# An introduction to $p$-adic $L$-functions

Joaquín Rodrigues Jacinto  Université Aix-Marseille  joaquin.rodrigues-
<EMAIL_ADDRESS>Chris Williams  University of Nottingham
<EMAIL_ADDRESS>
###### Abstract

In these expository notes, we give an introduction to $p$-adic $L$-functions
and the foundations of Iwasawa theory. Firstly, we give an (analytic) measure-
theoretic construction of Kubota and Leopoldt’s $p$-adic interpolation of the
Riemann zeta function, a $p$-adic analytic encoding of Kummer’s congruences.
Second, we give Coleman’s (arithmetic) construction of the $p$-adic Riemann
zeta function via cyclotomic units. Finally, we describe Iwasawa’s (algebraic)
construction via Galois modules over the Iwasawa algebra. The _Iwasawa Main
conjecture_ , now a theorem due to Mazur and Wiles, says that these
constructions agree. We will state the conjecture precisely, and give a proof
when $p$ is a Vandiver prime (which conjecturally covers every prime).
Throughout, we try to indicate how the various constructions and arguments
have been generalised, and how they connect to more modern research topics.

###### Contents

  1. 1 Introduction
  2. 2 Complex and $p$-adic $L$-functions
    1. 2.1 Classical $L$-functions
    2. 2.2 Motivating questions for Iwasawa theory
      1. 2.2.1 Special values and arithmetic data
      2. 2.2.2 Iwasawa Main Conjectures
      3. 2.2.3 Applications of $p$-adic methods to classical BSD
    3. 2.3 The Riemann zeta function
    4. 2.4 $p$-adic $L$-functions
      1. 2.4.1 $p$-adic $L$-functions, a first idea
      2. 2.4.2 Ideles, measures and Tate’s thesis
      3. 2.4.3 $p$-adic $L$-functions via measures
  3. 3 Measures and Iwasawa algebras
    1. 3.1 The Iwasawa algebra
    2. 3.2 $p$-adic analysis and Mahler transforms
    3. 3.3 An example: Dirac measures
    4. 3.4 A measure-theoretic toolbox
      1. 3.4.1 Multiplication by $x$
      2. 3.4.2 Multiplication by $z^{x}$
      3. 3.4.3 Restriction to open compact subsets
      4. 3.4.4 Restriction to $\mathbf{Z}_{p}^{\times}$
      5. 3.4.5 The action of $\mathbf{Z}_{p}^{\times}$, $\varphi$ and $\psi$
    5. 3.5 Pseudo-measures
    6. 3.6 Further remarks
  4. 4 The Kubota-Leopoldt $p$-adic $L$-function
    1. 4.1 The measure $\mu_{a}$
    2. 4.2 Restriction to $\mathbf{Z}_{p}^{\times}$
    3. 4.3 Rescaling and removing dependence on $a$
  5. 5 Interpolation at Dirichlet characters
    1. 5.1 Characters of $p$-power conductor
    2. 5.2 Non-trivial tame conductors
    3. 5.3 Analytic functions on $\mathbf{Z}_{p}$ via the Mellin transform
    4. 5.4 The values at $s=1$
      1. 5.4.1 The complex value at $s=1$
      2. 5.4.2 The $p$-adic value at $s=1$
  6. 6 The $p$-adic family of Eisenstein series
  7. 7 The Coleman map
    1. 7.1 Notation and Coleman’s theorem
    2. 7.2 Example: cyclotomic units
    3. 7.3 Proof of Coleman’s theorem
    4. 7.4 Definition of the Coleman map
    5. 7.5 The Kummer sequence, Euler systems and $p$-adic $L$-functions
  8. 8 Iwasawa’s theorem on the zeros of the $p$-adic zeta function
    1. 8.1 Measures on Galois groups
    2. 8.2 The ideal generated by the $p$-adic zeta function
    3. 8.3 Cyclotomic units and Iwasawa’s theorem
  9. 9 Proof of Iwasawa’s theorem
    1. 9.1 Equivariance properties of the Coleman map
      1. 9.1.1 The action of $\mathbf{Z}_{p}$
      2. 9.1.2 The action of Galois
    2. 9.2 The fundamental exact sequence
      1. 9.2.1 The logarithmic derivative
      2. 9.2.2 The fundamental exact sequence
    3. 9.3 Generators for the module of cyclotomic units
  10. 10 The Iwasawa Main Conjecture
    1. 10.1 Structure theory for $\Lambda$-modules
    2. 10.2 The $\Lambda$-modules arising from Galois theory
    3. 10.3 The Main Conjecture
    4. 10.4 The Iwasawa Main Conjecture for Vandiver primes
  11. A Iwasawa’s $\mu$-invariant
    1. A.1 Iwasawa’s theorem
      1. A.1.1 First step
      2. A.1.2 Second step
    2. A.2 Some consequences of Iwasawa’s theorem
  12. B Iwasawa theory for modular forms
    1. B.1 Recapping $\mathrm{GL}(1)$
    2. B.2 Analogues for $\mathrm{GL}(2)$
      1. B.2.1 Analytic
      2. B.2.2 Arithmetic
      3. B.2.3 Algebraic
    3. B.3 Iwasawa theory for elliptic curves
    4. B.4 Further generalisations

##  1 Introduction

The study of $L$-functions, and their special values, has been central in
number theory for 200 years. Their study spans from results in classical
number theory (Gauss’s class number formula and the proof of Dirichlet’s
theorem on primes in arithmetic progressions) to two of the Millennium
problems (the Riemann hypothesis and Birch and Swinnerton-Dyer conjecture).
They are also central in the Langlands program, a vast project connecting the
fields of number theory, geometry and representation theory.

The Birch and Swinnerton-Dyer conjecture is one example of a huge network of
conjectures on the special values of $L$-functions, including the Beilinson
and Bloch–Kato conjectures. At their heart, these problems relate complex
analytic information – the order of vanishing, and special values, of
meromorphic functions – to arithmetic data, such as invariants attached to
algebraic varieties and Galois representations. A fruitful approach to these
problems has been the use of $p$-adic methods. Naively, one might consider
that complex analysis is a ‘bad’ place to do arithmetic, as the integers are
discrete in ${\mathbf{C}}$. This is not the case when one instead considers
finite extensions of $\mathbf{Q}_{p}$. The $p$-adic setting brings extra
flexibility and methods with which to attack these open problems, including
$p$-adic $L$-functions, Euler systems, and Hida and Coleman families. Current
state-of-the-art results towards Birch–Swinnerton-Dyer and Bloch–Kato rely on
these $p$-adic tools.

The study of $p$-adic properties of special values of $L$-functions is
generally known as _Iwasawa theory_. In these notes, we give an introduction
to this subject, focusing on perhaps the most fundamental of all
$L$-functions: the Riemann zeta function $\zeta(s)$. We describe what a
$p$-adic $L$-function is, construct it in this setting, and then describe
Iwasawa’s Main Conjecture in this case. Our exposition is aimed primarily at
graduate students and researchers in number theory who are not experts in the
field, but should also be accessible to advanced undergraduates. Whilst none
of the results here are new – indeed, many of the theorems we present are now
decades old – we try and anchor the theory in the context of current research
activity, indicating how the various concepts we introduce have been
generalised, and where the reader should turn next to learn more.

Let us summarise the main results we cover. The Kubota–Leopoldt $p$-adic
$L$-function is the $p$-adic analogue of the Riemann zeta function. We will
see _three_ constructions of this object, each of a different flavour:

  * $\bullet$

Firstly, in Part I we construct an _analytic_ version of the Kubota–Leopoldt
$p$-adic $L$-function. This has the clearest connection to the classical
complex Riemann zeta function; it is a pseudo-measure
$\zeta_{p}^{\mathrm{an}}$ on $\mathbf{Z}_{p}^{\times}$ that interpolates the
(rational) values $\zeta(1-k)$ for all positive integers $k$. We explain what
this means in §2, and state the existence precisely in Theorem 2.14. The rest
of Part I (§2–6) is devoted to proving this theorem and its generalisation to
Dirichlet characters, and describing some consequences to families of
Eisenstein series.

  * $\bullet$

In §7, we give an _arithmetic_ version of the Kubota–Leopoldt $p$-adic
$L$-function. The _cyclotomic units_ are special elements in cyclotomic
fields. As one considers the tower $\mathbf{Q}_{p}(\mu_{p^{n}})$ of cyclotomic
extensions of $\mathbf{Q}_{p}$, the cyclotomic units fit together into a norm-
compatible system. The _Coleman map_ is a map from towers of units into spaces
of $p$-adic measures. Attached to the cyclotomic units, then, is a natural
pseudo-measure $\zeta_{p}^{\mathrm{arith}}$ on $\mathbf{Z}_{p}^{\times}$; and
one connection between arithmetic and analysis, fully explained in §7, is that
$\zeta_{p}^{\mathrm{an}}=\zeta_{p}^{\mathrm{arith}}$.

  * $\bullet$

Finally, in §10, we give an _algebraic_ construction of the Kubota–Leopoldt
$p$-adic $L$-function, as an ideal $[\zeta_{p}^{\mathrm{alg}}]$ in the Iwasawa
algebra. This ideal arises from the structure of a Galois module over the
Iwasawa algebra; whilst we will not make this explicit, the construction goes
through the $p$-adic Selmer group of a certain Galois representation.

The _Iwasawa Main Conjecture_ says this ideal is generated by the
analytic/arithmetic Kubota–Leopoldt $p$-adic $L$-function, connecting the
analytic, arithmetic and algebraic constructions, and ultimately connecting
special complex $L$-values and Selmer groups. We state this precisely in §10,
and prove it in the special case of Vandiver primes.

The three constructions we give here have all been generalised to other
situations, each spawning a field of study in their own right:

  * $\bullet$

The analytic construction is a prototype for constructions of $p$-adic
measures interpolating special $L$-values. There are very general conjectures
in this direction due to Coates–Perrin-Riou and Panchishkin [CPR89, Coa89,
Pan94]. Constructions of $p$-adic $L$-functions using modular symbols, the
doubling method, or $p$-adic interpolation of Eisenstein classes all fall into
this category.

  * $\bullet$

Cyclotomic units are the simplest example of an Euler system, and the Coleman
map is the prototype for Perrin-Riou’s big logarithm maps (see §7.5). Via
these maps, to any Euler system one can attach an arithmetic $p$-adic
$L$-function. A relationship between the analytic and arithmetic $p$-adic
$L$-functions is typically called an explicit reciprocity law.

  * $\bullet$

The algebraic construction leads to a more systematic study of Selmer groups
as modules over the Iwasawa algebra. It is expected that $p$-adic Selmer
groups are torsion modules over a corresponding Iwasawa algebra, giving an
algebraic $p$-adic $L$-function (the associated characteristic ideal). The
Iwasawa Main Conjecture in general says that the algebraic construction agrees
with the analytic/arithmetic ones.

The primary focus of these notes is the case of $\mathrm{GL}(1)$. We sketch
the generalisations of the analytic, arithmetic and algebraic theories to
$\mathrm{GL}(2)$ (the case of modular forms) in Appendix B.

### Further reading

For more information and detail on Part I of these notes, the reader could
consult Lang’s _Cyclotomic fields I and II_ [Lan90]. The construction of the
Kubota–Leopoldt $p$-adic $L$-function we give here is based on Colmez’s
beautiful lecture notes on the $p$-adic Riemann zeta function [Col] (in
French).

In the case of Iwasawa theory for $\mathrm{GL}(1)$, as primarily treated here,
these notes can serve as a prelude to a number of more advanced treatments,
such as Rubin’s proof of the Main Conjecture using the theory of Euler
systems. We must mention the book _Cyclotomic fields and zeta values_ by
Coates and Sujatha [CS06], which inspired our original course, and whose aim
was to present Rubin’s proof. A canonical book in the field is Washington’s
_An introduction to cyclotomic fields_ [Was97], which introduces further
topics in classical Iwasawa theory that there was not space to treat here. We
give a flavour of such topics in Appendix A.

We summarise generalisations of this theory to $\mathrm{GL}(2)$ (the case of
modular forms) in Appendix B. Since this is of fundamental interest in modern
research, we will make the $\mathrm{GL}(2)$ case the subject of a sequel set
of notes. In this sequel, we will describe in detail the analytic construction
of the $p$-adic $L$-function of a modular form via overconvergent modular
symbols, due to Pollack and Stevens [PS11], and sketch the work of Kato
[Kat04] on the arithmetic and algebraic side. Further topics of interest in
this direction include the close connection between $p$-adic $L$-functions and
$p$-adic Hida/Coleman families/the Coleman–Mazur eigencurve, and for these
topics, the reader is urged to consult Bellaïche’s _The Eigenbook_ [Bel21].

### Acknowledgements

These notes started life as the lecture notes for a course at the London
Taught Course Centre in 2017. We thank the organisers of the LTCC, and the
participants of that course, for their attention and enthusiasm. We would also
like to thank Martin Baric, Keith Conrad and Luis Santiago for their comments
and corrections on earlier drafts of these notes. We first learnt of this
construction of the Kubota–Leopoldt $p$-adic $L$-function from Pierre Colmez’s
notes [Col], and we are grateful to him for allowing us to reproduce them
here.

##  2 Complex and $p$-adic $L$-functions

This introductory section aims to motivate the definition and study of
$p$-adic $L$-functions. We start with a general discussion on complex
$L$-functions and then lean slowly to the $p$-adic world, focusing on the
example of most importance to us in these lectures, namely the Riemann zeta
function.

###  2.1 Classical $L$-functions

In order to understand what an $L$-function is, one should start by looking at
the following list of important examples.

  * –

The _Riemann zeta function_ , the most famous and fundamental of all
$L$-functions, is defined for $s\in{\mathbf{C}}$ by

$\zeta(s)=\sum_{n\geq 1}n^{-s}=\prod_{p}(1-p^{-s})^{-1},$

where the last product – an _Euler product_ – runs over all prime numbers $p$
and the second equality is a consequence of the unique prime factorisation of
integers. The sum converges absolutely for the real part of $s$ greater than
1, making $\zeta$ a holomorphic function in a right half-plane. It can be
meromorphically continued to the whole complex plane, and satisfies a
_functional equation_ , a symmetry relating the values $\zeta(s)$ and
$\zeta(1-s)$.

  * –

Let $F$ be a number field. The _zeta function of $F$_ is

$\zeta_{F}(s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{0\neq
I\subset\mathscr{O}{O}_{F}}N(I)^{-s}=\prod_{\mathfrak{p}}(1-N(\mathfrak{p})^{-s})^{-1},$

where the sum is over all non-zero ideals in the ring of integers, and the
product is over all non-zero prime ideals of $K$. Again, this converges
absolutely for $\mathrm{Re}(s)>1$, can be meromorphically continued to
${\mathbf{C}}$, and satisfies a functional equation relating $\zeta_{F}(s)$
and $\zeta_{F}(1-s)$. The existence of the Euler product again follows from
unique factorisation.

  * –

Let
$\chi:({\mathbf{Z}}/N{\mathbf{Z}})^{\times}\rightarrow{\mathbf{C}}^{\times}$
be a Dirichlet character. We extend $\chi$ to a function
$\chi:{\mathbf{Z}}\rightarrow{\mathbf{C}}$ by setting $\chi(m)=\chi(m\hskip
2.0pt(\mathrm{mod}\hskip 2.0ptN))$ if $m$ is prime to $N$, and $\chi(m)=0$
otherwise. The $L$-function of $\chi$ is

$L(\chi,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{n\geq
1}\chi(n)n^{-s}=\prod_{p}(1-\chi(p)p^{-s})^{-1}.$

Yet again, this converges for $\mathrm{Re}(s)>1$, admits meromorphic
continuation to ${\mathbf{C}}$ (analytic when $\chi$ is non-trivial), and
satisfies a functional equation relating the values at $s$ and $1-s$.

  * –

Let $E/{\mathbf{Q}}$ be an elliptic curve of conductor $N$. One can define an
$L$-function

$L(E,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{n\geq
1}a_{n}(E)n^{-s}=\prod_{p\nmid
N}(1-a_{p}(E)p^{-s}+p^{1-2s})^{-1}\prod_{p|N}L_{p}(s),$

where $a_{p}(E)=p+1-\\#E(\mathbf{F}_{p})$, and the $a_{n}(E)$ are defined
recursively from the $a_{p}(E)$. The factors $L_{p}(s)$ at bad primes $p|N$
are defined as $L_{p}(s)=1$ (resp. $(1-p^{-s})^{-1}$, resp. $(1+p^{-s})^{-1}$)
if $E$ has bad additive (resp. split multiplicative, resp. non-split
multiplicative) reduction at $p$. The function $L(E,s)$ converges for
$\mathrm{Re}(s)>3/2$, admits analytic continuation to ${\mathbf{C}}$, and
satisfies a functional equation relating the values at $s$ and $2-s$.

  * –

Let

$f=\sum_{n\geq 1}a_{n}(f)q^{n}\in S_{k}(\Gamma_{0}(N),\omega_{f})$

be a (normalised) newform of weight $k$, level $N$ and character $\omega_{f}$.
The $L$-function associated to $f$ is given by

$L(f,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{n\geq
1}a_{n}(f)n^{-s}=\prod_{p\nmid
N}(1-a_{p}(f)p^{-s}+\omega_{f}(p)p^{k-1-2s})^{-1}\prod_{p|N}(1-a_{p}(f)p^{-s})^{-1}.$

This converges for $\mathrm{Re}(s)>(k+1)/2$, admits analytic continuation to
${\mathbf{C}}$, and satisfies a functional equation relating the values at $s$
and $k-s$.

The above examples share common features. Any reasonably behaved $L$-function
should have the following basic properties (which can, nevertheless, be
extremely deep) :

  * (1)

An Euler product converging absolutely in a right-half plane;

  * (2)

A meromorphic continuation to the whole complex plane;

  * (3)

A functional equation relating $s$ and $k-s$ for some $k\in{\mathbf{R}}$.

######  Remark 2.1.

More generally, let
$\mathscr{G}_{{\mathbf{Q}}}=\mathrm{Gal}(\overline{{\mathbf{Q}}}/{\mathbf{Q}})$
denote the absolute Galois group of ${\mathbf{Q}}$ and let
$V\in\mathrm{Rep}_{L}\,\mathscr{G}_{{\mathbf{Q}}}$ be a $p$-adic Galois
representation (i.e. a finite dimensional vector space over a finite extension
$L$ of $\mathbf{Q}_{p}$ equipped with a continuous linear action of
$\mathscr{G}_{{\mathbf{Q}}}$). For $\ell\neq p$ a rational prime, one defines
a local factor at $\ell$ as

$L_{\ell}(V,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\det(\mathrm{Id}-\mathrm{Frob}_{\ell}^{-1}\ell^{-s}|V^{I_{\ell}})^{-1},$

where $\mathrm{Frob}_{\ell}$ denotes the arithmetic Frobenius at $\ell$, and
$I_{\ell}$ denotes the inertia group at $\ell$. One defines a local factor at
$p$ as

$L_{p}(V,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\det(\mathrm{Id}-\varphi^{-1}p^{-s}|\mathbf{D}_{\mathrm{cris}}(V))^{-1},$

where this time $\mathbf{D}_{\mathrm{cris}}(V)$ denotes the crystalline module
of $V|_{\mathscr{G}_{{\mathbf{Q}}_{p}}}$ from $p$-adic Hodge theory and
$\varphi$ denotes the crystalline Frobenius. One then defines the global
$L$-function of $V$ as the formal product

$L(V,s)=\prod_{\ell}L_{\ell}(V,s).$

When $V$ is the representation attached to an arithmetic object111For example,
a number field, a Dirichlet character, an elliptic curve, a modular form, or
much more generally – in the spirit of the Langlands program – an automorphic
representation of a reductive group. the $L$-function of the representation is
typically equal to the $L$-function attached to that object; for example,
taking $V=\mathbf{Q}_{p}(\chi)$ (that is, $V=\mathbf{Q}_{p}$ with
$\mathscr{G}_{{\mathbf{Q}}}$ acting through the character $\chi$ via class
field theory), one recovers the Dirichlet $L$-functions described above. See
[Bel09] for an introduction to these topics.

###  2.2 Motivating questions for Iwasawa theory

####  2.2.1 Special values and arithmetic data

There is much interest in the _special values_ of $L$-functions. There are
deep results and conjectures relating special values of $L$-functions to
important arithmetic information, of which a prototypical example is the
following:

######  Theorem 2.2 (Class number formula).

Let $F$ be a number field with $r_{1}$ real embeddings, $r_{2}$ pairs of
complex embeddings, $w$ roots of unity, discriminant $D$, and regulator $R$.
The zeta function $\zeta_{F}$ has a simple pole at $s=1$ with residue

$\mathrm{res}_{s=1}\zeta_{F}(s)=\frac{2^{r_{1}}(2\pi)^{r_{2}}R}{w\sqrt{|D|}}h_{F},$

where $h_{F}$ is the class number of $F$.

On the left-hand side, we have a special value of a complex meromorphic
function, from the world of analysis. On the right-hand side, we have
invariants attached to a number field, from the world of arithmetic. The class
number formula thus provides a beautiful and deep connection between two
different fields of mathematics.

A second famous example of links between special values of $L$-functions and
arithmetic information comes in the form of the _Birch and Swinnerton–Dyer
(BSD) conjecture_. Let $E/{\mathbf{Q}}$ be an elliptic curve. The set of
rational points $E({\mathbf{Q}})$ forms a finitely generated abelian group,
and Birch and Swinnerton–Dyer predicted that

$\mathrm{ord}_{s=1}L(E,s)=\mathrm{rank}_{{\mathbf{Z}}}E({\mathbf{Q}}).$

They also predicted a closer analogue of the class number formula: that the
leading term of the $L$-function can be described in terms of arithmetic
invariants attached to $E$.

Again, the left-hand side is from the world of analysis, and the right-hand
side is from the world of arithmetic, and this stunning prediction is
inherently surprising. The worlds are so different that the analytic
$L$-function defies easy study using arithmetic properties of the elliptic
curve. For example, the left-hand side was not even known to be well-defined
for several decades after the conjecture was formulated; this relies on
analytic continuation of the $L$-function, and even now the only proof we have
goes through another deep connection between arithmetic and analysis, namely
Wiles’ modularity theorem.

####  2.2.2 Iwasawa Main Conjectures

The full BSD conjecture remains open. One of the goals of Iwasawa theory is to
seek and prove $p$-adic analogues of BSD and its generalisations, replacing
complex analysis (which is poorly suited to arithmetic) with $p$-adic analysis
(where arithmetic arises naturally). For each prime $p$, there is a $p$-adic
_Iwasawa Main Conjecture (IMC)_ for the elliptic curve $E$, relating a
$p$-adic analytic $L$-function to certain $p$-adic arithmetic invariants of
$E$:

complex analytic $L$-functionBSDarithmetic invariants of $E$$p$-adic analytic
$L$-functionIMC$p$-adic invariants of $E$

One has many more tools available to attack the bottom row than the top,
including Euler systems, $p$-adic families and eigenvarieties, $p$-adic Hodge
theory and $(\varphi,\Gamma)$-modules, and more. As a result, the $p$-adic
conjectures are much more tractable than their complex counterparts. Indeed,
whilst classical complex BSD remains open, its $p$-adic counterpart – via the
IMC for elliptic curves – has been _proved_ in many cases by Skinner–Urban
(see [SU14]), following work of Kato (see [Kat04]).

####  2.2.3 Applications of $p$-adic methods to classical BSD

Each new $p$-adic Iwasawa Main Conjecture that is proved brings the worlds of
analysis and arithmetic a little closer together. They can also bring us
closer to our original goal of, for example, BSD. Indeed, the current state-
of-the-art results towards BSD have arisen as consequences of Iwasawa theory.

To elaborate, if we want to attack BSD, there are two natural subquestions.

  * (a)

We could try to prove that
$\mathrm{ord}_{s=1}L(E,s)\leq\mathrm{rank}_{{\mathbf{Z}}}E({\mathbf{Q}}).$ A
natural approach is to try to construct enough independent rational points on
the elliptic curve. The theory of _Heegner points_ is based on such an idea.
More recently, the $p$-adic theory of _Stark–Heegner points_ , initiated in
[Dar01], has been used with some success. These constructions tend to give
points of infinite order on $E({\mathbf{Q}})$ if and only if the $L$-function
vanishes to a certain order (for example, a Heegner point has infinite order
if and only if the order of vanishing is precisely 1). These constructions are
beautifully summarised in [Dar04].

  * (b)

Conversely, we could try and prove that
$\mathrm{ord}_{s=1}L(E,s)\geq\mathrm{rank}\ _{{\mathbf{Z}}}E({\mathbf{Q}}).$
In this case we want to _bound_ the number of points. One method for trying to
do this uses _Euler systems_ (see [Rub00] for a comprehensive introduction).
The primary application of the theory of Euler systems is in bounding certain
Galois cohomology groups, known as _Selmer groups_ , which are defined using
local behaviour and can be viewed as a cohomological interpretation of the
group of rational points on $E$. The difference between the Selmer group and
$E({\mathbf{Q}})$ is captured in the _Tate–Shafarevich group_
$\Sha(E/{\mathbf{Q}})$, which is a torsion abelian group that is conjecturally
finite. If the $p$-part of $\Sha(E/{\mathbf{Q}})$ is finite, then the
$p$-Selmer group and the group $E({\mathbf{Q}})$ have the same rank (as
abelian groups), so bounding the Selmer group is equivalent to bounding
$E({\mathbf{Q}})$.

The ideas above have led to special cases of the conjecture; in particular, we
now know it to be true (under some assumptions) when
$\mathrm{ord}_{s=1}L(E,s)\leq 1$ due to work of Kolyvagin, Gross–Zagier and
Murty–Murty (see [Kol88], [GZ86] and [MM91]). More recent research in this
area has led to results towards the converse [Ski20], as well as towards the
leading term formula [JSW17].

######  Remark 2.3.

The study of $p$-adic $L$-functions are common to both (a) and (b). Mazur,
Tate and Teitelbaum formulated a $p$-adic BSD conjecture (see [MTT86]), which
relates the rank of the $p$-Selmer group to the order of vanishing of a
$p$-adic $L$-function at $s=1$. Under finiteness of and existence of a precise
relation between the order of vanishing of the classical and $p$-adic
$L$-functions, both extremely difficult open problems, $p$-adic BSD and
classical BSD are equivalent. The $p$-adic BSD conjecture can be proved using
a version of the IMC for elliptic curves, hence follows from the
aforementioned works [SU14, Kat04].

In these notes, we will focus on the simplest example of the above picture,
namely the Main conjecture for the Riemann zeta function, as formulated by
Iwasawa himself. In the process, we will construct the $p$-adic analogue of
the zeta function on the way to stating the Main Conjecture, which we will
prove for a special case.

###  2.3 The Riemann zeta function

Since the Riemann zeta function will be a central player in the rest of these
notes, we take a brief detour to describe some of the classical theory
surrounding it. We start with the following general result.

######  Theorem 2.4.

Let $f:{\mathbf{R}}_{\geq 0}\longrightarrow{\mathbf{R}}$ be a
$\mathscr{C}^{\infty}$-function such that
$f,f^{\prime},f^{\prime\prime},\dots$ (i.e. $f$ and all of its derivatives)
all decay exponentially at infinity (a ‘Schwartz function’). Let

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

be the usual Gamma function. The function

$L(f,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}f(t)t^{s-1}dt,\hskip
12.0pts\in{\mathbf{C}},$

which converges to a holomorphic function for Re$(s)>0$, has an analytic
continuation to the whole complex plane, and

$L(f,-n)=(-1)^{n}\frac{d^{n}}{dt^{n}}f(0).$

We call $L(f,s)$ the _Mellin transform_ of $f$.

###### Proof.

To show analytic continuation, we claim that when Re$(s)>1$, we have

$L(f,s)=-L(f^{\prime},s+1),$

where $f^{\prime}=df/dt$. This is an exercise in integration by parts, using
the identity $\Gamma(s)=(s-1)\Gamma(s-1)$, and gives the analytic continuation
to all of ${\mathbf{C}}$ by iteration. Finally, iterating the same identity
$n+1$ times shows that

$\displaystyle L(f,-n)$ $\displaystyle=(-1)^{n+1}L(f^{(n+1)},1)$
$\displaystyle=(-1)^{n+1}\int_{0}^{\infty}f^{(n+1)}(t)dt$
$\displaystyle=(-1)^{n}f^{(n)}(0)$

from the fundamental theorem of calculus, giving the result. ∎

Now we pick a specific choice of $f$, namely, we let

$f(t)=\frac{t}{e^{t}-1}=\sum_{n\geq 0}B_{n}\frac{t^{n}}{n!},$

the generating function for the Bernoulli numbers $B_{n}$.

######  Remark 2.5.

The Bernoulli numbers are highly combinatorial, and satisfy recurrence
relations that ensures they are _rational_ numbers; for example, the first few
are

$B_{0}=1,B_{1}=-\frac{1}{2},B_{2}=\frac{1}{6},B_{3}=0,B_{4}=-\frac{1}{30},...$

For $k\geq 3$ odd, $B_{k}=0$.

We want to plug this function into Theorem 2.4, and for this, we require222We
thank Keith Conrad for pointing out this elegant proof.:

######  Lemma 2.6.

The function $f(t)$ and all of its derivatives decay exponentially at
infinity.

###### Proof.

For $t>0$, we may expand $f(t)$ as a geometric series

$\displaystyle f(t)$ $\displaystyle=t(e^{-t}+e^{-2t}+e^{-3t}+\cdots)=:tF(t).$

Note that $f^{\prime}(t)=F(t)+tF^{\prime}(t)$, and
$f^{\prime\prime}(t)=2F^{\prime}(t)+tF^{\prime\prime}(t)$; arguing inductively
we see

$\displaystyle f^{(n)}(t)$ $\displaystyle=nF^{(n-1)}(t)+tF^{(n)}(t)$
$\displaystyle=n(e^{-t}+e^{-2t}+\cdots)+(-1)^{n}t(e^{-t}+2^{n}e^{-2t}+3^{n}e^{-3t}+\cdots)$
$\displaystyle\sim(-1)^{n}te^{-t}\qquad\text{ as }t\to\infty.$

This decays exponentially. ∎

######  Lemma 2.7.

For the choice of $f$ as above, we have

$(s-1)\zeta(s)=L(f,s-1).$

###### Proof.

We use the classical formula for $\Gamma(s)$ above. Substituting $t$ for $nt$
and rearranging, we obtain

$n^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-nt}t^{s-1}dt.$

Now, when $\mathrm{Re}(s)$ is sufficiently large, we can write

$\zeta(s)=\sum_{n\geq 1}n^{-s}=\frac{1}{\Gamma(s)}\sum_{n\geq
1}\int_{0}^{\infty}e^{-nt}t^{s-1}dt=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\bigg{(}\sum_{n\geq
1}e^{-nt}\bigg{)}t\cdot t^{s-2}dt,$

and the result now follows from the identity

$\frac{1}{e^{t}-1}=\sum_{n\geq 1}e^{-nt}.\qed$

From the theorem above, we immediately obtain:

######  Corollary 2.8.

For $n\geq 0$, we have

$\zeta(-n)=-\frac{B_{n+1}}{n+1},$

In particular, $\zeta(-n)\in{\mathbf{Q}}$ for $n\geq 0$, and $\zeta(-n)=0$ if
$n\geq 2$ is even.

###  2.4 $p$-adic $L$-functions

As explained in the introduction, $p$-adic $L$-functions are excellent tools
to study difficult problems connecting special values of $L$-functions. In
this section, we explain what a $p$-adic $L$-function is and the properties it
should satisfy.

####  2.4.1 $p$-adic $L$-functions, a first idea

The complex $\zeta$-function is a function

$\zeta:{\mathbf{C}}\longrightarrow{\mathbf{C}}$

with complex analytic properties which is rational at negative integers. Since
${\mathbf{Z}}$ is a common subset of both ${\mathbf{C}}$ and
$\mathbf{Z}_{p}\subseteq\mathbf{C}_{p}$, it is natural to ask if there exists
a function

$\zeta_{p}:\mathbf{Z}_{p}\longrightarrow\mathbf{C}_{p}$

that is ‘$p$-adic analytic’ (in some sense to be defined) and which agrees
with the complex $L$-function at negative integers in the sense that

$\zeta_{p}(1-n)=(*)\cdot\zeta(1-n),$

for some explicit factor $(*)$. We would say that such a function
‘$p$-adically interpolates the special values of $\zeta(s)$’. Ideally, one
would like these properties to uniquely characterise $\zeta_{p}$.

####  2.4.2 Ideles, measures and Tate’s thesis

In practice, there is no _single_ analytic function on $\mathbf{Z}_{p}$ that
interpolates all of the special values333Rather, there are $p-1$ different
analytic functions $\zeta_{p,1},\hskip 1.0pt...,\hskip 1.0pt\zeta_{p,p-1}$ on
$\mathbf{Z}_{p}$, and $\zeta_{p,i}$ interpolates only the values $\zeta(-k)$
for which $k\equiv i\hskip 2.0pt(\mathrm{mod}\hskip 2.0ptp-1)$., as we will
explain in Section 5.3. Instead, a better way of thinking about $L$-functions
is to use a viewpoint initiated by Tate in his thesis [Tat50] (and later
independently by Iwasawa; see [Iwa52]). This viewpoint sees $L$-functions as
_measures on ideles_ , and allows one to package together _all_ Dirichlet
$L$-functions, including the Riemann zeta function, into a single object. We
will give a brief account of the classical theory here, but for fuller
accounts, one should consult the references above.

We begin with the following observations.

######  Lemma 2.9.

  * (i)

Any Dirichlet character
$\chi:({\mathbf{Z}}/N{\mathbf{Z}})^{\times}\longrightarrow{\mathbf{C}}^{\times}$
can be seen as a character

$\chi:\prod_{\ell\hskip
1.0pt\mathrm{prime}}{\mathbf{Z}}_{\ell}^{\times}\longrightarrow{\mathbf{C}}^{\times}.$

  * (ii)

There is an identification of ${\mathbf{C}}$ with
$\mathrm{Hom}_{\mathrm{cts}}({\mathbf{R}}_{>0},{\mathbf{C}}^{\times})$ by
sending $s$ to $x\mapsto x^{s}$.

###### Proof.

To see part (i), suppose that $N=\ell^{n}$ is a power of some prime $\ell$;
then we can see $\chi$ as a function on ${\mathbf{Z}}_{\ell}^{\times}$ via the
identification

${\mathbf{Z}}_{\ell}^{\times}\cong({\mathbf{Z}}/\ell^{n}{\mathbf{Z}})^{\times}\times(1+\ell^{n}{\mathbf{Z}}_{\ell}).$

The general case follows from the Chinese remainder theorem.

We turn to part (ii). For $s\in{\mathbf{C}}$, the function $x\mapsto x^{s}$ is
visibly a continuous character on ${\mathbf{R}}_{>0}$. We want to show that
all such characters are of this form. After taking a logarithm, this is
equivalent to showing that all continuous homomorphisms (of additive groups)
$g:{\mathbf{R}}\rightarrow{\mathbf{C}}$ are of the form $g(x)=xg(1)$, which is
easily shown by directly computing the values of $g$ on ${\mathbf{Q}}$ and
extending by continuity. ∎

By the identification of ${\mathbf{C}}$ with
$\mathrm{Hom}_{\mathrm{cts}}({\mathbf{R}}_{>0},{\mathbf{C}}^{\times})$ one can
view $\zeta$ as a function

$\displaystyle\zeta:\mathrm{Hom}_{\mathrm{cts}}({\mathbf{R}}_{>0},{\mathbf{C}}^{\times})$
$\displaystyle\longrightarrow{\mathbf{C}}$ $\displaystyle\big{[}x\mapsto
x^{s}\big{]}$ $\displaystyle\longmapsto\zeta(s).$

But we can add in Dirichlet characters using the following.

######  Proposition 2.10.

Under the identifications above, each pair $(\chi,s)$, where $\chi$ is a
Dirichlet character and $s\in{\mathbf{C}}$, corresponds to a (unique)
continuous character

$\displaystyle\kappa_{\chi,s}:{\mathbf{R}}_{>0}\times\prod_{\ell\hskip
1.0pt\mathrm{prime}}{\mathbf{Z}}_{\ell}^{\times}$
$\displaystyle\longrightarrow{\mathbf{C}}^{\times}$ $\displaystyle(x,y)$
$\displaystyle\longmapsto x^{s}\chi(y),$

where we equip the source with the product topology. All continuous characters
on this group are of this form.

###### Proof.

The first assertion is immediate from above. To see the converse, let $\kappa$
be such a character. Then we already know that the restriction of $\kappa$ to
${\mathbf{R}}_{>0}$ must be of the form $x\mapsto x^{s}$. Furthermore, we have
an isomorphism of topological groups

$\prod_{\ell\hskip
1.0pt\mathrm{prime}}{\mathbf{Z}}_{\ell}^{\times}\cong\lim_{\longleftarrow}({\mathbf{Z}}/M{\mathbf{Z}})^{\times},$

where the right hand side is equipped with the profinite topology, and by
taking a sufficiently small open neighbourhood of $1$ in
${\mathbf{C}}^{\times}$ we see that any continuous character $\kappa^{\prime}$
from this to ${\mathbf{C}}^{\times}$ must have open kernel. Hence the kernel
has finite index, and $\kappa$ descends to the (finite) quotient, which one
can check is of the form $({\mathbf{Z}}/N{\mathbf{Z}})^{\times}$ for some $N$,
giving rise to a Dirichlet character $\chi$ of conductor $N$. Then
$\kappa=\kappa_{\chi,s}$. ∎

The product space is more usually written as follows.

######  Definition 2.11.

Define the _ideles_ $\mathbb{A}^{\times}$ of ${\mathbf{Q}}$ to be

$\displaystyle\mathbb{A}^{\times}$
$\displaystyle=\mathbb{A}_{{\mathbf{Q}}}^{\times}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}={\mathbf{R}}^{\times}\times\sideset{}{{}^{\prime}}{\prod}_{\ell\hskip
1.0pt\mathrm{prime}}{\mathbf{Q}}_{\ell}^{\times}$
$\displaystyle=\big{\\{}(x_{{\mathbf{R}}},x_{2},x_{3},x_{5},...):x_{\ell}\in{\mathbf{Z}}_{\ell}^{\times}\text{
for all but finitely many }\ell\big{\\}}.$

(The prime on the product denotes _restricted product_ , which indicates the
almost everywhere integral property in the definition). It’s a good exercise
to check that:

######  Proposition 2.12 (Strong approximation).

There is a decomposition

$\mathbb{A}^{\times}\cong{\mathbf{Q}}^{\times}\times{\mathbf{R}}_{>0}\times\prod_{\ell\hskip
1.0pt\mathrm{prime}}{\mathbf{Z}}_{\ell}^{\times}.$

Hence all continuous characters

${\mathbf{Q}}^{\times}\backslash\mathbb{A}^{\times}\longrightarrow{\mathbf{C}}^{\times}$

are of the form $\kappa_{\chi,s}$ as above, where $\chi$ is a Dirichlet
character and $s\in{\mathbf{C}}$.

Now we can consider _all_ Dirichlet $L$-functions _at once_ via the function

$\displaystyle
L:\mathrm{Hom}_{\mathrm{cts}}({\mathbf{Q}}^{\times}\backslash\mathbb{A}^{\times},{\mathbf{C}}^{\times})$
$\displaystyle\longrightarrow{\mathbf{C}}$ $\displaystyle\kappa_{\chi,s}$
$\displaystyle\longmapsto L(\chi,s).$

In the framework of Tate, this function $L$ can be viewed as integrating
$\kappa_{\chi,s}$ against the _Haar measure_ on
${\mathbf{Q}}^{\times}\backslash\mathbb{A}^{\times}$. In his thesis, Tate
showed that properties such as the analytic continuation and functional
equations of Dirichlet $L$-functions by using harmonic analysis on measures.
Indeed, the idelic formulation gives a beautiful conceptual explanation for
the appearance of the $\Gamma$-functions and powers of $2\pi i$ in the
functional equation of the zeta function; these factors are the ‘Euler factors
at the archimedean place’. The measure-theoretic perspective has proven to be
a powerful method of defining and studying automorphic $L$-functions in wide
generality.

######  Remark 2.13.

  1. (1)

If $K$ is any number field, one can analogously define the ideles
$\mathbb{A}_{K}^{\times}$ of $K$ as the restricted product
$\prod_{v}K_{v}^{\times}$ over all places $v$ of $K$. Continuous homomorphisms
$K^{\times}\backslash\mathbb{A}_{K}^{\times}\rightarrow{\mathbf{C}}^{\times}$
are called _Hecke characters_ or _Größencharacters_. They are examples of
automorphic forms for $\mathrm{GL}_{1}/K$.

  2. (2)

By Class Field Theory, the idele class group
$K^{\times}\backslash\mathbb{A}_{K}^{\times}$ injects (with dense image) into
$\mathscr{G}_{K}^{\rm
ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Gal}(K^{\rm
ab}/K)$, where $K^{\rm ab}$ denotes the maximal abelian extension of $K$.
Since any character of $\mathscr{G}_{K}$ must factor through
$\mathscr{G}_{K}^{\rm ab}$, we have an injective restriction map
$\mathrm{Hom}_{\mathrm{cts}}(\mathscr{G}_{K},{\mathbf{C}}^{\times})\to\mathrm{Hom}_{\mathrm{cts}}(K^{\times}\backslash\mathbb{A}_{K}^{\times},{\mathbf{C}}^{\times})$,
where $\mathscr{G}_{K}=\mathrm{Gal}(\overline{K}/K)$ denotes the absolute
Galois group of $K$. Class Field Theory also implies that this map becomes an
isomorphism when one replaces the absolute Galois group of $K$ by its Weil
group. We can then package Dirichlet $L$-functions over $K$ into a complex
analytic function on the space of one dimensional complex representations of
the Weil group of $K$.

####  2.4.3 $p$-adic $L$-functions via measures

To obtain a $p$-adic version of this picture, a natural thing to do is to look
at continuous characters from
${\mathbf{Q}}^{\times}\backslash\mathbb{A}^{\times}$ into
$\mathbf{C}_{p}^{\times}$ (rather than ${\mathbf{C}}^{\times}$). Again, such a
function corresponds to a function on
${\mathbf{R}}_{>0}\times\prod{\mathbf{Z}}_{\ell}^{\times}$. Since
${\mathbf{R}}_{>0}$ is connected and $\mathbf{C}_{p}$ is totally disconnected,
the restriction of any such character to ${\mathbf{R}}_{>0}$ is trivial. Also
using topological arguments we find that the restriction to $\prod_{\ell\neq
p}{\mathbf{Z}}_{\ell}^{\times}$ factors through a finite quotient, so gives
rise to some Dirichlet character of conductor prime to $p$. This leaves the
restriction to $\mathbf{Z}_{p}^{\times}$, which is by far the most interesting
part.

In the measure-theoretic viewpoint of $L$-functions, it is then natural to
look for an analytic444Precisely, since
${{\mathbf{Z}}_{p}^{\times}}=\mu_{p-1}\times(1+p\mathbf{Z}_{p})$, the space
$\mathrm{Hom}_{\mathrm{cts}}(\mathbf{Z}_{p}^{\times},\mathbf{C}_{p}^{\times})$
can be identified with $p-1$ copies of the open unit ball in $\mathbf{C}_{p}$
(see the exercises). It carries the structure of a rigid analytic $p$-adic
space, and a function on this space is _rigid analytic_ if it can be written
as a convergent power series on each ball. Such an analytic function will be a
measure if these coefficients are bounded. function

$\zeta_{p}:\mathrm{Hom}_{\mathrm{cts}}(\mathbf{Z}_{p}^{\times},\mathbf{C}_{p}^{\times})\longrightarrow\mathbf{C}_{p}$

in such a way that

$\zeta_{p}(x\mapsto x^{k})=(*)\cdot\zeta(1-k),\hskip 12.0ptk\geq 1$

for an explicit factor $(*)$, that is, for a function on $p$-adic characters
interpolating the values $\zeta(-k)$ for $k\geq 0$. We say such a function is
a _measure on $\mathbf{Z}_{p}^{\times}$_. In equivalent and more elementary
terms, a measure on ${{\mathbf{Z}}_{p}^{\times}}$ is an element of the
continuous dual of the space of continuous functions on
${{\mathbf{Z}}_{p}^{\times}}$. We will prove:

######  Theorem 2.14 (Kubota-Leopoldt, Iwasawa).

There exists a (pseudo-)measure555 _Pseudo-measures_ will be defined in
Section 4. Roughly speaking, such an object is a measure that is allowed to
have simple poles. $\zeta_{p}$ on $\mathbf{Z}_{p}^{\times}$ such that, for all
$k>0$,

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\zeta_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\zeta_{p}(x\mapsto
x^{k})=\big{(}1-p^{k-1}\big{)}\zeta(1-k).$

######  Remark 2.15.

Note that we removed the Euler factor at $p$. This is a general phenomenon
appearing in the theory of $p$-adic $L$-functions.

From such an object, we can build the (meromorphic) functions on
$\mathbf{Z}_{p}$ we were initially looking for. But now, we have much more,
and the power of the measure-theoretic approach becomes obvious:

######  Theorem 2.16.

Let $\chi$ be a Dirichlet character of conductor $p^{n}$, $n\geq 0$, viewed as
a locally constant character on $\mathbf{Z}_{p}^{\times}$. Then, for all
$k>0$,

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{p}=\big{(}1-\chi(p)p^{k-1}\big{)}L(\chi,1-k).$

In other words, when viewed as a measure the Kubota–Leopoldt $p$-adic
$L$-function is a _single_ $p$-adic gadget that encodes the special values not
only of the Riemann zeta function, but also of _all_ of its twists by
characters of $p$-power conductor. This is pretty magic! Indeed, even though
one only uses the values $\zeta(-k)$ to construct the measure $\zeta_{p}$,
Theorem 2.16 affirms that its values at infinitely many different points are
still related to the complex $L$-function. We will also see a formula of
Leopoldt showing another striking resemblance when evaluating at the character
$x\mapsto\chi(x)$.

To complete the picture given in §2.4.2, one considers Dirichlet characters of
conductor prime to $p$. The ideas that go into the proof of Theorem 2.16 can
also be used to show:

######  Theorem 2.17.

Let $D>1$ be any integer coprime to $p$, and let $\eta$ denote a (primitive)
Dirichlet character of conductor $D$. There exists a unique measure
$\zeta_{\eta}$ on $\mathbf{Z}_{p}^{\times}$ with the following interpolation
property: for all primitive Dirichlet characters $\chi$ with conductor $p^{n}$
for some $n\geq 0$, we have, for all $k>0$,

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{\eta}=\big{(}1-\chi\eta(p)p^{k-1}\big{)}L(\chi\eta,1-k).$

######  Remark 2.18.

Let $({\mathbf{Z}}/D{\mathbf{Z}})^{\times\wedge}$ denote the space of
characters on $({\mathbf{Z}}/D{\mathbf{Z}})^{\times}$. The measures given by
Theorem 2.17 can be seen as functions on
$\mathrm{Hom}_{\mathrm{cts}}({{\mathbf{Z}}_{p}^{\times}},\mathbf{C}_{p}^{\times})\times({\mathbf{Z}}/D{\mathbf{Z}})^{\times\wedge}$
and they are compatible with respect to the natural maps
$({\mathbf{Z}}/E{\mathbf{Z}})^{\times\wedge}\to({\mathbf{Z}}/D{\mathbf{Z}})^{\times\wedge}$
for $E|D$. This shows that they define a function on

$\displaystyle\mathrm{Hom}_{\mathrm{cts}}({{\mathbf{Z}}_{p}^{\times}},\mathbf{C}_{p}^{\times})\times\varinjlim_{(D,p)=1}\big{(}{\mathbf{Z}}/D{\mathbf{Z}}\big{)}^{\times\wedge}$
$\displaystyle=\mathrm{Hom}_{\mathrm{cts}}({{\mathbf{Z}}_{p}^{\times}},\mathbf{C}_{p}^{\times})\times\big{(}\prod_{\ell\neq
p}{\mathbf{Z}}_{\ell}^{\times}\big{)}^{\wedge}$
$\displaystyle=\mathrm{Hom}_{\mathrm{cts}}({\mathbf{Q}}^{\times}\backslash\mathbb{A}_{\mathbf{Q}}^{\times},\mathbf{C}_{p}^{\times}).$

In other words, they give a measure on the idele class group of
${\mathbf{Q}}$.

######  Remark 2.19.

Note that if $k\equiv\ell\hskip 2.0pt(\mathrm{mod}\hskip 2.0ptp^{m-1}(p-1))$,
then $x^{k}\equiv x^{\ell}\hskip 2.0pt(\mathrm{mod}\hskip 2.0ptp^{m})$ for any
$x\in\mathbf{Z}_{p}^{\times}$. In particular, these theorems tell us that the
special values of $L$-functions satisfy congruences

$(1-\eta(p)p^{k-1})L(\eta,1-k)\equiv(1-\eta(p)p^{\ell-1})L(\eta,1-\ell)\hskip
2.0pt(\mathrm{mod}\hskip 2.0ptp^{m}).$

For the Riemann zeta function, these are the _Kummer congruences_ , which
played a role in his classification of irregular primes, and which provided
significant motivation for Theorem 2.14. This gives an alternative way of
viewing $p$-adic $L$-functions: as a $p$-adic analytic object that packages
together systematic congruences between $L$-values.

######  Remark 2.20.

The measure-theoretic interpretation of $p$-adic $L$-functions allows us to
generalise to number fields in a clean and conceptual way.

  * –

Let $F_{\infty}={\mathbf{Q}}(\mu_{p^{\infty}})$ denote the field extension of
${\mathbf{Q}}$ obtained by adjoining all $p$-power roots of unity. This is a
Galois extension of ${\mathbf{Q}}$ with
$\mathrm{Gal}(F_{\infty}/{\mathbf{Q}})\cong\mathbf{Z}_{p}^{\times}$ via the
cyclotomic character (see, for example, the notation at the start of Part II).
Under this isomorphism, we can see $\zeta_{p}$ as a pseudo-measure on
$\mathrm{Gal}(F_{\infty}/{\mathbf{Q}})$.

  * –

Note that $F_{\infty}$ is the maximal abelian extension of ${\mathbf{Q}}$ that
is unramified outside $p$. Indeed, the Kronecker-Weber theorem states that if
$K/{\mathbf{Q}}$ is abelian, then $K\subset{\mathbf{Q}}(\mu_{m})$ for some
minimal $m$. By the ramification properties of cyclotomic fields, if a prime
$\ell$ ramifies in $K$, then $\ell|m$, and hence if $K$ is unramified outside
$p$, there exists some $n$ such that
$K\subset{\mathbf{Q}}(\mu_{p^{n}})=F_{n}\subset F_{\infty}$.

  * –

Now let $K/{\mathbf{Q}}$ be a number field; then the $p$-adic analogue of the
zeta function $\zeta_{K}(s)$ should be a pseudo-measure on
$\mathrm{Gal}(K^{{\rm ab},p}/K)$, where $K^{{\rm ab},p}$ is the maximal
abelian extension of $K$ unramified outside primes above $p$. This is also the
natural setting for the construction of $p$-adic $L$-functions of other
arithmetic objects, such as elliptic curves or modular forms over $K$. It is
possible to translate measures on this Galois group into measures on
$(\mathscr{O}_{K}\otimes\mathbf{Z}_{p})^{\times}$ or analytic functions on
$\mathscr{O}_{K}\otimes\mathbf{Z}_{p}$, but this is not as clean; over
${\mathbf{Q}}$, things work nicely since the class number is 1 and the totally
positive global units are trivial. For a general number field $K$, the strong
approximation theorem takes a more complicated form, and we end up with a
collection of measures/analytic functions indexed by a class group. For an
example of the theory for modular forms over imaginary quadratic fields, see
[Loe14] (for measures/distributions) or [BW19, §3] (for analytic functions).

Part I: The Kubota–Leopoldt $p$-adic $L$-function

Let $p$ be an odd prime. In this part, we give a construction of the
Kubota–Leopoldt $p$-adic $L$-function and the $p$-adic $L$-functions of
Dirichlet characters. In Section 3, we introduce the necessary formalism of
$p$-adic measures and Iwasawa algebras, and show that there is an isomorphism
from the Iwasawa algebra of $\mathbf{Z}_{p}$ to the space
$\mathbf{Z}_{p}[\\![T]\\!]$ of power series over $\mathbf{Z}_{p}$, given by
the Mahler transform. In Section 4, we construct a pseudo-measure on
$\mathbf{Z}_{p}^{\times}$ that interpolates the values of the Riemann zeta
function at negative integers. In Section 5, we show moreover that this
pseudo-measure interpolates the values $L(\chi,-k)$ for $\chi$ a Dirichlet
character of $p$-power conductor. Further, if $\eta$ is a Dirichlet character
of conductor prime to $p$, we construct a measure on $\mathbf{Z}_{p}$ that
interpolates the values $L(\chi\eta,-k)$ as $\chi$ runs over Dirichlet
characters of $p$-power conductor. Finally, in Section 5.3 we rephrase the
construction in terms of analytic functions on $\mathbf{Z}_{p}$ via the Mellin
transform.

##  3 Measures and Iwasawa algebras

In §2, we explained that a natural way to construct $p$-adic $L$-functions is
to construct suitable $p$-adic measures on $\mathbf{Z}_{p}^{\times}$. In this
section, we introduce the formalism of the theory of $p$-adic analysis that we
will be using in the sequel. Whilst some of the results of this section may
appear a little dry in isolation, fluency in the measure-theoretic language
will greatly help us simplify calculations that would otherwise be very
technical.

###  3.1 The Iwasawa algebra

We fix a finite extension $L$ of $\mathbf{Q}_{p}$, equipped with the $p$-adic
valuation normalised such that $v_{p}(p)=1$; this will serve as the
coefficient field. We write $\mathscr{O}_{L}$ for its ring of integers. Let
$G$ be a profinite abelian group (e.g. $G=\mathbf{Z}_{p}$ or
$G=\mathbf{Z}_{p}^{\times}$, which are the examples of most interest to us).

We denote by $\mathscr{C}(G,L)$ the space of continuous functions $\phi:G\to
L$, equipped with the valuation $v_{\mathscr{C}}(\phi)=\inf_{x\in
G}v_{p}(\phi(x))$ (giving rise to the sup norm). This valuation makes
$\mathscr{C}(G,L)$ into an $L$-Banach space, i.e. a complete topological
$L$-vector space whose topology is defined by a valuation $v_{\mathscr{C}}$
satisfying

  * (i)

$v_{\mathscr{C}}(f)=+\infty$ if and only if $f=0$;

  * (ii)

$v_{\mathscr{C}}(f+g)\geq\mathrm{min}(v_{\mathscr{C}}(f),\
v_{\mathscr{C}}(g))$ for all $f,g\in\mathscr{C}(G,L)$;

  * (iii)

and $v_{\mathscr{C}}(\lambda f)=v_{p}(\lambda)+v_{\mathscr{C}}(f)$ for all
$\lambda\in L,f\in\mathscr{C}(G,L)$.

######  Definition 3.1.

We define the space $\mathscr{M}(G,L)$ of _$L$ -valued measures on $G$_ as the
dual $\mathrm{Hom}_{\rm cts}(\mathscr{C}(G,L),L)$ equipped with the strong
topology. If $\phi\in\mathscr{C}(G,L)$ and $\mu\in\mathscr{M}(G,L)$, the
evaluation of $\mu$ at $\phi$ will be denoted by

$\int_{G}\phi(x)\cdot\mu(x),$

or by $\int_{G}\phi\cdot\mu$ if the variable of integration is clear from the
context (in the literature, this is sometimes written alternatively as
$\int_{G}\phi\cdot d\mu$). We say that an element $\mu\in\mathscr{M}(G,L)$ is
an $\mathscr{O}_{L}$-valued measure, and write
$\mu\in\mathscr{M}(G,\mathscr{O}_{L})$, if $\mu$ takes values in
$\mathscr{O}_{L}$. Since $G$ is compact and measures are continuous (or,
equivalently, bounded), we have that
$\mathscr{M}(G,L)=\mathscr{M}(G,\mathscr{O}_{L})\otimes_{\mathscr{O}_{L}}L$.
We will be mainly concerned with $\mathscr{O}_{L}$-valued functions and
measures.

######  Remark 3.2.

We can think of measures as additive functions

$\mu:\\{\text{compact open subsets of $G$}\\}\longrightarrow\mathscr{O}_{L}.$

Indeed, let $\mu$ be such a function and let
$\phi\in\mathscr{C}(G,\mathscr{O}_{L})$. We will see how to integrate $\phi$
against $\mu$. Assume first $\phi$ is locally constant; then there is some
open subgroup $H$ of $G$ such that $\phi$ can be viewed as a function on
$G/H$. We define the integral of $\phi$ against $\mu$ to be

$\int_{G}\phi\cdot\mu\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{[a]\in
G/H}\phi(a)\mu(aH).$

In general, we can write $\phi=\lim_{n\rightarrow\infty}\phi_{n}$, where each
$\phi_{n}$ is locally constant. Then we can define

$\int_{G}\phi\cdot\mu\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{n\rightarrow\infty}\int_{G}\phi_{n}\cdot\mu,$

which exists and is independent of the choice of $\phi_{n}$. This defines an
element in $\mathscr{M}(G,\mathscr{O}_{L})$.

Conversely, if $\mu\in\mathscr{M}(G,\mathscr{O}_{L})$ and $U\subset G$ is an
open compact set, one defines
$\mu(U)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{G}\mathbf{1}_{U}(x)\cdot\mu(x)$,
where $\mathbf{1}_{U}(x)$ denotes the characteristic function of $U$.

######  Proposition 3.3.

We have an isomorphism

$\mathscr{M}(G,\mathscr{O}_{L})\cong\varprojlim_{H}\mathscr{O}_{L}[G/H],$

where the limit is over all open subgroups of $G$.

###### Proof.

Let $\mu$ be a measure, and let $H$ be an open subgroup of $G$. We define an
element $\lambda_{H}$ of $\mathscr{O}_{L}[G/H]$ by setting

$\lambda_{H}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{[a]\in
G/H}\mu(aH)[a].$

By the additivity property of $\mu$, we see that
$(\lambda_{H})_{H}\in\varprojlim\mathscr{O}_{L}[G/H]$, so we have a map from
measures to this inverse limit.

Conversely, given such an element $\lambda$ of the inverse limit, write
$\lambda_{H}$ for its image in $\mathscr{O}_{L}[G/H]$ under the natural
projection. Then

$\lambda_{H}=\sum_{[a]\in G/H}c_{a}[a].$

We define

$\mu(aH)=c_{a}.$

Since the $\lambda_{H}$ are compatible under projection maps, this defines an
additive function on the open compact subgroups of $G$, i.e. an element
$\mu\in\mathscr{M}(G,\mathscr{O}_{L})$. ∎

######  Definition 3.4.

We define the _Iwasawa algebra of $G$_ to be

$\Lambda(G)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{\begin{subarray}{c}\longleftarrow\\\
H\end{subarray}}\mathscr{O}_{L}[G/H].$

(Note that we suppress $L$ from the notation).

######  Remark 3.5.

The Iwasawa algebra $\Lambda(\mathbf{Z}_{p})$ has a natural
$\mathscr{O}_{L}$-algebra structure, and hence by transport of structure we
obtain such a structure on $\mathscr{M}(\mathbf{Z}_{p},\mathscr{O}_{L}).$ As
it happens classically when identifying the group algebra of a finite group
with the dual of its space of functions, the algebra structure on the space of
measures can be described directly via _convolution of measures_. For a
general profinite (abelian) group $G$, given two measures
$\mu,\lambda\in\mathscr{M}(G,\mathscr{O}_{L})$, one defines their convolution
$\mu*\lambda$ to be

$\int_{G}\phi\cdot(\mu*\lambda)=\int_{G}\left(\int_{G}\phi(x+y)\cdot\lambda(y)\right)\cdot\mu(x).$

One checks that this does give an algebra structure and that the isomorphism
above is an isomorphism of $\mathscr{O}_{L}$-algebras.

###  3.2 $p$-adic analysis and Mahler transforms

In this section we establish a link between $p$-adic measures on
$\mathbf{Z}_{p}$ and power series.

######  Definition 3.6.

For $x\in\mathbf{Z}_{p}$, let

$\begin{pmatrix}x\\\
n\end{pmatrix}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{x(x-1)\cdots(x-n+1)}{n!}\text{
for }n\geq 1,\qquad\text{and }\begin{pmatrix}x\\\ 0\end{pmatrix}=1.$

One easily checks that $x\mapsto{x\choose n}$ defines an element in
$\mathscr{C}(\mathbf{Z}_{p},\mathbf{Z}_{p})$ of valuation
$v_{\mathscr{C}}\big{(}{x\choose n}\big{)}=0$.

The following theorem is fundamental in all that follows. It says that the
functions ${x\choose n}$ form an orthonormal basis 666If $B$ is an $L$-Banach
space, an orthonormal basis of $B$ is a collection $(e_{i})_{i\in I}$ such
that $(a_{i})_{i\in I}\mapsto\sum_{i\in I}a_{i}e_{i}$ defines an isometry
between $\ell^{0}_{\infty}(I,L)$ and $B$, where $\ell^{0}_{\infty}(I,L)$ is
the set of sequences in $L$ indexed by $I$ that tend to 0 (in a sense that
depends on $I$). One can show that every $L$-Banach space $B$ with valuation
$v_{B}$ such that $v_{B}(B)=v_{p}(L)$ admits an orthonormal basis. for the
space $\mathscr{C}(\mathbf{Z}_{p},L)$.

######  Theorem 3.7 (Mahler).

Let $\phi:\mathbf{Z}_{p}\rightarrow L$ be a continuous function. There exists
a unique expansion

$\phi(x)=\sum_{n\geq 0}a_{n}(\phi)\begin{pmatrix}x\\\ n\end{pmatrix},$

where $a_{n}(\phi)\in L$ and $a_{n}(\phi)\rightarrow 0$ as
$n\rightarrow\infty$. Moreover,
$v_{\mathscr{C}}(\phi)=\inf_{n\in{\mathbf{N}}}v_{p}(a_{n}(\phi))$.

###### Proof.

See [Col10a, Théorème 1.2.3.]. ∎

######  Remark 3.8.

The coefficients $a_{n}(\phi)$ are called the _Mahler coefficients_ of $\phi$.
One can write down the Mahler coefficients of $\phi$ very simply; we define
the _discrete derivatives_ of $\phi$ by

$\phi^{[0]}=\phi,\hskip 12.0pt\phi^{[k+1]}(x)=\phi^{[k]}(x+1)-\phi^{[k]}(x),$

and then $a_{n}(\phi)=\phi^{[n]}(0).$

######  Definition 3.9.

Let $\mu\in\Lambda(\mathbf{Z}_{p})$ be a $p$-adic measure on $\mathbf{Z}_{p}$.
Define the _Mahler transform_ (or _Amice transform_) of $\mu$ to be

$\mathscr{A}_{\mu}(T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathbf{Z}_{p}}(1+T)^{x}\cdot\mu(x)=\sum_{n\geq
0}\left[\int_{\mathbf{Z}_{p}}\begin{pmatrix}x\\\
n\end{pmatrix}\cdot\mu\right]T^{n}\in\mathscr{O}_{L}[\\![T]\\!].$

######  Theorem 3.10.

The Mahler transform gives an $\mathscr{O}_{L}$-algebra isomorphism

$\Lambda(\mathbf{Z}_{p})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathscr{O}_{L}[\\![T]\\!].$

###### Proof.

We can explicitly define an inverse to the transform. Let $g(T)=\sum_{n\geq
0}c_{n}T^{n}\in\mathscr{O}_{L}[\\![T]\\!]$. Let $H\subset\mathbf{Z}_{p}$ be an
open subgroup, and for each $[a]\in\mathbf{Z}_{p}/H$ let $\mathbf{1}_{aH}$
denote the characteristic function of the coset $aH\subset\mathbf{Z}_{p}$.
This is a continuous function on $\mathbf{Z}_{p}$, and hence has a Mahler
expansion

$\mathbf{1}_{aH}(x)=\sum_{n\geq 0}a_{n}^{[a]}\begin{pmatrix}x\\\
n\end{pmatrix},$

with $a_{n}^{[a]}\in\mathbf{Z}_{p}\subset\mathscr{O}_{L}$. Then define

$\mu_{[a]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{n\geq
0}a_{n}^{[a]}c_{n},$

and

$\mu_{H}=\sum_{[a]\in\mathbf{Z}_{p}/H}\mu_{[a]}[a].$

It is an easy check that $(\mu_{H})_{H}$ is an element of the Iwasawa algebra
and the resulting function
$\mathscr{O}_{L}[\\![T]\\!]\rightarrow\Lambda(\mathbf{Z}_{p})$ is an inverse
to the Mahler transform. ∎

######  Definition 3.11.

If $g\in\mathscr{O}_{L}[\\![T]\\!]$, we write
$\mu_{g}\in\Lambda(\mathbf{Z}_{p})$ for the corresponding
($\mathscr{O}_{L}$-valued) measure on $\mathbf{Z}_{p}$ (so that
$\mathscr{A}_{\mu_{g}}=g)$.

######  Remark 3.12.

Let $g\in\mathscr{O}_{L}[\\![T]\\!]$ with associated measure $\mu_{g}$. From
the definitions, it is evident that

$\int_{\mathbf{Z}_{p}}\mu_{g}=g(0).$

###  3.3 An example: Dirac measures

We illustrate the above theory in an example.

######  Definition 3.13.

Let $a\in\mathbf{Z}_{p}$. The _Dirac measure_
$\delta_{a}\in\mathscr{M}(\mathbf{Z}_{p},\mathscr{O}_{L})$ is the linear
functional ‘evaluation at $a$’, that is, the measure defined by

$\displaystyle\delta_{a}:\mathscr{C}(\mathbf{Z}_{p},\mathscr{O}_{L})$
$\displaystyle\longrightarrow\mathscr{O}_{L}$ $\displaystyle\phi$
$\displaystyle\longmapsto\phi(a).$

Under the identification of measures with additive functions on open compact
subsets of $\mathbf{Z}_{p}$, we find that this corresponds to the function

$\widetilde{\delta}_{a}(X)=\left\\{\begin{array}[]{ll}1&\text{if }a\in X\\\
0&\text{if }a\notin X,\end{array}\right.$

as can be seen directly from the proof of the identification.

At finite level $\delta_{a}$ corresponds to the basis element
$[a+p^{n}\mathbf{Z}_{p}]\in\mathscr{O}_{L}[\mathbf{Z}_{p}/p^{n}\mathbf{Z}_{p}]$.
In the inverse limit this yields an element of the Iwasawa algebra that we
denote $[a]$.

Finally, we compute the Mahler transform of $\delta_{a}$. If
$a\in\mathbf{Z}_{p}$ then, by definition, this is

$\mathscr{A}_{\delta_{a}}(T)=\sum_{n\geq 0}\begin{pmatrix}a\\\
n\end{pmatrix}T^{n}=(1+T)^{a}.$

###  3.4 A measure-theoretic toolbox

There are natural operations one might consider on measures, and via the
Mahler transform these give rise to operators on power series. The following
operations can be considered as a ‘toolbox’ for working with measures and
power series; as we shall see in the sequel, the ability to manipulate
measures in this way has important consequences. For further details (and more
operations), see [Col10a].

####  3.4.1 Multiplication by $x$

Given a measure $\mu$ on $\mathbf{Z}_{p}$, we naturally wish to compute
$\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu$ for $k$ a positive integer. To allow us
to do that, we define $x\mu$ to be the measure defined by

$\int_{\mathbf{Z}_{p}}f(x)\cdot x\mu=\int_{\mathbf{Z}_{p}}xf(x)\cdot\mu.$

We can ask what this operation does on Mahler transforms; we find:

######  Lemma 3.14.

We have

$\mathscr{A}_{x\mu}=\partial\mathscr{A}_{\mu},$

where $\partial$ denotes the differential operator $(1+T)\frac{d}{dT}$.

###### Proof.

The result follows directly from computing

$x\begin{pmatrix}x\\\ n\end{pmatrix}=(x-n)\begin{pmatrix}x\\\
n\end{pmatrix}+n\begin{pmatrix}x\\\ n\end{pmatrix}=(n+1)\begin{pmatrix}x\\\
n+1\end{pmatrix}+n\begin{pmatrix}x\\\ n\end{pmatrix}.\qed$

From the above lemma and Remark 3.12, we immediately obtain:

######  Corollary 3.15.

For $\mu\in\Lambda(\mathbf{Z}_{p})$, we have

$\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu=\big{(}\partial^{k}\mathscr{A}_{\mu}\big{)}(0).$

####  3.4.2 Multiplication by $z^{x}$

Let $z\in\mathscr{O}_{L}$ be such that $|z-1|<1$. Then the Mahler transform of
$z^{x}\mu$ is

$\mathscr{A}_{z^{x}\mu}(T)=\mathscr{A}_{\mu}\big{(}(1+T)z-1\big{)}.$

Indeed, from the definition of the Mahler transform, we see that

$\mathscr{A}_{\mu}\big{(}(1+T)z-1\big{)}=\int_{\mathbf{Z}_{p}}\big{(}(1+T)z\big{)}^{x}\cdot\mu,$

and this is precisely the Mahler transform of $z^{x}\mu$ (one has to be
slightly careful about convergence issues).

####  3.4.3 Restriction to open compact subsets

Consider an open compact subset $X\subset\mathbf{Z}_{p}$. If we define
$\mathbf{1}_{X}$ to be the characteristic function of this subset, we can
consider the restriction $\mathrm{Res}_{X}(\mu)$ of $\mu$ to $X$ defined by

$\int_{X}f\cdot\mathrm{Res}_{X}(\mu)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathbf{Z}_{p}}f\mathbf{1}_{X}\cdot\mu.$

In the case $X=b+p^{n}\mathbf{Z}_{p}$, we can write this characteristic
function explicitly as

$\mathbf{1}_{b+p^{n}\mathbf{Z}_{p}}(x)=\frac{1}{p^{n}}\sum_{\xi\in\mu_{p^{n}}}\xi^{x-b},$

and then using the above, we calculate the Mahler transform of
$\mathrm{Res}_{b+p^{n}\mathbf{Z}_{p}}(\mu)$ to be

$\mathscr{A}_{\mathrm{Res}_{b+p^{n}\mathbf{Z}_{p}}(\mu)}(T)=\frac{1}{p^{n}}\sum_{\xi\in\mu_{p^{n}}}\xi^{-b}\mathscr{A}_{\mu}\big{(}(1+T)\xi-1\big{)}.$

####  3.4.4 Restriction to $\mathbf{Z}_{p}^{\times}$

From the above applied to $b=0$ and $n=1$, it is immediate that

$\mathscr{A}_{\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(\mu)}(T)=\mathscr{A}_{\mu}(T)-\frac{1}{p}\sum_{\xi\in\mu_{p}}\mathscr{A}_{\mu}\big{(}(1+T)\xi-1\big{)}.$
(3.1)

In order to calculate a formula for the restriction to an arbitrary open
compact subset $X\subseteq\mathbf{Z}_{p}$, we can write $X$ (or its
complement, as we did with $\mathbf{Z}_{p}^{\times}$) as a disjoint union of
sets of the form $b+p^{n}\mathbf{Z}_{p}$ and apply the formulas obtained
before.

####  3.4.5 The action of $\mathbf{Z}_{p}^{\times}$, $\varphi$ and $\psi$

We introduce an action of $\mathbf{Z}_{p}^{\times}$ that serves as a precursor
to a Galois action later on. Let $a\in\mathbf{Z}_{p}^{\times}$. We can define
a measure $\sigma_{a}(\mu)$ by

$\int_{\mathbf{Z}_{p}}f(x)\cdot\sigma_{a}(\mu)=\int_{\mathbf{Z}_{p}}f(ax)\cdot\mu.$

This has Mahler transform

$\mathscr{A}_{\sigma_{a}(\mu)}=\mathscr{A}_{\mu}\big{(}(1+T)^{a}-1\big{)}.$

In a similar manner, we can define an operator $\varphi$ acting as
‘$\sigma_{p}$’ by

$\int_{\mathbf{Z}_{p}}f(x)\cdot\varphi(\mu)=\int_{\mathbf{Z}_{p}}f(px)\cdot\mu,$

and this corresponds to
$\mathscr{A}_{\varphi(\mu)}=\varphi(\mathscr{A}_{\mu})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{A}_{\mu}\big{(}(1+T)^{p}-1\big{)}$.
Finally, we also define the analogous operator for $p^{-1}$; we define a
measure $\psi(\mu)$ on $\mathbf{Z}_{p}$ by defining

$\int_{\mathbf{Z}_{p}}f(x)\cdot\psi(\mu)=\int_{p\mathbf{Z}_{p}}f(p^{-1}x)\cdot\mu.$

Note that $\psi\circ\varphi=\mathrm{id}$, whilst
$\varphi\circ\psi(\mu)=\mathrm{Res}_{p\mathbf{Z}_{p}}(\mu)$. Indeed, we have

$\int_{\mathbf{Z}_{p}}f(x)\cdot\psi\circ\varphi(\mu)=\int_{\mathbf{Z}_{p}}\mathbf{1}_{p\mathbf{Z}_{p}}(x)f(p^{-1}x)\cdot\varphi(\mu)=\int_{\mathbf{Z}_{p}}\mathbf{1}_{p\mathbf{Z}_{p}}(px)f(x)\cdot\varphi(\mu)=\int_{\mathbf{Z}_{p}}f(x)\cdot\mu,$
$\int_{\mathbf{Z}_{p}}f(x)\cdot\varphi\circ\psi(\mu)=\int_{\mathbf{Z}_{p}}f(px)\cdot\psi(\mu)=\int_{p\mathbf{Z}_{p}}f(x)\cdot\mu=\int_{\mathbf{Z}_{p}}f(x)\cdot\mathrm{Res}_{p\mathbf{Z}_{p}}(\mu).$

In particular, we have

$\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(\mu)=(1-\varphi\circ\psi)(\mu).$ (3.2)

The operator $\psi$ also gives an operator on any
$F(T)\in\mathscr{O}_{L}[\\![T]\\!]$ under the Amice transform, and using the
restriction formula above, we see that it is the unique operator satisfying

$\varphi\circ\psi(F)(T)=\frac{1}{p}\sum_{\xi\in\mu_{p}}F\big{(}(1+T)\xi-1\big{)}.$

The following result will be useful in Part II.

######  Corollary 3.16.

A measure $\mu\in\Lambda(\mathbf{Z}_{p})$ is supported on
$\mathbf{Z}_{p}^{\times}$ if and only if $\psi(\mathscr{A}_{\mu})=0$.

###### Proof.

Let $\mu\in\Lambda(\mathbf{Z}_{p})$. Then $\mu$ is supported on
$\mathbf{Z}_{p}^{\times}$ if and only if
$\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(\mu)=\mu$, or equivalently if and only
if $\mathscr{A}_{\mu}=\mathscr{A}_{\mu}-\varphi\circ\psi(\mathscr{A}_{\mu})$,
which happens if and only if $\psi(\mathscr{A}_{\mu})=0$, since the operator
$\varphi$ is injective. ∎

######  Remark 3.17.

We have an injection
$\iota:\Lambda(\mathbf{Z}_{p}^{\times})\hookrightarrow\Lambda(\mathbf{Z}_{p})$
given by

$\int_{\mathbf{Z}_{p}}\phi\cdot\iota(\mu)=\int_{\mathbf{Z}_{p}^{\times}}\phi\big{|}_{\mathbf{Z}_{p}^{\times}}\cdot\mu,$

and as $\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}\circ\iota$ is the identity on
$\Lambda(\mathbf{Z}_{p}^{\times})$, we can identify
$\Lambda(\mathbf{Z}_{p}^{\times})$ with its image as a subset of
$\Lambda(\mathbf{Z}_{p})$. By Corollary 3.16, a measure
$\mu\in\Lambda(\mathbf{Z}_{p})$ lies in $\Lambda(\mathbf{Z}_{p}^{\times})$ if
and only if $\psi(\mu)=0$. Whilst we identify
$\Lambda(\mathbf{Z}_{p}^{\times})$ with a subset of $\Lambda(\mathbf{Z}_{p})$,
it is important to remark that it is _not_ a subalgebra. Indeed, convolution
of measures on a group $G$ uses the group structure of $G$; for
$\mathbf{Z}_{p}^{\times}$ this is multiplicative, and for $\mathbf{Z}_{p}$
this is additive (cf. Remark 3.5). If $\lambda$ and $\mu$ are two measures on
$\mathbf{Z}_{p}^{\times}$, writing $\mu*_{\mathbf{Z}_{p}^{\times}}\lambda$ for
the convolution over $\mathbf{Z}_{p}^{\times}$, we have

$\int_{{{\mathbf{Z}}_{p}^{\times}}}f(x)\cdot(\mu*_{\mathbf{Z}_{p}^{\times}}\lambda)=\int_{{{\mathbf{Z}}_{p}^{\times}}}\bigg{(}\int_{{{\mathbf{Z}}_{p}^{\times}}}f(xy)^{k}\cdot\mu(x)\bigg{)}\cdot\lambda(y)$

###  3.5 Pseudo-measures

The Mahler transform gives a correspondence between $p$-adic measures and
$p$-adic analytic functions on the open unit ball (explained in Remark 3.23
below). The Riemann zeta function, however, is not analytic everywhere, as it
has a simple pole at $s=1$. To reflect this, we also need to be able to handle
simple poles on the $p$-adic side. We do this via the theory of pseudo-
measures.

######  Definition 3.18.

Let $G$ be an abelian profinite group, and let $Q(G)$ denote the ring of
fractions of the Iwasawa algebra $\Lambda(G)$. A _pseudo-measure_ on $G$ is an
element $\lambda\in Q(G)$ such that

$([g]-[1])\lambda\in\Lambda(G)$

for all $g\in G$.

We will be most interested in pseudo-measures on $G=\mathbf{Z}_{p}^{\times}$.
The following lemma shows that a pseudo-measure $\mu$ on
$\mathbf{Z}_{p}^{\times}$ is uniquely determined by the values
$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\mu$ for $k>0$.

######  Lemma 3.19.

  * (i)

Let $\mu\in\Lambda(\mathbf{Z}_{p}^{\times})$ such that

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\mu=0$

for all $k>0$. Then $\mu=0$.

  * (ii)

Let $\mu\in\Lambda(\mathbf{Z}_{p}^{\times})$ such that

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\mu\neq 0$

for all $k>0$. Then $\mu$ is not a zero divisor in
$\Lambda(\mathbf{Z}_{p}^{\times}).$

  * (iii)

Part (i) holds if, more generally, $\mu$ is a pseudo-measure.

###### Proof.

To prove part (i), note that the vanishing condition forces the Mahler
transform $\mathscr{A}_{\mu}(T)$ of $\mu$ to be constant, since each non-
trivial binomial polynomial is divisible by $x$. As $\mu$ is a measure on
$\mathbf{Z}_{p}^{\times}$, we also have $\psi(\mathscr{A}_{\mu})(T)=0$. We
deduce $\mathscr{A}_{\mu}(T)=0$, so $\mu=0$.

For part (ii), suppose there exists a measure $\lambda$ such that
$\mu\lambda=0$, where the product is the convolution product on
$\mathbf{Z}_{p}^{\times}$ (cf. Remark 3.17). Then

$0=\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot(\mu\lambda)=\int_{\mathbf{Z}_{p}^{\times}}\bigg{(}\int_{\mathbf{Z}_{p}^{\times}}(xy)^{k}\cdot\mu(x)\bigg{)}\cdot\lambda(y)=\bigg{(}\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\mu\bigg{)}\bigg{(}\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\lambda\bigg{)},$

which forces $\lambda=0$ by part (i).

Finally, let $\mu$ be a pseudo-measure satisfying the vanishing condition. Let
$a\neq 1$ be an integer prime to $p$; then there is a natural measure

$[a]-[1]\in\Lambda(\mathbf{Z}_{p}^{\times}),$

with $\int_{\mathbf{Z}_{p}^{\times}}f\cdot([a]-[1])=f(a)-f(1).$ Consider the
measure $\lambda=([a]-[1])\mu\in\Lambda(\mathbf{Z}_{p}^{\times})$. Then
$\lambda$ satisfies the condition of part (i), so $\lambda=0$. But $[a]-[1]$
satisfies the condition of part (ii), so it is not a zero-divisor, and this
forces $\mu=0$, as required. ∎

Finally, we give a simpler process for writing down pseudo-measures on
$\mathbf{Z}_{p}^{\times}$.

######  Definition 3.20.

The _augmentation ideal_
$I((\mathbf{Z}_{p}/p^{n})^{\times})\subset\mathscr{O}_{L}[(\mathbf{Z}_{p}/p^{n})^{\times}]$
is the kernel of the natural ‘degree’ map

$\mathscr{O}_{L}[({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}]\to\mathscr{O}_{L},\qquad\sum_{a}c_{a}[a]\mapsto\sum_{a}c_{a}.$

These fit together into a degree map
$\Lambda(\mathbf{Z}_{p}^{\times})\to\mathscr{O}_{L}$; we call its kernel the
augmentation ideal
$I(\mathbf{Z}_{p}^{\times})\subset\Lambda(\mathbf{Z}_{p}^{\times})$. One may
check directly there is an isomorphism

$I(\mathbf{Z}_{p}^{\times})\cong\varprojlim
I((\mathbf{Z}_{p}/p^{n})^{\times}).$

######  Lemma 3.21.

Let $a$ be a topological generator of $\mathbf{Z}_{p}^{\times}$, and
$\mu\in\Lambda(\mathbf{Z}_{p}^{\times})$ a measure. Then

$\mu^{\prime}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\mu}{[a]-[1]}\in
Q(\mathbf{Z}_{p}^{\times})$

is a pseudo-measure.

###### Proof.

As $p$ is odd, $(\mathbf{Z}_{p}/p^{n})^{\times}$ is cyclic, generated by
$\overline{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=a\hskip
2.0pt(\mathrm{mod}\hskip 2.0ptp^{n})$, and we have

$I((\mathbf{Z}_{p}/p^{n})^{\times})=([\overline{a}]-[1])\mathscr{O}_{L}[(\mathbf{Z}_{p}/p^{n})^{\times}].$

In the inverse limit we see that

$I(\mathbf{Z}_{p}^{\times})=([a]-[1])\Lambda(\mathbf{Z}_{p}^{\times}).$

Thus if $g\in\Lambda(\mathbf{Z}_{p}^{\times})$, we have $[g]-[1]\in
I(\mathbf{Z}_{p}^{\times})$, and we must have

$[g]-[1]=\nu([a]-[1])$

for some $\nu\in\Lambda(\mathbf{Z}_{p}^{\times})$. Then

$([g]-[1])\mu^{\prime}=\nu([a]-[1])\mu^{\prime}=\nu\cdot\mu\in\Lambda(\mathbf{Z}_{p}^{\times}),$

that is, $\mu^{\prime}$ is a pseudo-measure. ∎

###  3.6 Further remarks

######  Remark 3.22.

Power series rings have been generalised to what now are called Fontaine
rings. It turns out that Galois representations are connected to certain
modules over these rings called $(\varphi,\Gamma)$-modules. The operations
described above are examples of the basic operations we have on
$(\varphi,\Gamma)$-modules and their interpretation with $p$-adic analysis
inspired the proof of the $p$-adic Langlands correspondence for
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$ [Col10b].

######  Remark 3.23.

The Mahler transform allows one to view measures as (bounded) rigid analytic
functions on the $p$-adic open unit ball. Indeed, let
$B(0,1)=\\{z\in\mathbf{C}_{p}\,|\,|z|<1\\}$ be the open unit ball. Its space
of bounded $L$-valued rigid functions, i.e.
$\mathcal{O}_{B(0,1)}^{+}(B(0,1)_{L})$, is given by
$\mathcal{O}_{L}[\\![T]\\!]\otimes_{\mathcal{O}_{L}}L$ and hence $p$-adic
measures on $\mathbf{Z}_{p}$ can be viewed as rigid analytic functions on
$B(0,1)$. Some of the operations introduced so far arise by pullback from
continuous maps from the open unit ball to its. For example, $\varphi$ (resp.
$\sigma_{a}$ for $a\in\mathbf{Z}_{p}^{\times}$) is given by taking pullack of
$z\mapsto(1+z)^{p}-1$ (resp. $z\mapsto(1+z)^{a}-1$).

######  Remark 3.24.

In this section, we have seen one of the major benefits of passing to inverse
limits, and considering measures on $\mathbf{Z}_{p}$ (rather than just
${\mathbf{Z}}/p^{n}{\mathbf{Z}}$): namely, the connection to power series
rings. This has profound consequences later on, when considering modules over
the algebra $\Lambda(\mathbf{Z}_{p})$. In particular, passing to the inverse
limit ‘rigidifies’ the picture, and as a consequence, there is a structure
theorem for finitely generated $\Lambda(\mathbf{Z}_{p})$-modules similar to
that for modules over a principal ideal domain: see §10.1.

##  4 The Kubota-Leopoldt $p$-adic $L$-function

In this section, we prove the following:

######  Theorem 4.1.

There is a unique pseudo-measure $\zeta_{p}$ on $\mathbf{Z}_{p}^{\times}$ such
that for all $k>0$, we have

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\zeta_{p}=(1-p^{k-1})\zeta(1-k).$

This pseudo-measure, denoted $\zeta_{p}^{\mathrm{an}}$ in §1, is the
_Kubota–Leopoldt $p$-adic $L$-function_.

###  4.1 The measure $\mu_{a}$

Recall that we can write the Riemann zeta function in the form

$(s-1)\zeta(s)=\frac{1}{\Gamma(s-1)}\int_{0}^{\infty}f(t)t^{s-2}dt,$

where $f(t)=t/(e^{t}-1)$, and that
$\zeta(-k)=(d^{k}f/dt^{k})(0)=(-1)^{k}B_{k+1}/(k+1).$ We want to remove the
smoothing factor at $s=1$. For this, let $a$ be an integer coprime to $p$ and
consider the related function

$f_{a}(t)=\frac{1}{e^{t}-1}-\frac{a}{e^{at}-1}.$

This is also $\mathscr{C}^{\infty}$ and has exponential decay at infinity, so
we can apply Theorem 2.4 and consider the function $L(f_{a},s)$. The presence
of $a$ removes the factor of $s-1$, at the cost of introducing a different
smoothing factor.

######  Lemma 4.2.

We have

$L(f_{a},s)=(1-a^{1-s})\zeta(s),$

which has an analytic continuation to ${\mathbf{C}}$, and

$f_{a}^{(k)}(0)=(-1)^{k}(1-a^{1+k})\zeta(-k).$

###### Proof.

This follows from similar calculations as those in the proof of Lemma 2.7. ∎

We now introduce the $p$-adic tools we have developed so far into the picture
and, starting from the function $f_{a}(t)$, we will slowly manipulate it until
we construct a (pseudo-)measure with the desired interpolation properties.
Note the following very simple observation.

######  Lemma 4.3.

Under the substitution $e^{t}=T+1$, the derivative $d/dt$ becomes the operator
$\partial=(1+T)\frac{d}{dT}$. In particular, if we define

$F_{a}(T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{T}-\frac{a}{(1+T)^{a}-1},$

we have

$f_{a}^{(k)}(0)=\big{(}\partial^{k}F_{a}\big{)}(0).$ (4.1)

The left-hand side of (4.1) computes the $L$-value $\zeta(-k)$ by Lemma 4.2.
The right-hand side is similar to Corollary 3.15, which expressed the integral
$\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu$ in terms of the Mahler transform
$\mathscr{A}_{\mu}$. This motivates us to seek a measure $\mu_{a}$ with
$\mathscr{A}_{\mu_{a}}=F_{a}$. This is possible by:

######  Proposition 4.4.

The function $F_{a}(T)$ is an element of $\mathbf{Z}_{p}[\\![T]\\!]$.

###### Proof.

We can expand

$(1+T)^{a}-1=\sum_{n\geq 1}{a\choose n}T^{n}=aT\big{[}1+Tg(T)\big{]},$

where $g(T)=\sum_{n\geq 2}\frac{1}{a}{a\choose n}T^{n-2}$ has coefficients in
$\mathbf{Z}_{p}$ since we have chosen $a$ coprime to $p$. Hence, expanding the
geometric series, we find

$\frac{1}{T}-\frac{a}{(1+T)^{a}-1}=\frac{1}{T}\sum_{n\geq 1}(-T)^{n}g(T)^{n},$

which is visibly an element of $\mathbf{Z}_{p}[\\![T]\\!]$. ∎

######  Definition 4.5.

Let $\mu_{a}$ be the measure on $\mathbf{Z}_{p}$ whose Mahler transform is
$F_{a}(T)$.

We have proved:

######  Proposition 4.6.

For $k\geq 0$, we have

$\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu_{a}=(-1)^{k}(1-a^{k+1})\zeta(-k).$

###  4.2 Restriction to $\mathbf{Z}_{p}^{\times}$

Recall from the introduction that we want the $p$-adic analogue of the Riemann
zeta function to be a measure on $\mathbf{Z}_{p}^{\times}$, not all of
$\mathbf{Z}_{p}$. We have already defined a restriction operator in equation
(3.2), which on Mahler transforms acts as $1-\varphi\circ\psi$.

######  Proposition 4.7.

We have

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\mu_{a}=(-1)^{k}(1-p^{k})(1-a^{k+1})\zeta(-k).$
(4.2)

(In other words, restricting to $\mathbf{Z}_{p}^{\times}$ removes the Euler
factor at $p$).

###### Proof.

We first show that $\psi(\mu_{a})=\mu_{a}$ by considering the action on power
series. Indeed, we have by definition

$\displaystyle(\varphi\circ\psi)\left(\frac{1}{T}\right)$
$\displaystyle=p^{-1}\sum_{\xi^{p}=1}\frac{1}{(1+T)\xi-1}=\frac{1}{(1+T)^{p}-1}=\varphi\left(\frac{1}{T}\right),$

as can be seen by calculating the partial fraction expansion. By injectivity
of $\varphi$, we deduce that $\psi(\frac{1}{T})=\frac{1}{T}$ and hence
$\psi(\mu_{a})=\mu_{a}$ since $\psi$ commutes with the action of
${{\mathbf{Z}}_{p}^{\times}}$.

Since $\mathrm{Res}_{{{\mathbf{Z}}_{p}^{\times}}}=1-\varphi\circ\psi$, we
deduce that

$\int_{{{\mathbf{Z}}_{p}^{\times}}}x^{k}\cdot\mu_{a}=\int_{\mathbf{Z}_{p}}x^{k}\cdot(1-\varphi\circ\psi)\mu_{a}=\int_{\mathbf{Z}_{p}}x^{k}\cdot(1-\varphi)\mu_{a}=(1-p^{k})\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu_{a},$

as required. ∎

###  4.3 Rescaling and removing dependence on $a$

Finally we remove the dependence on $a$. Thus far, the presence of $a$ has
acted as a ‘smoothing factor’ which removes the pole of the Riemann zeta
function; so to remove it, we must be able to handle such poles on the
$p$-adic side. We use the notion of pseudo-measures from §3.5.

######  Definition 4.8.

Let $a$ be an integer that is prime to $p$, and let $\theta_{a}$ denote the
element of $\Lambda(\mathbf{Z}_{p}^{\times})$ corresponding to $[a]-[1]$. Note
that

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\theta_{a}=a^{k}-1$

from the definitions. However, in (4.2) it is $a^{k+1}-1$ that appears. To
bridge this gap, note that on $\mathbf{Z}_{p}^{\times}$, we have a well-
defined operation ‘multiplication by $x^{-1}$’ given by

$\int_{\mathbf{Z}_{p}^{\times}}f(x)\cdot
x^{-1}\mu\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathbf{Z}_{p}^{\times}}x^{-1}f(x)\cdot\mu,$

and that

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot
x\mu_{a}=(-1)^{k}(a^{k}-1)(1-p^{k-1})\zeta(1-k).$

(We comment further on this multiplication by $x^{-1}$ in Remarks 8.1 and
9.7). Finally define

$\zeta_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{x^{-1}\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}\mu_{a}}{\theta_{a}}\in
Q(\mathbf{Z}_{p}^{\times}).$

######  Proposition 4.9.

Suppose $a^{p-1}\not\equiv 1\hskip 2.0pt(\mathrm{mod}\hskip 2.0ptp^{2})$. Then
the element $\zeta_{p}$ is a well-defined pseudo-measure that is:

  * (i)

independent of the choice of $a$,

  * (ii)

and satisfies the interpolation property that for all $k>0$, we have

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\zeta_{p}=(1-p^{k-1})\zeta(1-k).$

###### Proof.

The element $\theta_{a}$ is not a zero-divisor by Lemma 3.19, so $\zeta_{p}$
is well-defined. The condition on $a$ ensures it is a topological generator of
$\mathbf{Z}_{p}^{\times}$; so $\zeta_{p}$ is a pseudo-measure by Lemma 3.21.

To prove independence, if $a$ and $b$ are two integers coprime to $p$, then

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot(\theta_{a}x^{-1}\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}\mu_{b})=(-1)^{k}(1-a^{k})(1-b^{k})(1-p^{k-1})\zeta(1-k)=\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot(\theta_{b}x^{-1}\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}\mu_{a})$

for all $k>0$, so that

$\theta_{a}\mu_{b}=\theta_{b}\mu_{a},$

by Lemma 3.19, giving the required independence. We obtain the interpolation
property

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\zeta_{p}=(-1)^{k}(1-p^{k-1})\zeta(1-k).$

as a formal consequence of the definition and the algebra structure on
$\Lambda(\mathbf{Z}_{p}^{\times})$. To get the stated interpolation, we may
remove the $(-1)^{k}$ as $\zeta(1-k)\neq 0$ if and only if $k$ is even. ∎

We’ve now proved the existence of the pseudo-measure required by Theorem 4.1.
Uniqueness follows from Lemma 3.19(iii); so Theorem 4.1 is proved. ∎

##  5 Interpolation at Dirichlet characters

Throughout the construction of the Kubota–Leopoldt $p$-adic $L$-function,
we’ve kept half an eye on the interpolation property and links to the values
of the Riemann zeta function, so the interpolation of these values should not
have come as a surprise. However, now some real magic happens. Since the
introduction, we’ve failed to mention Dirichlet $L$-functions once – but,
miraculously, the Kubota–Leopoldt $p$-adic $L$-function also interpolates
Dirichlet $L$-values as well.

###  5.1 Characters of $p$-power conductor

We start studying the interpolation properties when twisting by a Dirichlet
character of conductor a power of $p$.

######  Theorem 5.1.

Let $\chi$ be a (primitive) Dirichlet character of conductor $p^{n}$ for some
integer $n\geq 1$ (seen as a locally constant character of
$\mathbf{Z}_{p}^{\times}$). Then, for $k>0$, we have

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{p}=L(\chi,1-k).$

The rest of this subsection will contain the proof of this result. The proof
is somewhat calculation-heavy, but – given familiarity with the dictionary
between measures and power series – is not conceptually difficult.

In particular: the Riemann zeta function was the complex Mellin transform of a
real analytic function, which – via Theorem 2.4 – gave us a formula for its
special values. Under the transformation $e^{t}=T+1$, we obtained a $p$-adic
power series; and under the measures–power series correspondence given by the
Mahler transform, this gave us a measure on $\mathbf{Z}_{p}$, from which we
constructed $\zeta_{p}$. To obtain interpolation at Dirichlet characters, we
pursue this in reverse, as summarised in the following diagram:

$\textstyle{(1-a^{1-s})\zeta(s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Mellin$\textstyle{(1-\chi(a)a^{1-s})L(\chi,s)}$$\textstyle{f_{a}(t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{t}=T+1}$$\textstyle{F_{a}(T)\in\mathcal{O}_{L}[\\![T]\\!]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Mahler$\textstyle{f_{a,\chi}(t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{t}=T+1}$Mellin$\textstyle{F_{a,\chi}(T)\in\mathcal{O}_{L}[\\![T]\\!]}$$\textstyle{\mu_{a}\in\Lambda(\mathbf{Z}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$“twist
by
$\chi$”$\textstyle{\mu_{a,\chi}\in\Lambda(\mathbf{Z}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Mahler$\textstyle{\zeta_{p}}$

Firstly, we introduce a twisting operation on measures. If $\mu$ is a measure
on $\mathbf{Z}_{p}$, we define a measure $\mu_{\chi}$ on $\mathbf{Z}_{p}$ by

$\int_{\mathbf{Z}_{p}}f(x)\cdot\mu_{\chi}=\int_{\mathbf{Z}_{p}}\chi(x)f(x)\cdot\mu.$

In particular, under this we have

$\displaystyle\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{p}=\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot(\zeta_{p})_{\chi}=\big{(}\partial^{k}\mathscr{A}_{(\zeta_{p})_{\chi}}\big{)}(0),$

where the last equality follows from Corollary 3.15. Thus we want to determine
the Mahler transform of $\mu_{\chi}$ in terms of $\mathscr{A}_{\mu}$, for
which we use our measure-theoretic toolkit. This requires a classical
definition.

######  Definition 5.2.

Let $\chi$ be a primitive Dirichlet character of conductor $p^{n}$, $n\geq 1$.
Define the _Gauss sum of $\chi$_ as

$G(\chi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)\varepsilon_{p^{n}}^{c},$

where $(\varepsilon_{p^{n}})_{n\in{\mathbf{N}}}$ denotes a system of primitive
$p$-power roots of unity in $\overline{{\mathbf{Q}}}_{p}$ such that
$\varepsilon_{p^{n+1}}^{p}=\varepsilon_{p^{n}}$ for all $n\geq 0$ (if we fix
an isomorphism $\overline{{\mathbf{Q}}}_{p}\cong{\mathbf{C}}$, then one can
take
$\varepsilon_{p^{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{2\pi
i/p^{n}}$).

######  Remark 5.3.

We will use the following basic properties of Gauss sums:

  1. (1)

$G(\chi)G(\chi^{-1})=\chi(-1)p^{n}.$

  2. (2)

$G(\chi)=\chi(a)\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)\varepsilon_{p^{n}}^{ac}$
for any $a\in{{\mathbf{Z}}_{p}^{\times}}$.

######  Lemma 5.4.

The Mahler transform of $\mu_{\chi}$ is

$\mathscr{A}_{\mu_{\chi}}(T)=\frac{1}{G(\chi^{-1})}\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)^{-1}\mathscr{A}_{\mu}\big{(}(1+T)\varepsilon_{p^{n}}^{c}-1\big{)}.$

###### Proof.

Since $\chi$ is constant modulo $p^{n}$, the measure $\mu_{\chi}$ is simply

$\mu_{\chi}=\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)\mathrm{Res}_{c+p^{n}\mathbf{Z}_{p}}(\mu).$

Using this expression and the formula for the Mahler transform of the
restriction of a measure to $c+p^{n}\mathbf{Z}_{p}$, we find that

$\mathscr{A}_{\mu_{\chi}}(T)=\frac{1}{p^{n}}\sum_{b\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(b)\sum_{\xi\in\mu_{p^{n}}}\xi^{-b}\mathscr{A}_{\mu}\big{(}(1+T)\xi-1\big{)}.$

Writing $\mu_{p^{n}}=\\{\varepsilon_{p^{n}}^{c}:c=0,...,p^{n}-1\\},$ and
rearranging the sums, we have

$\displaystyle\mathscr{A}_{\mu_{\chi}}(T)$
$\displaystyle=\frac{1}{p^{n}}\sum_{c\hskip 2.0pt(\mathrm{mod}\hskip
2.0ptp^{n})}\sum_{b\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(b)\varepsilon_{p^{n}}^{-bc}\mathscr{A}_{\mu}\big{(}(1+T)\varepsilon_{p^{n}}^{c}-1\big{)}$
$\displaystyle=\frac{1}{p^{n}}\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}G(\chi)\chi(-c)^{-1}\mathscr{A}_{\mu}\big{(}(1+T)\varepsilon_{p^{n}}^{c}-1\big{)}$
$\displaystyle=\frac{1}{G(\chi^{-1})}\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)^{-1}\mathscr{A}_{\mu}\big{(}(1+T)\varepsilon_{p^{n}}^{c}-1\big{)},$

where the second equality follows from Remark 5.3(2) and the last one from
Remark 5.3(1). This finishes the proof. ∎

We now consider the case where $\mu=\mu_{a}$ of Definition 4.5, the measure
from which we built the Kubota–Leopoldt $p$-adic $L$-function, and which has
Mahler transform

$\mathscr{A}_{\mu_{a}}(T)=\frac{1}{T}-\frac{a}{(1+T)^{a}-1}.$

Applying the above transformation, we obtain a measure $\mu_{\chi,a}$ with
Mahler transform

$F_{\chi,a}(T)=\frac{1}{G(\chi^{-1})}\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)^{-1}\left[\frac{1}{(1+T)\varepsilon_{p^{n}}^{c}-1}-\frac{a}{(1+T)^{a}\varepsilon_{p^{n}}^{ac}-1}\right].$

Via the standard substitution $e^{t}=T+1$, this motivates the study of the
function

$f_{\chi,a}(t)=\frac{1}{G(\chi^{-1})}\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)^{-1}\left[\frac{1}{e^{t}\varepsilon_{p^{n}}^{c}-1}-\frac{a}{e^{at}\varepsilon_{p^{n}}^{ac}-1}\right],$

by way of analogy with the case of the Riemann zeta function.

######  Lemma 5.5.

We have

$L(f_{\chi,a},s)=\chi(-1)\big{(}1-\chi(a)a^{1-s}\big{)}L(\chi,s),$

where $L(f_{\chi,a},s)$ is as defined in Theorem 2.4. Hence, for $k\geq 0$, we
have

$\displaystyle f_{\chi,a}^{(k)}(0)$
$\displaystyle=(-1)^{k}\chi(-1)\big{(}1-\chi(a)a^{k+1}\big{)}L(\chi,-k)$
$\displaystyle=-\big{(}1-\chi(a)a^{k+1}\big{)}L(\chi,-k).$

###### Proof.

We follow a similar strategy as in the case of the Riemann zeta function. In
particular, we can expand as a geometric series, obtaining

$\frac{1}{e^{t}\varepsilon_{p^{n}}^{c}-1}=\sum_{k\geq
1}e^{-kt}\varepsilon_{p^{n}}^{-kc}.$

Then we have

$L(f_{\chi,a},s)=\frac{1}{\Gamma(s)G(\chi^{-1})}\int_{0}^{\infty}\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)^{-1}\sum_{k\geq
1}\bigg{(}e^{-kt}\varepsilon_{p^{n}}^{-kc}-e^{-akt}\varepsilon_{p^{n}}^{-akc}\bigg{)}t^{s-1}dt.$

Note that

$\sum_{c\in({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}}\chi(c)^{-1}\varepsilon_{p^{n}}^{-akc}=\chi(-ak)G(\chi^{-1}),$

and similarly for the first term, so that the expression collapses to

$L(f_{\chi,a},s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\sum_{k\geq
1}\chi(-k)\big{(}e^{-kt}-\chi(a)e^{-akt}\big{)}t^{s-1}dt.$

For Re$(s)\gg 0$, we can rearrange the sum and the integral, and then we can
evaluate the $k$th term of the sum easily to $(1-\chi(a)a^{1-s})k^{-s}$,
giving

$L(f_{\chi,a},s)=\chi(-1)\big{(}1-\chi(a)a^{1-s}\big{)}\sum_{k\geq
1}\chi(-k)k^{-s}=\chi(-1)\big{(}1-\chi(a)a^{1-s}\big{)}L(\chi,s),$

showing the equality of $L$-functions. To prove the final statement about
special values, observe that a simple computation shows that
$f_{\chi,a}(-t)=-\chi(-1)f_{\chi,a}(t)$, which implies (looking at the series
expansions) that $f_{\chi,a}^{(k)}(0)=0$ unless $\chi(-1)(-1)^{k}=-1$. This
concludes the proof. ∎

Note that, along the proof of Lemma 5.5, we have also shown the following
useful fact.

######  Lemma 5.6.

If $\chi$ is an even character, that is if $\chi(-1)=1$, then $L(\chi,-k)=0$
whenever $k$ is even. If $\chi$ is an odd character, then $L(\chi,-k)=0$
whenever $k$ is odd.

We can now prove Theorem 5.1.

###### Proof.

_(Theorem 5.1)._ Since $\chi$ is 0 on $p\mathbf{Z}_{p}$, we have

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\mu_{a}=\int_{\mathbf{Z}_{p}}\chi(x)x^{k}\cdot\mu_{a}=\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu_{\chi,a},$

where $\mu_{\chi,a}$ is the twist of $\mu_{a}$ by $\chi$. We know this
integral to be

$\big{(}\partial^{k}F_{\chi,a}\big{)}(0)=f_{\chi,a}^{(k)}(0),$

under the standard transform $e^{t}=T+1$. Hence, by Lemma 5.5, we find

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\mu_{a}=-(1-\chi(a)a^{k+1})L(\chi,-k),$

so that

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot
x^{-1}\mu_{a}=-(1-\chi(a)a^{k})L(\chi,1-k).$

By definition, we have

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\theta_{a}=-(1-\chi(a)a^{k}),$

and hence we find

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{p}=L(\chi,1-k),$

as was to be proved. ∎

###  5.2 Non-trivial tame conductors

We can go even further. The theorem above deals with the case of ‘tame
conductor 1’, in that we have constructed a $p$-adic measure that interpolates
all of the $L$-values $L(\chi,1-k)$ for $k>0$ and cond$(\chi)=p^{n}$ with
$n\geq 0$ (where trivial conductor corresponds to the Riemann zeta function).
More generally, we have the following result.

######  Theorem 5.7.

Let $D>1$ be any integer coprime to $p$, and let $\eta$ denote a (primitive)
Dirichlet character of conductor $D$. There exists a unique measure
$\zeta_{\eta}\in\Lambda(\mathbf{Z}_{p}^{\times})$ such that, for all primitive
Dirichlet characters $\chi$ with conductor $p^{n}$, $n\geq 0$, and for all
$k>0$, we have

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{\eta}=\big{(}1-\chi\eta(p)p^{k-1}\big{)}L(\chi\eta,1-k).$

######  Remark 5.8.

  * (i)

In this case, we obtain a genuine measure rather than a pseudo-measure. As
$L$-functions of non-trivial Dirichlet characters are everywhere holomorphic,
there is no need for the smoothing factor involving $a$.

  * (ii)

Implicit in this theorem is the fact that the relevant Iwasawa algebra is
defined over a (fixed) finite extension $L/\mathbf{Q}_{p}$ containing the
values of $\eta$.

Since many of the ideas involved in proving the above theorem are present in
the case of trivial tame conductor, the proof of Theorem 5.7 is a good
exercise. As such, we give only the main ideas involved in the proof. Note
first that the calculation relating $L(f_{\chi,a},s)$ to $L(\chi,s)$ above was
entirely classical, in the sense that $p$ did not appear anywhere;
accordingly, we can perform a similar calculation in the general case. Since
there is no need for the smoothing factor $a$, we can then consider the
function

$f_{\eta}(t)=\frac{-1}{G(\eta^{-1})}\sum_{c\in({\mathbf{Z}}/D{\mathbf{Z}})^{\times}}\frac{\eta(c)^{-1}}{e^{t}\varepsilon_{D}^{c}-1}.$

(This scaling by $-1$ also appears in the trivial tame conductor situation,
but it is incorporated into $\theta_{a}$). Defining $F_{\eta}(T)$ by
substituting $T+1$ for $e^{t}$ and expanding the geometric series, we find

$F_{\eta}(T)=\frac{-1}{G(\eta^{-1})}\sum_{c\in({\mathbf{Z}}/D{\mathbf{Z}})^{\times}}\eta(c)^{-1}\sum_{k\geq
0}\frac{\varepsilon_{D}^{kc}}{(\varepsilon_{D}^{c}-1)^{k+1}}T^{k}.$

This is an element of $\mathscr{O}_{L}[\\![T]\\!]$ for some sufficiently large
finite extension $L$ of $\mathbf{Q}_{p}$, since the Gauss sum is a $p$-adic
unit (indeed, we have $G(\eta)G(\eta^{-1})=\eta(-1)D$ and $D$ is coprime to
$p$) and $\varepsilon_{D}^{c}-1\in\mathscr{O}_{L}^{\times}$ (since it has norm
dividing $D$). There is therefore a measure
$\mu_{\eta}\in\Lambda(\mathbf{Z}_{p})$, the Iwasawa algebra over
$\mathscr{O}_{L}$, corresponding to $F_{\eta}$ under the Mahler transform.

######  Lemma 5.9.

We have $L(f_{\eta},s)=-\eta(-1)L(\eta,s).$ Hence

$\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu_{\eta}=L(\eta,-k)$

for $k\geq 0$.

###### Proof.

This is proved in a similar manner to above, equating $\partial$ with $d/dt$
and using the general theory described in Theorem 2.4. ∎

######  Lemma 5.10.

We have $\psi(F_{\eta})=\eta(p)F_{\eta}$. Hence

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\mu_{\eta}=\big{(}1-\eta(p)p^{k}\big{)}L(\eta,-k).$

###### Proof.

We show first that

$\frac{1}{p}\sum_{\xi\in\mu_{p}}\frac{1}{(1+T)\xi\varepsilon_{D}^{c}-1}=\frac{1}{(1+T)^{p}\varepsilon_{D}^{pc}-1}.$
(5.1)

Expanding each summand as a geometric series, the left hand side is

$\frac{-1}{p}\sum_{\xi\in\mu_{p}}\sum_{n\geq
0}(1+T)^{n}\varepsilon_{D}^{nc}\xi^{n}=-\sum_{n\geq
0}(1+T)^{pn}\varepsilon_{D}^{pcn},$

and summing the geometric series now gives the right hand side of (5.1). It
follows that

$\displaystyle(\varphi\circ\psi)(F_{\eta})$
$\displaystyle=\frac{-1}{pG(\eta)^{-1}}\sum_{\xi\in\mu_{p}}\sum_{c\in({\mathbf{Z}}/D{\mathbf{Z}})^{\times}}\frac{\eta(c)^{-1}}{(1+T)\xi\varepsilon_{D}^{c}-1}$
$\displaystyle=\frac{-1}{G(\eta^{-1})}\sum_{c\in({\mathbf{Z}}/D{\mathbf{Z}})^{\times}}\frac{\eta(c)^{-1}}{(1+T)^{p}\varepsilon_{D}^{pc}-1}$
$\displaystyle=\eta(p)\varphi(F_{\eta}).$

The first claim now follows by the injectivity of $\varphi$. For the second,
we have

$\displaystyle\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(\mu_{\eta})$
$\displaystyle=(1-\varphi\circ\psi)(\mu_{\eta})=\mu_{\eta}-\eta(p)\varphi(\mu_{\eta}),$

and

$\int_{\mathbf{Z}_{p}}x^{k}\cdot\varphi(\mu_{\eta})=p^{k}\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu_{\eta}.$

The result now follows from Lemma 5.9. ∎

Now let $\chi$ be a Dirichlet character of conductor $p^{n}$ for some $n\geq
0$, and let
$\theta\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\chi\eta$ the
product (a Dirichlet character of conductor $Dp^{n}$). Using Lemma 5.4, we
find easily that:

######  Lemma 5.11.

The Mahler transform of
$\mu_{\theta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\mu_{\eta})_{\chi}$
is

$F_{\theta}(T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{A}_{\mu_{\theta}}(T)=\frac{-1}{G(\theta^{-1})}\sum_{c\in({\mathbf{Z}}/Dp^{n}{\mathbf{Z}})^{\times}}\frac{\theta(c)^{-1}}{(1+T)\varepsilon_{Dp^{n}}^{c}-1}.$

Via a calculation essentially identical to the cases already seen, we can
prove

$\displaystyle\int_{\mathbf{Z}_{p}}\chi(x)x^{k}\cdot\mu_{\eta}$
$\displaystyle=\int_{\mathbf{Z}_{p}}x^{k}\cdot\mu_{\theta}=L(\theta,-k),$

that

$\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(\mu_{\theta})=\big{(}1-\theta(p)\varphi\big{)}(\mu_{\theta}),$

and that accordingly

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\mu_{\eta}=\big{(}1-\theta(p)p^{k}\big{)}L(\theta,-k).$

Finally, to complete the proof of Theorem 5.7 and to ensure compatibility with
the construction of $\zeta_{p}$, we introduce a shift by 1. The following is
directly analogous to the construction of $\zeta_{p}$; note again that
$\zeta_{\eta}$ is truly a measure, not a pseudo-measure.

######  Definition 5.12.

Define
$\zeta_{\eta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=x^{-1}\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(\mu_{\eta})$.

We see that

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{\eta}=\big{(}1-\theta(p)p^{k-1}\big{)}L(\theta,1-k).$

which completes the proof of Theorem 5.7.

###  5.3 Analytic functions on $\mathbf{Z}_{p}$ via the Mellin transform

The reader should hopefully now be convinced that the language of measures is
a natural one in which to discuss $p$-adic $L$-functions. In this subsection,
we use this (more powerful) language to answer the question we originally
posed in the introduction: namely, we define analytic functions on
$\mathbf{Z}_{p}$ that interpolate the values $\zeta(1-k)$ for $k>0$. In
passing from measures to analytic functions on $\mathbf{Z}_{p}$, we lose the
clean interpolation statements. In particular, there is no _single_ analytic
function on $\mathbf{Z}_{p}$ interpolating the values $\zeta(1-k)$ for all
$k>0$, but rather $p-1$ different ‘branches’ of the Kubota–Leopoldt $p$-adic
$L$-function on $\mathbf{Z}_{p}$, each interpolating a different range.

The reason we cannot define a single $p$-adic $L$-function on $\mathbf{Z}_{p}$
is down to the following technicality. We’d _like_ to be able to define
“$\zeta_{p}(s)=\int_{\mathbf{Z}_{p}^{\times}}x^{-s}\cdot\zeta_{p}$” for
$s\in\mathbf{Z}_{p}$. The natural way to define the exponential $x\mapsto
x^{s}$ is as

$x^{s}=\mathrm{exp}(s\cdot\mathrm{log}(x)),$

but unfortunately in the $p$-adic world the exponential map does not converge
on all of $\mathbf{Z}_{p}$, so this is not well-defined for general
$x\in\mathbf{Z}_{p}^{\times}$. Instead:

######  Lemma 5.13.

The $p$-adic exponential map converges on $p\mathbf{Z}_{p}$. Hence, for any
$s\in\mathbf{Z}_{p}$, the function
$1+p\mathbf{Z}_{p}\rightarrow\mathbf{Z}_{p}$ given by $x\mapsto
x^{s}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{\exp}(s\cdot\mathrm{log}(x))$
is well-defined.

###### Proof.

This is a standard result in the theory of local fields. See, for example,
[Cas86]. ∎

######  Definition 5.14.

Recall that we assume $p$ to be odd and that we have a decomposition
$\mathbf{Z}_{p}^{\times}\cong\mu_{p-1}\times(1+p\mathbf{Z}_{p})$. Let

$\displaystyle\omega:\mathbf{Z}_{p}^{\times}$
$\displaystyle\longrightarrow\mu_{p-1},$
$\displaystyle\langle\cdot\rangle:\mathbf{Z}_{p}^{\times}$
$\displaystyle\longrightarrow 1+p\mathbf{Z}_{p},$

where $\omega(x):=$ Teichmüller lift of the reduction modulo $p$ of $x$ and
$\langle x\rangle:=\omega^{-1}(x)x$ denote the projections to the first and
second factors respectively. If $x\in\mathbf{Z}_{p}^{\times}$, then we can
write $x=\omega(x)\langle x\rangle$.

By Lemma 5.13, the function $\langle x\rangle^{s}$ is well-defined. When $p$
is odd, for each $i=1,..,p-1$ we can define an injection

$\displaystyle\mathbf{Z}_{p}$
$\displaystyle\lhook\joinrel\relbar\joinrel\rightarrow\mathrm{Hom}_{\mathrm{cts}}(\mathbf{Z}_{p}^{\times},\mathbf{C}_{p}^{\times})$
$\displaystyle s$ $\displaystyle\longmapsto\big{[}x\mapsto\omega(x)^{i}\langle
x\rangle^{s}\big{]},$

and hence we can define an analytic function

$\displaystyle\zeta_{p,i}:\mathbf{Z}_{p}$
$\displaystyle\longrightarrow\mathbf{C}_{p}$ $\displaystyle s$
$\displaystyle\longmapsto\int_{\mathbf{Z}_{p}^{\times}}\omega(x)^{i}\langle
x\rangle^{1-s}\cdot\zeta_{p}.$

This function _does not_ interpolate as wide a range of values as the measure
$\zeta_{p}$, since the character $x^{k}$ can be written in the form
$\omega(x)^{i}\langle x\rangle^{k}$ if and only if $k\equiv i\hskip
2.0pt(\mathrm{mod}\hskip 2.0ptp-1)$, and in this case $x^{k}$ is the value of
$\omega(x)^{i}\langle x\rangle^{1-s}$ at the value $s=1-k$. Then we have the
following result.

######  Theorem 5.15.

For all $k\geq 0$ with $k\equiv i\hskip 2.0pt(\mathrm{mod}\hskip 2.0ptp-1)$,
we have

$\zeta_{p,i}(1-k)=(1-p^{k-1})\zeta(1-k).$

More generally, we can twist by Dirichlet characters as we have done before.

######  Definition 5.16.

Let $\theta=\chi\eta$ be a Dirichlet character, where $\eta$ has conductor $D$
prime to $p$ and $\chi$ has conductor $p^{n}$ for $n\geq 0$. Define

$L_{p}(\theta,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathbf{Z}_{p}^{\times}}\chi(x)\langle
x\rangle^{1-s}\cdot\zeta_{\eta},\hskip 12.0pts\in\mathbf{Z}_{p}.$

######  Remark 5.17.

  1. (1)

An equivalent definition is

$L_{p}(\theta,s)=\int_{\mathbf{Z}_{p}^{\times}}\chi\omega^{-1}(x)\langle
x\rangle^{-s}\cdot\mu_{\eta}.$ (5.2)

In [Was97], the analytic functions $L_{p}(\theta,s)$ are constructed directly
without using measures, and the more direct approach differs from the one
obtained using our measure-theoretic approach by precisely this factor of
$\omega$. This twist by 1 also appears naturally when we study the Iwasawa
Main Conjecture.

  2. (2)

Directly from the definitions, we have $\zeta_{p,i}(s)=L_{p}(\omega^{i},s)$.
Hence for arbitrary $k>0$, we have

$\zeta_{p,i}(1-k)=(1-\omega^{i-k}(p)p^{k-1})L(\omega^{i-k},1-k).$

Of course, $\omega^{i-k}$ is just the trivial character when $i\equiv k\hskip
2.0pt(\mathrm{mod}\hskip 2.0ptp-1)$, so we recover Theorem 5.15 from Theorem
5.18 below.

######  Theorem 5.18.

For all $k\geq 1$, we have

$L_{p}(\theta,1-k)=\big{(}1-\theta\omega^{-k}(p)p^{k-1}\big{)}L(\theta\omega^{-k},1-k).$

###### Proof.

We use the description of (5.2). From the definitions, we have
$\chi\omega^{-1}(x)\langle
x\rangle^{k-1}=\chi\omega^{-k}(x)\cdot\omega^{k-1}(x)\langle
x\rangle^{k-1}=\chi\omega^{-k}(x)x^{k-1}$, so that

$\displaystyle\int_{\mathbf{Z}_{p}^{\times}}\chi(x)\langle
x\rangle^{k-1}\cdot\mu_{\eta}$
$\displaystyle=\int_{\mathbf{Z}_{p}^{\times}}\chi\omega^{-k}(x)x^{k-1}\cdot\mu_{\eta}$
$\displaystyle=\big{(}1-\theta\omega^{-k}(p)p^{k-1}\big{)}L(\theta\omega^{-k},1-k),$

as required. ∎

In general, for any measure $\mu$ on $\mathbf{Z}_{p}^{\times}$ one can define

$\mathrm{Mel}_{\mu,i}(s)=\int_{\mathbf{Z}_{p}^{\times}}\omega(x)^{i}\langle
x\rangle^{s}\cdot\mu,$

the _Mellin transform_ of $\mu$ at $i$. We have then
$\zeta_{p,i}(s)=\mathrm{Mel}_{\zeta_{p},i}(1-s)$. This transform gives a way
to pass from $p$-adic measures on $\mathbf{Z}_{p}$ to analytic function on
$\mathbf{Z}_{p}$.

###  5.4 The values at $s=1$

In the following we give an example of further remarkable links between the
classical and $p$-adic zeta functions. Let $\theta$ be a non-trivial Dirichlet
character, which as usual we write in the form $\chi\eta$, where $\chi$ has
conductor $p^{n}$ and $\eta$ has conductor $D$ prime to $p$. By Theorem 5.7,
for any $k>0$, we have

$\int_{\mathbf{Z}_{p}^{\times}}\chi(x)x^{k}\cdot\zeta_{\eta}=L(\theta,1-k)$

We say that the range of interpolation is $\\{...,-3,-2,-1,0\\}.$ It’s natural
to ask what happens outside this range of interpolation. In particular, what
happens when we take $k=0$? Since this is _outside_ the range of interpolation
this value may have _a priori_ nothing to do with classical $L$-values.
Indeed, the classical value $L(\theta,1)$ is transcendental777This follows
from Baker’s theorem and Theorem 5.19, part (i)., and if it is transcendental
one cannot see it as a $p$-adic number in a natural way. However, just because
we cannot directly equate the two values does not mean there is no
relationship between them. It turns out that there is a formula for the
$p$-adic $L$-function at $s=1$ which is strikingly similar to its classical
analogue.

######  Theorem 5.19.

Let $\theta$ be a non-trivial even Dirichlet character of conductor $N$, and
let $\xi$ denote a primitive $N$th root of unity. Then:

  * (i)

(Classical value at $s=1$). We have

$L(\theta,1)=-\frac{1}{G(\theta^{-1})}\sum_{a\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta^{-1}(a)\log\big{(}1-\xi^{a}\big{)}.$

  * (ii)

($p$-adic value at $s=1$). We have

$L_{p}(\theta,1)=-\big{(}1-\theta(p)p^{-1}\big{)}\frac{1}{G(\theta^{-1})}\sum_{a\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta^{-1}(a)\log_{p}\big{(}1-\xi^{a}).$

If $\theta$ is an odd character, both sides of the $p$-adic formula vanish. In
any case, the formulae are identical up to replacing $\log$ with its $p$-adic
avatar and, as usual, deleting the Euler factor at $p$. This result can be
used to prove a $p$-adic analogue of the class number formula. For
completeness, we prove these results below.

####  5.4.1 The complex value at $s=1$

###### Proof.

_(Theorem 5.19, classical value)._ Write

$L(\theta,1)=\sum_{a\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta(a)\sum_{n\equiv
a\hskip 2.0pt(\mathrm{mod}\hskip 2.0ptD)}n^{-s}.$

Using the fact that

$\frac{1}{N}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})}\xi^{(a-n)c}=\left\\{\begin{array}[]{r
c l}0\;\;\;\text{ if $n\not\equiv a$ mod $N$}\\\ 1\;\;\;\text{ if $n\equiv a$
mod $N$,}\end{array}\right.$

we show that the above formula equals

$\displaystyle\sum_{a\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta(a)\frac{1}{N}\sum_{n\geq
1}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})}\xi^{(a-n)c}n^{-s}$
$\displaystyle=\frac{1}{N}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})}\left(\sum_{a\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta(a)\xi^{ac}\right)\sum_{n\geq
1}\xi^{-nc}n^{-s}$
$\displaystyle=\frac{G(\theta)}{N}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})}\theta^{-1}(c)\sum_{n\geq
1}\xi^{-nc}n^{-s},$

the last equality following from one of the standard identities for Gauss sums
(cf. Remark 5.3(2)). Evaluating this expression at $s=1$ (checking that there
is no convergence problem since $\theta$ is not trivial), using the Taylor
series expansion of the logarithm, and applying Remark 5.3(2), we obtain the
result. ∎

######  Remark 5.20.

We see from the formulas of the proof of Theorem 5.19(1) that the parity of
the character $\theta$ plays an important role on the behaviour of the zeta
function at $s=1$. Making some elementary calculations we can deduce that, if
$\theta$ is even, then

$L(\theta,1)=-\frac{1}{G(\theta^{-1})}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta^{-1}(c)\log|1-\xi^{c}|.$

If $\theta$ is odd, we can use the functional equation to obtain

$L(\theta,1)=i\pi\frac{1}{G(\theta^{-1})}B_{1,\theta^{-1}},$

where $B_{k,\theta}$ denotes the $k$th twisted Bernoulli number (see [Was97,
Chapter 4]).

####  5.4.2 The $p$-adic value at $s=1$

Recall the power series

$F_{\theta}(T)=\frac{-1}{G(\theta^{-1})}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\frac{\theta(c)^{-1}}{(1+T)\xi^{c}-1}$

which gives rise to a measure $\mu_{\theta}$ on $\mathbf{Z}_{p}$ that
interpolates the special values of $L(\theta,s)$. Accordingly, by the measure-
theoretic arguments we’ve employed repeatedly above, we have

$\displaystyle L_{p}(\theta,1)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{{{\mathbf{Z}}_{p}^{\times}}}x^{-1}\cdot\mu_{\theta}=\mathscr{A}_{x^{-1}\mathrm{Res}_{{{\mathbf{Z}}_{p}^{\times}}}(\mu_{\theta})}(0).$

We first compute $\mathscr{A}_{x^{-1}\mu_{\theta}}$.

######  Lemma 5.21.

There exists a constant $C$ such that

$\mathscr{A}_{x^{-1}\mu_{\theta}}(T)=-\frac{1}{G(\theta^{-1})}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta^{-1}(c)\log\big{(}(1+T)\xi^{c}-1\big{)}+C.$

###### Proof.

This follows immediately from the formula

$\partial\log\big{(}(1+T)\xi^{c}-1\big{)}=\frac{(1+T)\xi^{c}}{(1+T)\xi^{c}-1}=1+\frac{1}{(1+T)\xi^{c}-1}$

and the fact that $\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})}\theta^{-1}(c)=0$. ∎

######  Lemma 5.22.

We have

$\mathscr{A}_{\mathrm{Res}_{{{\mathbf{Z}}_{p}^{\times}}}(\mu_{\theta})}(T)=\mathscr{A}_{x^{-1}\mu_{\theta}}(T)-\theta(p)p^{-1}\mathscr{A}_{x^{-1}\mu_{\theta}}\big{(}(1+T)^{p}-1\big{)}.$

###### Proof.

This is immediate from the formula

$\mathscr{A}_{\mathrm{Res}_{{{\mathbf{Z}}_{p}^{\times}}}(\mu_{\chi})}(T)=(1-\varphi\circ\psi)\mathscr{A}_{\mu_{\chi}}(T)$

and the fact that

$\displaystyle\psi(x^{-1}\mu_{\theta})$
$\displaystyle=p^{-1}x^{-1}\psi(\mu_{\theta})=\chi(p)p^{-1}x^{-1}\mu_{\theta}.\qed$

We can now complete the proof of Theorem 5.19.

###### Proof.

_(Theorem 5.19, $p$-adic value)_. Evaluating at $T=0$ the formula of Lemma
5.22 and using Lemma 5.21 we obtain

$\displaystyle L_{p}(\theta,1)$
$\displaystyle=-(1-\theta(p)p^{-1})\mathscr{A}_{x^{-1}\mu_{\theta}}(0)$
$\displaystyle=-(1-\theta(p)p^{-1})\frac{1}{G(\theta^{-1})}\sum_{c\in({\mathbf{Z}}/N{\mathbf{Z}})^{\times}}\theta^{-1}(c)\log_{p}(\xi^{c}-1),$

as required. ∎

##  6 The $p$-adic family of Eisenstein series

We now take a brief detour to illustrate another example of $p$-adic variation
in number theory, namely the $p$-adic variation of modular forms. In
constructing the Kubota–Leopoldt $p$-adic $L$-function, we have seen many of
the key ideas that go into the simplest example of this, namely the $p$-adic
family of Eisenstein series, which we will illustrate below. For simplicity,
in this section we’ll take $p$ an odd prime.

Let $k\geq 4$ be an even integer. The _Eisenstein series of level $k$_,
defined as

$G_{k}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{\begin{subarray}{c}c,d\in{\mathbf{Z}}\\\
(c,d)\neq(0,0)\end{subarray}}\frac{1}{(cz+d)^{k}},\hskip
12.0ptz\in\mathcal{H}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{z\in{\mathbf{C}}:\mathrm{Im}(z)>0\\}$

can be viewed as a two-dimensional analogue of the zeta value $\zeta(k)$. It
is an example of a _modular form of weight $k$_. In the classical theory of
modular forms, one computes the normalised Fourier expansion of such an object
to be

$\displaystyle E_{k}(z)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{G_{k}(z)(k-1)!}{2\cdot(2\pi
i)^{k}}=\frac{\zeta(1-k)}{2}+\sum_{n\geq 1}\sigma_{k-1}(n)q^{n},$

where $\sigma_{k-1}(n)=\sum_{0<d|n}d^{k-1}$ and $q=e^{2i\pi z}$. In
particular, it is a power series with rational coefficients. (This is a
standard exercise; see [DS05, Chapter 1.1] for details).

From the definition, we see the Kubota–Leopoldt $p$-adic $L$-function as a
pseudo-measure that, when evaluated at $x^{k}$ with $k\geq 4$ even, gives (up
to an Euler factor) the constant coefficient of the Eisenstein series of
weight $k$. The idea now is to find measures giving similar interpolations of
the other coefficients. Fortunately, these are much easier to deal with: we
only need interpolations of the functions $d\mapsto d^{k-1}$, where $k$ is
varying $p$-adically. When $d$ is coprime to $p$, we can define this measure
simply to be $\delta_{d}$, the Dirac measure at $d$ (recalling this is defined
by evaluation at $d$).

When $d$ is divisible by $p$, however, we run into an immutable obstacle.
There is no Dirac measure on $\mathbf{Z}_{p}^{\times}$ corresponding to
evaluation at $p$, since $p\notin\mathbf{Z}_{p}^{\times}$. Moreover, the
function $p\mapsto p^{k}$ can _never_ be interpolated continuously
$p$-adically; it simply behaves too badly for this to be possible. Suppose
there was indeed a measure $\theta_{p}$ with

$\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\theta_{p}=p^{k},$

and then suppose $k_{n}$ is a strictly increasing sequence of integers
$p$-adically tending to $k$. Then

$p^{k_{n}}=\int_{\mathbf{Z}_{p}^{\times}}x^{k_{n}}\cdot\theta_{p}\longrightarrow\int_{\mathbf{Z}_{p}^{\times}}x^{k}\cdot\theta_{p}=p^{k},$

which is clearly impossible since $p^{k_{n}}$ tends to 0. We get around this
issue by taking $p$-stabilisations to kill the coefficients at $p$.

######  Definition 6.1.

We define the _$p$ -stabilisation of $E_{k}$_ to be

$E_{k}^{(p)}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=E_{k}(z)-p^{k-1}E_{k}(pz).$

An easy check shows that

$E_{k}^{(p)}=\frac{(1-p^{k-1})\zeta(1-k)}{2}+\sum_{n\geq
1}\sigma_{k-1}^{p}(n)q^{n},$

where

$\sigma_{k-1}^{p}(n)=\sum_{\begin{subarray}{c}0<d|n\\\ p\nmid
d\end{subarray}}d^{k-1}.$

Note $E_{k}^{(p)}$ is a modular form of weight $k$ and level
$\Gamma_{0}(p)=\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{SL}_{2}({\mathbf{Z}}):p|c\\}$.

We’ve done all the work in proving the following result.

######  Theorem 6.2.

There exists a power series

$\ \mathbf{E}(z)=\sum_{n\geq 0}A_{n}q^{n}\in
Q(\mathbf{Z}_{p}^{\times})[\\![q]\\!]$

such that:

  * (a)

$A_{0}$ is a pseudo-measure, and $A_{n}\in\Lambda(\mathbf{Z}_{p}^{\times})$
for all $n\geq 1$;

  * (b)

For all even $k\geq 4$, we have

$\int_{{{\mathbf{Z}}_{p}^{\times}}}x^{k-1}\cdot\mathbf{E}(z)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{n\geq
0}\bigg{(}\int_{{{\mathbf{Z}}_{p}^{\times}}}x^{k-1}\cdot
A_{n}\bigg{)}q^{n}=E_{k}^{(p)}(z).$

###### Proof.

Clearly, $A_{0}$ is simply the pseudo-measure $x\zeta_{p}/2$ (shifting by 1
again in the opposite direction). We then define

$A_{n}=\sum_{\begin{subarray}{c}0<d|n\\\ p\nmid
d\end{subarray}}\delta_{d}\in\Lambda(\mathbf{Z}_{p}^{\times}).$

By the interpolation property of the Kubota–Leopoldt $p$-adic $L$-function,
$A_{0}$ interpolates the constant term of the Eisenstein series. We also have

$\displaystyle\int_{\mathbf{Z}_{p}^{\times}}x^{k-1}\cdot A_{n}$
$\displaystyle=\sum_{\begin{subarray}{c}0<d|n\\\ p\nmid
d\end{subarray}}\int_{\mathbf{Z}_{p}^{\times}}x^{k-1}\cdot\delta_{d}$
$\displaystyle=\sum_{\begin{subarray}{c}0<d|n\\\ p\nmid
d\end{subarray}}d^{k-1}=\sigma_{k-1}^{(p)}(n),$

so we get the required interpolation property. ∎

######  Remark.

  1. (1)

These results are often presented in a different (equivalent) way. One defines
the _weight space_

$\mathcal{W}(\mathbf{C}_{p})=\mathrm{Hom}_{\mathrm{cts}}(\mathbf{Z}_{p}^{\times},\mathbf{C}_{p}^{\times})$

and shows that, topologically, it is the union of $p-1$ open unit balls in
$\mathbf{C}_{p}$ (centered around the $(p-1)$th roots of unity). The integers
are naturally a subset of $\mathcal{W}(\mathbf{C}_{p})$ via the maps $x\mapsto
x^{k}$, and two integers $k,k^{\prime}$ lie in the same unit ball if and only
if $k\equiv k^{\prime}\hskip 2.0pt(\mathrm{mod}\hskip 2.0ptp-1)$. This space
can be given more structure; there is a rigid analytic space $\mathcal{W}$
such that the elements of $\mathcal{W}(\mathbf{C}_{p})$ are the
$\mathbf{C}_{p}$-points of $\mathcal{W}$. By a Theorem of Amice, giving a
measure on $\mathbf{Z}_{p}^{\times}$ is equivalent to giving a bounded rigid
analytic function on $\mathcal{W}$. Defining $\mathscr{O}(\mathcal{W})$ to be
the space of rigid analytic functions on $\mathcal{W}$, we can view
$\mathbf{E}$ as a power series in $\mathscr{O}(\mathcal{W})[\\![q]\\!]$. We
see it as a $p$-adic interpolation of the Eisenstein series over the weight
space.

  2. (2)

The power series $\mathbf{E}(z)$ is an example of a _$\Lambda$ -adic modular
form_. In particular, it can be colloquially described as the statement:

> “Eisenstein series vary $p$-adically continuously as you change the weight;
> if $k$ and $k^{\prime}$ are close $p$-adically, then the Fourier expansions
> of $E_{k}$ and $E_{k^{\prime}}$ are close $p$-adically.”

The theory of $p$-adic modular forms, and in particular the construction and
study $p$-adic families of Eisenstein series, was introduced by Serre [Ser73]
to give a new construction of the $p$-adic zeta function of a totally real
number field. Pioneering work of Hida went much further than this, showing
that similar families (known as _Hida families_) exist for far more general
modular forms, and his work has been vastly generalised to the theory of
Coleman families and eigenvarieties, with important application to the
construction of $p$-adic $L$-functions. For a flavour of the theory of Hida
and Coleman families, see the book [Hid93], or the more recent [Bel21]. The
original paper constructing the eigencurve was [CM98]; state-of-the-art
results can be found in the recent work [BP].

Part II: Iwasawa’s Main Conjecture

Part II is devoted to the motivation, formulation and study of Iwasawa’s Main
Conjecture. We will start by studying the Coleman map, a map between towers of
local units and $p$-adic measures. This gives a connection between the tower
of cyclotomic units – historically important for their connection to class
numbers – and the Kubota–Leopoldt $p$-adic $L$-function $\zeta_{p}$ from Part
I, and hence a new arithmetic construction of $\zeta_{p}$ (Theorem 7.14). This
construction can be seen as an arithmetic manifestation of the Euler product
expression of the zeta function, and this point of view has led to beautiful
generalisations now known as the theory of Euler Systems. We will then prove a
theorem of Iwasawa (Theorem 9.21), which relates the zeros of the $p$-adic
$L$-function to arithmetic information in terms of units. Using these two
results and class field theory, we will naturally arrive at the formulation
and proof of (a special case of) the Main Conjecture (Theorem 10.8).

######  Notation.

Our study of the Iwasawa Main Conjecture requires certain amount of notation,
which we introduce straight away for convenience. The following should be used
as an index of the key notation, and the reader is urged to consult the
definition of new objects as they appear in the text.

Let $p$ be an odd prime. Throughout this section, we work with coefficient
field $L=\mathbf{Q}_{p}$. For $n\in{\mathbf{N}}$, write

$F_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}={\mathbf{Q}}(\mu_{p^{n}}),\;\;\;F_{n}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}={\mathbf{Q}}(\mu_{p^{n}})^{+};$
$\mathscr{V}_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{O}_{F_{n}}^{\times},\;\;\;\mathscr{V}_{n}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{O}_{F_{n}^{+}}^{\times};$
$K_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{Q}_{p}(\mu_{p^{n}}),\;\;\;K_{n}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{Q}_{p}(\mu_{p^{n}})^{+};$
$\mathscr{U}_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{O}_{K_{n}}^{\times},\;\;\;\mathscr{U}_{n}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{O}_{K_{n}^{+}}^{\times}.$

The extensions $F_{n}/{\mathbf{Q}}$, $K_{n}/\mathbf{Q}_{p}$,
$F^{+}_{n}/{\mathbf{Q}}$ and $K_{n}^{+}/\mathbf{Q}_{p}$ are Galois and totally
ramified at $p$ (the first two of degree $(p-1)p^{n-1}$ and the last two of
degree $\frac{p-1}{2}p^{n-1}$) and we denote $\mathfrak{p}_{n}$ the unique
prime ideal above the rational prime $p$. We let

$F_{\infty}={\mathbf{Q}}(\mu_{p^{\infty}})=\bigcup_{n\geq
1}F_{n},\;\;\;F_{\infty}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(F_{\infty})^{+}=\bigcup_{n\geq
1}F_{n}^{+},$

and
$\Gamma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Gal}(F_{\infty}/{\mathbf{Q}})$,
$\Gamma^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Gal}(F_{\infty}^{+}/{\mathbf{Q}})=\Gamma/\langle
c\rangle$, where $c$ denotes the complex conjugation. Since
$\mathrm{Gal}(F_{n}/{\mathbf{Q}})$ sends a primitive $p^{n}$th root of unity
to a primitive $p^{n}$th root of unity, one deduces an isomorphism

$\chi_{n}:\mathrm{Gal}(F_{n}/{\mathbf{Q}})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}$

determined by the identity

$\sigma(\xi)=\xi^{\chi_{n}(\sigma)},$

for $\sigma\in\mathrm{Gal}(F_{n}/{\mathbf{Q}})$ and $\xi\in\mu_{p^{n}}$ any
primitive $p^{n}$th root of unity. By infinite Galois theory,

$\Gamma=\mathrm{Gal}(F_{\infty}/{\mathbf{Q}})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varprojlim_{n}\mathrm{Gal}(F_{n}/{\mathbf{Q}})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\varprojlim_{n}({\mathbf{Z}}/p^{n}{\mathbf{Z}})^{\times}\cong\mathbf{Z}_{p}^{\times},$
(6.1)

via the _cyclotomic character_
$\chi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varprojlim\chi_{n}$.
Observe that $\chi$ induces an isomorphism
$\Gamma^{+}\cong{{\mathbf{Z}}_{p}^{\times}}/\\{\pm 1\\}$.

We also define

$\mathscr{U}_{n,1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{u\in\mathscr{U}_{n}\;:\;u\equiv
1\text{ (mod
}\mathfrak{p}_{n})\\},\;\;\;\mathscr{U}_{n,1}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{U}_{n,1}\cap\mathscr{U}_{n}^{+}.$
(6.2)

The subsets $\mathscr{U}_{n,1}$ and $\mathscr{U}_{n,1}^{+}$ are important as
they have the structure of $\mathbf{Z}_{p}$-modules (indeed, if
$u\in\mathscr{U}_{n,1}$ or $\mathscr{U}_{n,1}^{+}$ and $a\in\mathbf{Z}_{p}$,
then $u^{a}=\sum_{k\geq 0}{a\choose k}(u-1)^{k}$ converges). By contrast, the
full local units $\mathscr{U}_{n}$ and $\mathscr{U}_{n}^{+}$ are only
${\mathbf{Z}}$-modules.

In general, our notation satisfies the following logic: if $X_{n}$ is any
subgroup of $\mathscr{U}_{n}$, then we let
$X_{n}^{+}=X_{n}\cap\mathscr{U}_{n}^{+}$, $X_{n,1}=X_{n}\cap\mathscr{U}_{n,1}$
and $X_{n,1}^{+}=X_{n}^{+}\cap\mathscr{U}_{n,1}^{+}$. Observe that, since
$\mathscr{V}_{n}\subseteq\mathscr{U}_{n}$, the same applies for any subgroup
$X_{n}$ of $\mathscr{V}_{n}$.

It will be essential for our constructions and methods to consider these
modules at all levels simultaneously. In that mood, we define

$\mathscr{U}_{\infty}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varprojlim_{n}\mathscr{U}_{n},\;\;\;\mathscr{U}_{\infty,1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varprojlim_{n}\mathscr{U}_{n,1};$
$\mathscr{U}_{\infty}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varprojlim_{n}\mathscr{U}_{n}^{+},\;\;\;\mathscr{U}_{\infty,1}^{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varprojlim_{n}\mathscr{U}_{n,1}^{+};$

where all limits are taken with respect to the norm maps. All of these
infinite level modules are compact $\mathbf{Z}_{p}$-modules (since they are
inverse limits of compact $\mathbf{Z}_{p}$-modules) and moreover they are all
endowed with natural continuous actions of
$\Gamma=\mathrm{Gal}(F_{\infty}/{\mathbf{Q}})$ or
$\Gamma^{+}=\mathrm{Gal}(F_{\infty}^{+}/{\mathbf{Q}})$. Accordingly, they are
endowed with continuous actions of the Iwasawa algebras $\Lambda(\Gamma)$ or
$\Lambda(\Gamma^{+})$ (which is the primary reason for passing to infinite
level objects).

We fix once and for all a compatible system of roots of unity
$(\xi_{p^{n}})_{n\in{\mathbf{N}}}$, that is, a sequence where $\xi_{p^{n}}$ is
a primitive $p^{n}$th root of unity such that $\xi_{p^{n+1}}^{p}=\xi_{p^{n}}$
for all $n\in{\mathbf{N}}$. We let $\pi_{n}=\xi_{p^{n}}-1$, which is a
uniformiser of $K_{n}$.

##  7 The Coleman map

In this section we prove a theorem of Coleman that relates local units to
power series over $\mathbf{Z}_{p}$. Using this result, we construct in §7 the
_Coleman map_ , a machine for constructing $p$-adic $L$-functions from the
data of a compatible system of units. We will explain how the Kubota–Leopoldt
$p$-adic $L$-function can be constructed from towers of cyclotomic units using
the Coleman map. Coleman’s map thus provides an important bridge between
analytic objects ($p$-adic $L$-functions) and algebraic structures (the
arithmetic of cyclotomic fields), and will serve as the key step in our
formulation of the Main Conjecture.

In §7.5, we discuss a program started by Perrin-Riou to generalise Coleman’s
work. Given a $p$-adic Galois representation, Perrin-Riou’s big logarithm maps
construct a $p$-adic $L$-function from the data of certain compatible system
of cohomology classes that coincides with Coleman’s map when the Galois
representation is $\mathbf{Q}_{p}(1)$. Coleman’s work is therefore a prototype
for studying $p$-adic $L$-functions in a larger and more conceptual framework.

###  7.1 Notation and Coleman’s theorem

We take

$K_{n}=\mathbf{Q}_{p}(\mu_{p^{n}}),\hskip
12.0ptK_{\infty}=\mathbf{Q}_{p}(\mu_{p^{\infty}})$

to be the local versions of $F_{n}={\mathbf{Q}}(\mu_{p^{n}})$ and
$F_{\infty}={\mathbf{Q}}(\mu_{p^{\infty}})$. We also defined

$\mathscr{U}_{n}=\mathscr{O}_{K_{n}}^{\times}$

to be the module of local units at level $n$, took a compatible system
($\xi_{p^{n}}$) of primitive $p^{n}$th roots of unity, and defined
$\pi_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\xi_{p^{n}}-1$,
a uniformiser for $K_{n}$.

######  Proposition 7.1.

Let $u\in\mathscr{U}_{n}$ be a local unit at level $n$. There exists a power
series $f\in\mathbf{Z}_{p}[\\![T]\\!]$ such that $f(\pi_{n})=u.$

###### Proof.

This is essentially immediate from the fact that $\pi_{n}$ is a uniformiser.
Indeed, $K_{n}$ is totally ramified, so one can choose some
$a_{0}\in\mathbf{Z}_{p}$ such that

$a_{0}\equiv u\hskip 2.0pt(\mathrm{mod}\hskip 2.0pt\pi_{n}),$

and then $a_{1}\in\mathbf{Z}_{p}$ such that

$a_{1}\equiv\frac{u-a_{0}}{\pi_{n}}\hskip 2.0pt(\mathrm{mod}\hskip
2.0pt\pi_{n}),$

and so on, and then define $f(T)=\sum_{n}a_{n}T^{n}.$ By construction, this
satisfies the required property. ∎

The problem with this proposition is that such a power series $f$ is far from
being unique, since we had an abundance of choices for each coefficient. In
the usual spirit of Iwasawa theory, Coleman realised that it was possible to
solve this problem by passing to the infinite tower $K_{\infty}$. Recall that
we defined

$\mathscr{U}_{\infty}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varprojlim_{n}\mathscr{U}_{n},$

where the projective limit is taken with respect to the norm maps
$N_{n,n-1}:K_{n}\rightarrow K_{n-1}$. Coleman’s theorem says that for each
$u\in\mathscr{U}_{\infty}$, there is a _unique_ power series $f_{u}$
satisfying the condition of the above proposition for all $n$.

######  Theorem 7.2 (Coleman).

There exists a unique injective multiplicative map

$\displaystyle\mathscr{U}_{\infty}$
$\displaystyle\to\mathbf{Z}_{p}[\\![T]\\!]$ $\displaystyle u$
$\displaystyle\mapsto f_{u}$

such that $f_{u}(\pi_{n})=u_{n}$ for all $u\in\mathscr{U}_{\infty}$ and $n\geq
1$.

We will prove this in §7.3 below. First, though, we study an important
application.

###  7.2 Example: cyclotomic units

Let us now explain how this theorem is related o the Kubota–Leopoldt $p$-adic
$L$-function. Let $a\in{\mathbf{Z}}$ prime to $p$, and define

$c_{n}(a)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\xi_{p^{n}}^{a}-1}{\xi_{p^{n}}-1}\in\mathscr{U}_{n}.$

######  Lemma 7.3.

We have
$c(a)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(c_{n}(a))_{n}\in\mathscr{U}_{\infty}.$

###### Proof.

This is equivalent to proving that $N_{n,n-1}(c_{n}(a))=c_{n-1}(a).$ Since the
minimal polynomial of $\zeta_{p^{n}}$ over $K_{n-1}$ is
$X^{p}-\zeta_{p^{n-1}}$, for any $b$ prime to $p$ we see that

$N_{n,n-1}(\zeta_{p^{n}}^{b}-1)=\prod_{\eta\in\mu_{p}}(\zeta_{p^{n}}^{b}\eta-1)=\zeta_{p^{n}}^{bp}-1=\zeta_{p^{n-1}}^{b}-1,$

where in the penultimate equality we have used the identity
$X^{p}-1=\prod_{\eta\in\mu_{p}}(X\eta-1)$. Applying this with $b=a$ shows the
numerator of $c(a)$ is norm-compatible, and with $b=1$ the denominator. We
conclude as norm is multiplicative. ∎

It is possible to write down $f_{c(a)}\in\mathbf{Z}_{p}[\\![T]\\!]$ directly
by inspection. Indeed, we see that

$f_{c(a)}(T)=\frac{(1+T)^{a}-1}{T}$

satisfies the required property (and $f_{c(a)}$ is even a polynomial). We now
connect this to the construction studied in §4. Recall the operator
$\partial=(1+T)\tfrac{d}{dT}$ from Lemma 3.14.

######  Proposition 7.4.

We have

$\partial\log f_{c(a)}(T)=a-1-F_{a}(T),$

where $F_{a}(T)$ is the power series defined in Lemma 4.3.

###### Proof.

We compute directly that

$\displaystyle\partial\log f_{c(a)}$
$\displaystyle=\partial\log\big{(}(1+T)^{a}-1\big{)}-\partial\log(T)$
$\displaystyle=\frac{a(1+T)^{a}}{(1+T)^{a}-1}-\frac{T+1}{T}$
$\displaystyle=a-1+\frac{a}{(1+T)^{a}-1}-\frac{1}{T}$
$\displaystyle=a-1-F_{a}(T).\qed$

######  Lemma 7.5.

We have

$\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(\mu_{\partial\log
f_{c(a)}})=\mathrm{Res}_{\mathbf{Z}_{p}^{\times}}(-\mu_{a}),$

where $\mu_{a}$ is the measure of Definition 4.5.

###### Proof.

In terms of power series, the restriction to $\mathbf{Z}_{p}^{\times}$
corresponds to applying the operator $(1-\varphi\circ\psi)$. As
$1-\varphi\circ\psi$ kills the term $a-1$, we find that