_{\leftarrow}}_{n} 𝒪_{n}^{\times}$ under norm maps.

We let $σ \in \tilde{Γ}$ act on $f \in 𝒪 ⟦ T ⟧$ by

(𝜎𝑓) (T) = f ({(1 + T)}^{χ (σ)} - 1),

where $χ : \tilde{Γ} \to ℤ_{p}^{\times}$ denotes the $p$ -adic cyclotomic character. The group $Gal (E_{\infty} ∕ ℚ_{p}) ≅ ⟨ φ ⟩ \times \tilde{Γ}$ then acts on $𝒪 ⟦ T ⟧$ through the action of powers of Frobenius on coefficients and the action of $\tilde{Γ}$ described above.

Notation 5.4.11. §

Set

M = {f \in Λ^{\times} ∣ 𝒩 (f) = φ (f)} .

Definition 5.4.12. §

The Coleman power series attached to $u = {(u_{n})}_{n} \in U_{\infty}$ is the unique $f \in M$ such that $f (ζ_{p^{n}} - 1) = φ^{n} (u_{n})$ for all $n \geq 0$ .

Corollary 5.4.13. §

The map $U_{\infty} \to M$ that takes a norm compatible sequence to its associated Coleman power series is a continuous $Gal (E_{\infty} ∕ ℚ_{p})$ -equivariant isomorphism.

Proof.

That the map is an injective homomorphism is a consequence of uniqueness of the power series $f$ attached to $u$ by Theorem 5.4.9, and its image is in $M$ by said theorem.

For any $f \in M$ , if we set $u_{n} = φ^{- n} (f (ζ_{p^{n}} - 1))$ , then

φ^{n} (u_{n}) = f (ζ_{p^{n}} - 1) = φ^{- 1} 𝒩 (f) (ζ_{p^{n}} - 1) = φ^{- 1} \prod_{i = 0}^{p - 1} f (ζ_{p^{n + 1}} ζ_{p^{n}}^{i} - 1) = φ^{n} (N_{F_{n + 1} ∕ F_{n}} u_{n + 1}) .

Thus $f$ is the power series attached to ${(u_{n})}_{n} \in U_{\infty}$ . Continuity follows from the construction of the map and is easily checked. □

Lemma 5.4.14. §

For all $f \in Λ$ , we have

𝒮 ([p] (f)) = 𝑝𝑓 .

Proof.

By definition, we have that

([p] \circ 𝒮 \circ [p] (f)) (T) = \sum_{i = 0}^{p - 1} f ([p] (ζ_{p}^{i} (1 + T) - 1)) = 𝑝𝑓 ([p] (T)) = [p] (𝑝𝑓) (T) .

The result then follows by injectivity of $[p]$ . □

Notation 5.4.15. §

Let

\begin{array}{rcl} Λ^{𝒮 = 0} = {f \in Λ ∣ 𝒮 (f) = 0} & and & Λ^{𝒮 = 𝑝𝜑} = {f \in Λ ∣ 𝒮 (f) = 𝑝𝜑𝑓} . \end{array}

Proposition 5.4.16. §

The sequence

0 \to ℤ_{p} \to Λ^{𝒮 = 𝑝𝜑} \to_{}^{1 - [p] φ} Λ^{𝒮 = 0} \to_{}^{f \mapsto {Tr}_{E ∕ ℚ_{p}} f (0)} ℤ_{p} \to 0

is exact.

Proof.

Any constant $a \in ℤ_{p}$ satisfies $𝑝𝜑𝑎 = 𝑝𝑎 = (𝒮 \circ [p]) (a) = 𝒮 (a)$ , so sits inside $Λ^{𝒮 = 𝑝𝜑}$ . If $f \in Λ^{𝒮 = 𝑝𝜑}$ , then

𝒮 ((1 - [p] φ) (f)) = 𝑝𝜑 (f) - 𝑝𝜑 (f) = 0

by Lemma 5.4.14, so $(1 - [p] φ) (f) \in Λ^{𝒮 = 0}$ . Thus, the sequence is well-defined.

Note that $(1 - [p] φ) (a) = a - φ (a) = 0$ for $a \in ℤ_{p}$ and $(1 - [p] φ) (f) (0) = f (0) - φ (f (0))$ for $f \in Λ$ , which is carried to $0$ under ${Tr}_{E ∕ ℚ_{p}}$ . Thus, the sequence is a complex.

Injectivity of the first map is obvious, so we consider exactness at $Λ^{𝒮 = 𝑝𝜑}$ . If $f \in Λ^{𝒮 = 𝑝𝜑}$ satisfies $[p] φ (f) = f$ , then $f (0) \in ℤ_{p}$ , and we may replace $f$ by $g = p^{- m} (f - f (0)) \in Λ^{𝒮 = 𝑝𝜑}$ for $m \geq 0$ maximal, supposing $g \neq 0$ . We then have

g \equiv b T^{i} mod (p, T^{i + 1})

for some $b \in 𝒪^{\times}$ and $i \geq 1$ . But this congruence forces $φ (g) (T^{p}) \equiv 0 mod (p, T^{i + 1})$ , a contradiction. Thus, we have $f = f (0) \in ℤ_{p}$ .

We next consider exactness at $Λ^{𝒮 = 0}$ . Suppose that $g \in Λ$ with ${Tr}_{E ∕ ℚ_{p}} g (0) = 0$ . Then $g (0) = (1 - [p] φ) (b)$ for some $b \in 𝒪$ by Hilbert’s theorem 90, and

(1 - [p] φ) (a T^{i}) \equiv a T^{i} mod (p T^{i}, T^{i + 1})

for all $a \in 𝒪$ and $i \geq 1$ , so we can find a sequence in the image of $1 - [p] φ$ that converges to $g$ recursively, and thus $g = (1 - p [φ]) (f)$ for some $f \in Λ$ . If moreover $g \in Λ^{𝒮 = 0}$ , then

𝒮 (f) = 𝒮 (g) + 𝒮 ([p] φ (f)) = 𝒮 ([p] φ (f)) = 𝑝𝜑 (f) .

Let $ξ \in μ_{q - 1} (E)$ satisfy ${Tr}_{E ∕ ℚ_{p}} ξ = 1$ . Note that $ξ (1 + T) \in Λ^{𝒮 = 0}$ , since

[p] \circ 𝒮 (ξ (1 + T)) = \sum_{i = 0}^{p - 1} ζ_{p}^{i} ξ (1 + T) = 0,

and $[p]$ is injective. Thus, the final map is surjective. □

Notation 5.4.17. §

a.

Define $D : Λ \to Λ$ on $f \in Λ$ by $D (f) = (1 + T) f^{'} (T)$ .

b.

Define $\log : Λ^{\times} \to E ⟦ T ⟧$ to be the homomorphism satisfying

\log (1 + f) = \sum_{i = 1}^{\infty} \frac{{(- 1)}^{i - 1} f^{i}}{i}

for $f \in (p, T)$ and $\log (ξ) = 0$ for $ξ$ any root of unity in $𝒪$ .

c.

Define $D \log : Λ^{\times} \to Λ$ on $f \in Λ^{\times}$ by $D \log (f) = (1 + T) \frac{f^{'} (T)}{f (T)}$ .

Remark 5.4.18. §

Note that $D \log = D \circ \log$ . We also consider $D^{k} \log = D^{k - 1} \circ D \log$ for $k \geq 1$ .

Lemma 5.4.19. §

For any $f \in Λ^{\times}$ , the quantity

\frac{1}{p} \log (\frac{f^{p}}{[p] φ (f)})

lies in $Λ$ .

Proof.

We have

φ (f) ({(1 + T)}^{p} - 1) \equiv φ (f) (T^{p}) \equiv f^{p} (T) mod p,

so $\frac{f^{p}}{[p] φ (f)} = 1 + 𝑝𝑔$ for some $g \in Λ$ . We have

\frac{1}{p} \log (1 + 𝑝𝑔) = \sum_{i = 1}^{\infty} \frac{{(- 1)}^{i - 1} p^{i - 1} g^{i}}{i},

and the latter quantity clearly lies in $Λ$ , since $i \geq v_{p} (i) + 1$ for all $i \geq 1$ . □

Notation 5.4.20. §

Define $L : Λ^{\times} \to Λ$ on $f \in Λ^{\times}$ by

L (f) = \log f - \frac{1}{p} \log ([p] φ (f)) .

Proposition 5.4.21. §

We have a commutative square

The logarithmic derivative square. A full diagram description follows.

Diagram description: The logarithmic derivative square

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: script M; column 2: capital Lambda superscript (script S equals 0).
Row 2, from left to right: column 1: capital Lambda superscript (script S equals p varphi); column 2: capital Lambda superscript (script S equals 0).

Arrows and lines:

An arrow from script M to capital Lambda superscript (script S equals 0) (row 1, column 2), labelled script L.
An arrow from script M to capital Lambda superscript (script S equals p varphi), labelled D log.
An arrow from capital Lambda superscript (script S equals 0) (row 1, column 2) to capital Lambda superscript (script S equals 0) (row 2, column 2), labelled D.
An arrow from capital Lambda superscript (script S equals p varphi) to capital Lambda superscript (script S equals 0) (row 2, column 2), labelled 1 minus [p] varphi.

of continuous $Gal (E_{\infty} ∕ ℚ_{p})$ -equivariant $ℤ_{p}$ -linear homomorphisms.

Proof.

For any $g \in Λ$ , we have

\begin{matrix} \begin{aligned} ([p] \circ 𝒮) (D (g)) & = \sum_{i = 0}^{p - 1} ζ_{p}^{i} (1 + T) g^{'} (ζ_{p}^{i} (1 + T) - 1) \\ = \sum_{i = 0}^{p - 1} D (g (ζ_{p}^{i} (1 + T) - 1)) \\ = D (([p] \circ 𝒮) (g)), \end{aligned} \end{matrix}

(5.4.1)

employing the chain rule in the second equality. Since $[p]$ is an injective endomorphism, it follows that $D (Λ^{𝒮 = 0}) \subseteq Λ^{𝒮 = 0}$ . For $h \in M$ , that $D \log h \in Λ^{𝒮 = 𝑝𝜑}$ follows from the string of equalities

\begin{array}{l} [p] (𝒮 (D \log h)) & = D (([p] \circ 𝒮) (\log h)) \\ = D \log (([p] \circ 𝒩) (h)) \\ = D \log (φ [p] (h)) \\ = [p] (𝑝𝜑 (D \log h)), \end{array}

the first using (5.4.1), the second using that $\log$ is a homomorphism, the third using $h \in M$ , and the fourth by definition of the endomorphism $[p]$ . We also have $L (h) \in Λ^{𝒮 = 0}$ , since

\begin{array}{l} [p] \circ 𝒮 (L (h)) & = \sum_{i = 0}^{p - 1} L (h) (ζ_{p}^{i} (1 + T) - 1) \\ = \sum_{i = 0}^{p - 1} \log (h) (ζ_{p}^{i} (1 + T) - 1) - \log ([p] φ (h)) = 0, \end{array}

the last step using $[p] (T) = \prod_{i = 0}^{p - 1} (ζ_{p}^{i} (1 + T) - 1)$ .

Next, for any $h \in Λ^{\times}$ , we have by the chain rule that

D \log ([p] \circ h) = (1 + T) \frac{{(h \circ [p])}^{'} (T)}{h \circ [p] (T)} = {(1 + T)}^{p} \frac{h^{'} ([p] (T))}{h ([p] (T))} = p [p] (D \log h) .

It follows from this that the diagram commutes. Galois-equivariance of $\log$ and then $L$ is clear, and Galois-equivariance of $D$ and then $D \log$ follows from the chain rule (for the $\tilde{Γ}$ -action). □

Lemma 5.4.22. §

For every $n \geq 1$ , we have

\frac{1}{p} 𝒮 ([p] {(T)}^{n} \frac{1 + T}{T}) = T^{n} \frac{1 + T}{T} .

Proof.

Set $P = [p] (T)$ , and note that $P = \prod_{i = 0}^{p - 1} (ζ_{p}^{i} (1 + T) - 1)$ , and apply $D \log$ to both sides. We then have

p \frac{1 + P}{P} = \sum_{i = 0}^{p - 1} \frac{ζ_{p}^{i} (1 + T)}{ζ_{p}^{i} (1 + T) - 1} .

We have

[p] (𝒮 (P^{n} \cdot \frac{1 + T}{T})) = P^{n} \sum_{i = 0}^{p - 1} \frac{ζ_{p}^{i} (1 + T)}{ζ_{p}^{i} (1 + T) + 1} = p P^{n} \frac{1 + P}{P} = [p] (p T^{n} \frac{1 + T}{T}),

which yields the result. □

Notation 5.4.23. §

Let $Ω = 𝔽_{q} ⟦ T ⟧$ , and define $\partial : Ω^{\times} \to T Ω$ by

\partial (f) = T \frac{f^{'} (T)}{f (T)}

for $f \in Ω$ .

Lemma 5.4.24. §

We have

T Ω = \partial (Ω^{\times}) + {f^{p} ∣ f \in T Ω} .

Proof.

We claim that

\partial (Ω^{\times}) = {\sum_{i = 1}^{\infty} a_{i} T^{i} \in Ω ∣ a_{𝑝𝑖} = a_{i}^{p} for all i \geq 1} .

Let us denote the latter set by $S$ . For this, we note that any $u \in Ω^{\times}$ may be written uniquely as an infinite product

u = c \prod_{n = 1}^{\infty} (1 - b_{n} T^{n})

with $c \in 𝔽_{q}^{\times}$ and $b_{n} \in 𝔽_{q}$ for $n \geq 1$ . Note that

\partial (1 - b_{n} T^{n}) = \frac{- n b_{n} T^{n}}{1 + b_{n} T^{n}} = - n \sum_{i = 1}^{\infty} b_{n}^{i} T^{𝑛𝑖} \in S,

so $\partial (u) \in S$ as $\partial$ is a continuous homomorphism. Conversely, any element of $S$ may be written as an infinite sum of terms of the form $- n \sum_{i = 1}^{\infty} a^{i} T^{𝑛𝑖}$ for some $a \in 𝔽_{q}$ and $n$ prime to $p$ (by using it to specify the coefficient of $T^{n}$ and thereby of the $T^{n p^{i}}$ for $i \geq 1$ ), and such an element is equal to $\partial (1 - a T^{n})$ . Thus, we have the claim.

Note that we can pick the coefficients $a_{i}$ with $p ∤ i$ of a power series in $S$ arbitrarily. The fact that $\partial (Ω^{\times}) = S$ implies the result since the $p$ th powers of elements of $T Ω$ are exactly the power series in $T^{p} 𝔽_{q} ⟦ T^{p} ⟧$ , for which we can pick the coefficients $a_{i}$ with $p ∣ i$ arbitrarily. □

Proposition 5.4.25. §

The map $D \log : M \to Λ^{𝒮 = 𝑝𝜑}$ is surjective and has kernel the group $μ_{q - 1}$ of roots of unity of $𝒪$ of prime-to- $p$ order.

Proof.

Clearly, the kernel of $D \log$ on $Λ^{\times}$ is $𝒪^{\times}$ . Note that $a \in 𝒪^{\times} \cap M$ if and only if $a^{p} = φ (a)$ . It is easy to see that no element of $1 + p 𝒪$ can have this property, while every element of $μ_{q - 1}$ does. Thus, the kernel is as stated.

We claim first that it suffices to check the surjectivity of $D \log$ modulo $p$ . Let $Ω = 𝔽_{q} ⟦ T ⟧$ , and note that the formula for $D \log$ makes sense on $Ω^{\times}$ . Let $f \in Λ^{𝒮 = 𝑝𝜑}$ . Suppose by induction that there exists $h_{k} \in M$ such that $D \log (h_{k}) \equiv f mod p^{k} Λ$ . Then set

f^{'} = \frac{1}{p^{k}} (f - D \log (h_{k})) \in Λ^{𝒮 = 𝑝𝜑},

and choose $h^{'} \in M$ such that $D h^{'} \equiv f^{'} mod p Λ$ . Setting $h_{k + 1} = h_{k} + p^{k} h^{'}$ , we then have

D \log (h_{k + 1}) \equiv f mod p^{k + 1} Λ .

If we set $h = \lim_{k \to \infty} h_{k}$ , then $D \log (h) = f$ . Thus, we have the claim.

Next, we note that the reduction modulo $p$ map $M \to Ω^{\times}$ is surjective. This is straightforward: if $\bar{f} \in Ω^{\times}$ , then choose any lift $f$ of it to $Λ^{\times}$ and consider $g = \lim_{k \to \infty} φ^{- k} 𝒩^{k} f$ , which also lifts $\bar{f}$ but now lies in $M$ . To see that $D \log$ is surjective, it is then enough to see that the image $Φ$ of $Λ^{𝒮 = 𝑝𝜑}$ under reduction modulo $p$ is contained in $\frac{1 + T}{T} \partial (Ω^{\times})$ .

Let $v \in Φ$ . By Lemma 5.4.24, we have that

\frac{T}{1 + T} v = \partial (u) + f^{p}

for some $u \in Ω^{\times}$ and $f \in T Ω$ . Note that $\frac{1}{p} φ^{- 1} 𝒮 : Λ \to Λ$ reduces to an operator $𝔰 : Ω \to Ω$ that fixes both $v$ and $\frac{1 + T}{T} \partial (u)$ , so fixes $\frac{1 + T}{T} f^{p}$ . On the other hand, Lemma 5.4.22 tells us that $𝔰 (\frac{1 + T}{T} f^{p}) = \frac{1 + T}{T} f$ , since $[p] (φ (f)) = f^{p}$ in $Ω$ . But for $\frac{1 + T}{T} f = \frac{1 + T}{T} f^{p}$ to hold for $f \in T Ω$ , we must have $f = 0$ . Therefore, $v = \frac{1 + T}{T} \partial (u)$ , finishing the proof. □

Corollary 5.4.26. §

The map $D : Λ^{𝒮 = 0} \to Λ^{𝒮 = 0}$ is a bijection.

Proof.

The kernel of $D$ on $Λ$ is $ℤ_{p}$ , but $𝒮 (a) = 𝑝𝑎$ for all $a \in ℤ_{p}$ , so $D$ is an injection. Since $D \log$ is a surjection by Proposition 5.4.25, the exact sequence of Proposition 5.4.16 reduces us to the claim that the composite map $g \mapsto {Tr}_{E ∕ ℚ_{p}} 𝐷𝑔 (0)$ is surjective. For $a \in 𝒪$ , one may observe that $𝒮 (a (1 + T)) = 0$ , so $a (1 + T) \in Λ^{𝒮 = 0}$ , and $D (a (1 + T)) (0) = a$ . The corollary now follows by the surjectivity of trace in unramified extensions. □

We now have the following consequence of what we have proven.

Proposition 5.4.27. §

The diagram

0 \to μ_{q - 1} \times ℤ_{p} (1) \to_{}^{(ξ, a) \mapsto ξ {(1 + T)}^{a}} M \to_{}^{L} Λ^{𝒮 = 0} \to_{}^{f \mapsto {Tr}_{E ∕ ℚ_{p}} f^{'} (0)} ℤ_{p} (1) \to 0

is an exact sequence in the category of compact abelian groups with continuous $Gal (E_{\infty} ∕ ℚ_{p})$ -actions.

Proof.

We use Proposition 5.4.21, Proposition 5.4.25, and Corollary 5.4.26 to replace the middle terms in the exact sequence of Proposition 5.4.16. The fact that $D \log : M \to Λ^{𝒮 = 𝑝𝜑}$ has kernel $μ_{q - 1}$ is taken care of by adding it to the first term to preserve exactness. Note for this that the Coleman power series attached to the norm compatible sequence ${(ζ_{p^{n}}^{a})}_{n}$ for $a \in ℤ_{p}$ is exactly ${(1 + T)}^{a}$ , and the map is clearly $Gal (E_{\infty} ∕ ℚ_{p})$ -equivariant. Also, note that $𝐷𝑓 (0) = f^{'} (0)$ , so the last map is as stated, and

f {({(1 + T)}^{χ (a)} - 1)}^{'} |_{T = 0} = χ (a) f^{'} (0),

so it is also $Gal (E_{\infty} ∕ ℚ_{p})$ -equivariant. □

Recall that an $𝒪$ -valued measure on $ℤ_{p}$ is identified with an element of $𝒪 ⟦ ℤ_{p} ⟧$ which is isomorphic to $Λ$ under the continuous $𝒪$ -linear map that takes the group element of $[i]$ to ${(1 + T)}^{i}$ .

Notation 5.4.28. §

We define an operator $Φ : Λ \to Λ$ on $f \in Λ$ by

Φ (f) = f - \frac{1}{p} [p] (𝒮 (f)) .

By definition, $f \in Λ$ satisfies $Φ (f) = f$ if and only if $f \in Λ^{𝒮 = 0}$ .

Proposition 5.4.29. §

A measure $μ$ on $ℤ_{p}$ is the extension by zero of a measure on $ℤ_{p}^{\times}$ if and only if the power series attached to $μ$ lies in $Λ^{𝒮 = 0}$ . In other words, the continuous $𝒪$ -linear isomorphism $𝒪 ⟦ ℤ_{p} ⟧ \to_{}^{\sim} Λ$ sending the group element $1$ to $T + 1$ restricts to an isomorphism $𝒪 ⟦ ℤ_{p}^{\times} ⟧ \to_{}^{\sim} Λ^{𝒮 = 0}$ of topological $𝒪$ -modules.

Proof.

Let $f$ be the power series attached to $μ$ , and let ${\bar{f}}_{n} \in 𝒪 [T] ∕ ({(1 + T)}^{p^{n}} - 1)$ denote its image. For the distribution ${(μ_{n})}_{n}$ attached to $μ$ , we have that

f_{n} = \sum_{k = 0}^{p^{n} - 1} μ_{n} (k) {(1 + T)}^{k} \in 𝒪 [T]

lifts ${\bar{f}}_{n}$ . Here, $μ_{n} (k) = \int_{ℤ_{p}} χ_{k + p^{n} ℤ_{p}} 𝑑𝜇$ for $χ_{k + p^{n} ℤ_{p}}$ the chararacteristic function of $k + p^{n} ℤ_{p}$ . So, $μ$ is the extension by zero of a measure on $ℤ_{p}^{\times}$ if and only if $μ_{n} (k) = 0$ for all $0 \leq k \leq p^{n} - 1$ with $p ∣ k$ . We claim this occurs if and only if $Φ (f) = f$ , which will finish the proof.

Note that for any $a \in 𝒪$ , we have

Φ (a {(1 + T)}^{k}) = a {(1 + T)}^{k} - \frac{1}{p} \sum_{i = 0}^{p - 1} a ζ_{p}^{𝑖𝑘} {(1 + T)}^{k} = {\begin{matrix} a {(1 + T)}^{k} & if p ∤ k \\ 0 & if p ∣ k . \end{matrix}

Then

Φ (f_{n}) = \sum_{\begin{array}{c} k = 0 \\ p ∤ k \end{array}}^{p^{n} - 1} μ_{n} (k) {(1 + T)}^{k} \in 𝒪 [T],

and it is clear that

Φ (f_{n}) \equiv f_{n} mod ({(1 + T)}^{p^{n}} - 1)

if and only if $μ_{n} (k) = 0$ for all $p ∣ k$ . This holds for all $n$ if and only if $μ$ is the extension by zero of a measure on $ℤ_{p}^{\times}$ . □

Note that $𝒪 ⟦ ℤ_{p}^{\times} ⟧ \to_{}^{\sim} Λ^{𝒮 = 0}$ is $Gal (E_{\infty} ∕ ℚ_{p})$ -equivariant, using the action of $σ \in Gal (E_{\infty} ∕ E)$ by multiplication by the group element of $χ (σ)$ on $𝒪 ⟦ ℤ_{p}^{\times} ⟧$

Definition 5.4.30. §

The Coleman map $Col : U_{\infty} \to 𝒪 ⟦ ℤ_{p}^{\times} ⟧$ is the map that takes $u \in U_{\infty}$ to the element of $𝒪 ⟦ ℤ_{p}^{\times} ⟧$ corresponding to $L (f)$ , where $f$ is the Coleman power series attached to $u$ .

Set $ζ = {(ζ_{p^{n}})}_{n} \in U_{\infty}$ . For $ξ \in μ_{q - 1}$ , let $\tilde{ξ} \in U_{\infty}$ be the unique norm compatible sequence of elements of $μ_{q - 1}$ with norm $ξ \in 𝒪^{\times}$ .

Theorem 5.4.31. §

There is an exact sequence

0 \to μ_{q - 1} \times ℤ_{p} (1) \to_{}^{(ξ, a) \mapsto \tilde{ξ} ζ^{a}} U_{\infty} \to_{}^{Col} 𝒪 ⟦ ℤ_{p}^{\times} ⟧ \to_{}^{λ \mapsto {Tr}_{E ∕ ℚ_{p}} \int_{ℤ_{p}} 𝑥𝑑𝜆 (x)} ℤ_{p} (1) \to 0

of continuous $Gal (E_{\infty} ∕ ℚ_{p})$ -equivariant homomorphisms.

Proof.

The Coleman map is the composite

Col : U_{\infty} \to_{}^{\sim} M \to_{}^{L} Λ^{𝒮 = 0} \to_{}^{\sim} 𝒪 ⟦ ℤ_{p}^{\times} ⟧

of the Coleman power series isomorphism with $L$ and the isomorphism of Proposition 5.4.29. We use this to replace the middle part of the exact sequence of Proposition 5.4.27 with $Col$ . That the first map is then as stated is immediate. That the final map is as stated comes from the fact that for $f \in Λ^{𝒮 = 0}$ corresponding to $λ \in 𝒪 ⟦ ℤ_{p}^{\times} ⟧$ (which yields a measure on $ℤ_{p}$ by extension by zero) and the distribution ${(λ_{n})}_{n}$ , we have

f^{'} (0) = \lim_{n \to \infty} \sum_{k = 0}^{p^{n} - 1} k λ_{n} (k) {(1 + T)}^{k - 1} |_{T = 0} = \lim_{n \to \infty} \sum_{k = 0}^{p^{n} - 1} k λ_{n} (k) = \int_{ℤ_{p}} 𝑥𝑑𝜆 (x) .

□

Lemma 5.4.32. §

Let $μ$ be an $𝒪$ -valued measure on $ℤ_{p}$ , and let $f \in Λ$ be the corresponding power series. For all $k \geq 0$ , we have

\int_{ℤ_{p}} x^{k} 𝑑𝜇 (x) = (D^{k} f) (0) .

Proof.

We have a linear functional defined by

L (g) = \int_{ℤ_{p}} 𝑥𝑔 (x) 𝑑𝜇 (x)

for all $g \in C (ℤ_{p}, ℂ_{p})$ . We then have

| L (g) | \leq \max_{a \in ℤ_{p}} | g (a) |

for all $g$ , so $L$ is bounded and thus gives a measure $μ_{1}$ , with a corresponding power series $h \in Λ$ .

We claim that $h = 𝐷𝑓$ . To see this, write $f = \sum_{n = 0}^{\infty} c_{n} T^{n} \in Λ$ , where $c_{n} = \int_{ℤ_{p}} (\binom{x}{n}) 𝑑𝜇 (x)$ . Note that

𝐷𝑓 = \sum_{n = 0}^{\infty} (n c_{n} + (n + 1) c_{n + 1}) T^{n} .

Write $h = \sum_{n = 0}^{\infty} e_{n} T^{n}$ . Then

e_{n} = \int_{ℤ_{p}} x (\binom{x}{n}) 𝑑𝜇 (x) .

Since $x (\binom{x}{n}) = (n + 1) (\binom{x}{n + 1}) + n (\binom{x}{n})$ , we have $e_{n} = (n + 1) c_{n + 1} + n c_{n}$ and therefore the claim.

Now, to prove the lemma, it suffices (by repeated application of the claim) to show that

\int_{ℤ_{p}} x^{k} 𝑑𝜇 = \int_{ℤ_{p}} d μ_{k},

where $μ_{k}$ is the measure corresponding to $D^{k} f$ . We can see this by induction, it being a consequence of the claim for $k = 1$ . That is, if we know if for all measures with $k - 1$ in place of $k$ , then

\int_{ℤ_{p}} x^{k - 1} d μ_{1} = \int_{ℤ_{p}} d μ_{k},

since $D^{k - 1} (𝐷𝑓) = D^{k} f$ . But by the claim, we have

\int_{ℤ_{p}} x^{k - 1} d μ_{1} = \int_{ℤ_{p}} x^{k} 𝑑𝜇,

so we are done. □

Definition 5.4.33. §

For $k \geq 1$ , the $k$ th Coates-Wiles homomorphism $δ_{k} : U_{\infty} \to 𝒪$ takes $u \in U_{\infty}$ to $D^{k} \log (f) (0)$ , where $f$ is the Coleman power series attached to $u$ .

Lemma 5.4.34. §

Let $χ : \tilde{Γ} \to_{}^{\sim} ℤ_{p}^{\times}$ be the $p$ -adic cyclotomic character. Then

δ_{k} (σ (u)) = χ {(σ)}^{k} δ_{k} (u)

for all $u \in U_{\infty}$ and $σ \in \tilde{Γ}$ .

Proof.

Note that for any $g \in E ⟦ T ⟧$ and $a \in ℤ_{p}$ , we have

D (g ({(1 + T)}^{a} - 1)) = a {(1 + T)}^{a} g^{'} ({(1 + T)}^{a} - 1) = a (𝐷𝑔) ({(1 + T)}^{a} - 1) .

So, by recursion we see that

D^{k} (g ({(1 + T)}^{a} - 1)) = a^{k} (𝐷𝑔) ({(1 + T)}^{a} - 1) .

(5.4.2)

We can apply this with $g = \log f$ for $f$ the Coleman power series attached to $u \in U_{\infty}$ and $a = χ (σ)$ for $σ \in \tilde{Γ}$ . For this, note that the Coleman power series attached to $σ (u)$ is $σ (f) (T) = f ({(1 + T)}^{χ (σ)} - 1)$ . Therefore, we have

D^{k} (\log σ (f)) = χ {(σ)}^{k} (D^{k} \log f) (σ (T)),

and plugging in $0$ , we get the desired formula. □

Proposition 5.4.35. §

For $u \in U_{\infty}$ , let $λ_{u} = Col (u)$ , which we view as an $𝒪$ -valued measure on $ℤ_{p}^{\times}$ . We then have

\int_{ℤ_{p}^{\times}} x^{k} d λ_{u} = (1 - p^{k - 1} φ) δ_{k} (u)

for all $k \geq 1$ .

Proof.

Let $f \in Λ^{\times}$ denote the power series attached to $u$ , and note that

\begin{array}{l} \int_{ℤ_{p}^{\times}} x^{k} d λ_{u} & = (D^{k} L (f)) (0) \\ = D^{k} \log (f) (0) - p^{- 1} D^{k} \log (φ (f) \circ [p]) (0) \\ = δ_{k} (u) - p^{k - 1} D^{k} \log φ (f) (0) \\ = (1 - p^{k - 1} φ) δ_{k} (u), \end{array}

the second-to-last step following from (5.4.2). □

Chapter 5 Kubota-Leopoldt p-adic L-functions

5.1. p-adic measures

Definition 5.1.1. §

Remark 5.1.2. §

Notation 5.1.3. §

Remark 5.1.4. §

Definition 5.1.5. §

Remark 5.1.6. §

Example 5.1.7. §

Remark 5.1.8. §

Proposition 5.1.9. §

Remark 5.1.10. §

Remark 5.1.11. §

Corollary 5.1.12. §

Corollary 5.1.13. §

Theorem 5.1.14 (Mahler). §

Corollary 5.1.15. §

Definition 5.1.16. §

Lemma 5.1.17. §

Remark 5.1.18. §

5.2. p-adic L-functions

Definition 5.2.1. §

Definition 5.2.2. §

Lemma 5.2.3. §

Definition 5.2.4. §

Proposition 5.2.5. §

Remark 5.2.6. §

Definition 5.2.7. §

Remark 5.2.8. §

Remark 5.2.9. §

Definition 5.2.10. §

Remark 5.2.11. §

Remark 5.2.12. §

Definition 5.2.13. §

Remark 5.2.14. §

Proposition 5.2.15. §

Corollary 5.2.16. §

5.3. Iwasawa power series

Definition 5.3.1. §

Notation 5.3.2. §

Proposition 5.3.3. §

Remark 5.3.4. §

Definition 5.3.5. §

Notation 5.3.6. §

Remark 5.3.7. §

Notation 5.3.8. §

Definition 5.3.9. §

Remark 5.3.10. §

Lemma 5.3.11. §

Proposition 5.3.12. §

Theorem 5.3.13. §

Proposition 5.3.14. §

Corollary 5.3.15. §

Corollary 5.3.16. §

5.4. Coleman theory

Notation 5.4.1. §

Lemma 5.4.2. §

Proposition 5.4.3. §

Definition 5.4.4. §

Lemma 5.4.5. §

Proposition 5.4.6. §

Corollary 5.4.7. §

Corollary 5.4.8. §

Theorem 5.4.9 (Coleman). §

Notation 5.4.10. §

Notation 5.4.11. §

Definition 5.4.12. §

Corollary 5.4.13. §

Lemma 5.4.14. §

Notation 5.4.15. §

Proposition 5.4.16. §

Notation 5.4.17. §

Remark 5.4.18. §

Lemma 5.4.19. §

Notation 5.4.20. §

Proposition 5.4.21. §

Lemma 5.4.22. §

Notation 5.4.23. §

Lemma 5.4.24. §

Chapter 5
Kubota-Leopoldt $p$ -adic $L$ -functions

5.1. $p$ -adic measures

5.2. $p$ -adic $L$ -functions