(Oxford mathematical monographs) haruzo hida hilbert modular forms and iwasawa theory oxford university press, USA (2006)

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Hilbert Modular Forms and

Iwasawa Theory
HARUZO HIDA
Department of Mathematics, UCLA,
Los Angeles, CA 90095-1555, USA

CLARENDON PRESS · OXFORD
2006


3

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1 3 5 7 9 10 8 6 4 2


PREFACE
When I was a toddler, my parents brought me to an esoteric Buddhist temple
(Kongobu-ji “temple” of the Shingon Buddhist sect) in the southern hilly part of
Osaka in Japan, where I saw a prototypical example of the set of twin mandala
depicting Buddha’s twin universe of the inside and the outside, following the

Shingon philosophy. I was utterly impressed by, or even obsessed with, the picture; afterwards, I was often bothered by nightmarish dreams somehow finding
myself in one of the ghostly mandalas. This is something like placing oneself in
between two mirrors, and then finding infinitely many copies of oneself, and then
one losing one’s identity of one’s whereabouts. One’s present state of existence
is in confusion, common to ordinary people.
When I started learning mathematics in the junior year of undergraduate
study at Kyoto, I read a couple of books, starting with a book on linear partial differential equations, which is the first serious book in mathematics I ever
read (because of the student movement at the time, the university was virtually
closed for my freshman and sophomore years; so, I was given almost no general
undergraduate education including mathematics). I found in the books, a sort of
universe neatly arranged, something like the mandala, but somehow, I felt that
the Buddha sitting at the center (who presides over his world) was missing from
the book. I then read, as the third book of mathematics, Shimura’s introduction to modular and automorphic forms [IAT], where I clearly saw a focus; so,
I decided to pursue number theory, in particular, the theory of modular forms
and automorphic forms. From that time on, I have been determined to create my
own twin mandalas depicting my own mathematical twin worlds. I have revealed
my determination/obsession only to a very small number of people in my life up
until now, because I did not like to appear eccentric. If I remember correctly,
in a queue at a cafeteria at Universit´e de Paris-Sud (Orsay) in 1984, I started a
conversation with my fellow young French mathematicians about what kind of
mathematicians we would like to be, and succinctly, I explained to them about
the mandala and my obsession, and to my surprise, some of them (including
Perrin-Riou and Tilouine) seemed somehow to understood the point, at least to
some extent.
When I arrived at Princeton (Institute for Advanced Study) as a postdoctoral
fellow in 1979, I was fairly desperate, because I had not been able to find even
a clue about how to create a new universe cut out of, say, all elliptic modular
forms (which appeared to me like looking into a pitch-dark well too deep to see
through). I was solving small problems and giving answers as had been predicted.
Small-problem solving gives me some pleasure but not much. After having spent

a couple of months in Princeton, I was really desperate; so, I decided to do


vi

Preface

one more problem solving, finishing up the project (I started with Koji Doi) of
relating congruences among Hecke eigenforms to (now called) the adjoint square
L-value at s = 1. Trying to prove that mod p congruence of a Hecke eigenform with another implies that p is a factor of the L-value, somehow I found a
p-adic projector (acting on modular forms) I named e for some reasons (which
cut the clear surface out of the dark-well water) as the holomorphic projection
of the L2 -space of functions on the hyperbolic Poincar´e upper half-plane kills all
nonholomorphic functions, though I had only a guess of the precise meaning of
the projector at the time. I admit that the non-p-ordinary modular forms are
as equally important as the p-ordinary modular forms (which is in the image of
e), as nonholomorphic automorphic forms are as important as holomorphic ones.
The point is that this p-ordinary projector creates a world where p-adic deformation theory can be built in the neatest way. In this book, I try to describe the
world of p-ordinary Hilbert modular forms and their deformation for which many
theorems can be established easily, leaving the hard work of extending them to
more general nonordinary automorphic forms to mathematicians more efficient
and ambitious.
In this book, several results on ordinary modular forms are presented. First
of all, I describe, in Chapter 3, Fujiwara’s (highly nontrivial) generalization [Fu]
(to the Hilbert modular forms) of the proof by Wiles and Taylor of the identification of an appropriate Hecke algebra and the corresponding universal Galois
deformation ring (of Mazur). As a preparation to this, I give a detailed exposition
of the theory of automorphic forms on a definite quaternion algebra, including
the level-raising argument of R. Taylor. I do not touch the level-lowering arguments which might still be premature in book form. Thus the identification of
the Hecke algebra and the Galois deformation ring treated in this book is limited
to minimally ramified deformations. After finishing this, we discuss three major

applications that I found:
1. A description of Greenberg’s L-invariant of the adjoint square L-function,
and its generic nonvanishing;
2. A solution to the integral basis problem of Eichler;
3. A proof of the torsion property of the (modular) adjoint square Selmer
groups, and related Iwasawa modules.
I have been studying all these topics since 1996 after I learned of Fujiwara’s
work. I have written some papers on the subjects (at least for elliptic modular
forms), but the treatment in this book is new and also covers more general
cases.
Some early chapters are from my graduate courses in 2002–2005 at UCLA and
also from my lectures in Peking University in February 2004 and at the morning center of Mathematics at Beijing in August, 2004. I have been encouraged
by many people (especially those who supported me in my desperate period).


Preface

vii

I would like to thank all these people including the audience in my lectures and
the people at the above institutions.
Haruzo Hida,
Los Angeles,
October, 2005


viii

Preface

Suggestions to the reader
In the text, articles are quoted by abbreviating the author’s name, for example,
three articles by Hida–Tilouine are quoted as [HT], [HT1] and [HT2]. There
is one exception: articles written by myself are quoted, for example, as [H04a]
and [H98] indicating also the year published (or the year written in the case of
preprints). For these examples, [H04a] and [H98] are published in 2004 and in
1998, respectively. Books are quoted by abbreviating their title. For example,
one of my earlier books with the title: Geometric Modular Forms and Elliptic
Curves is quoted as [GME]. Our style of reference is slightly unconventional but
has been used in my earlier books [MFG], [GME] and [PAF], and the abbreviation
is (basically) common to all of the above three books.
As for the notation and the terminology, we describe here some standard ones
used at many places in this book. The symbol Zp denotes the p-adic integer ring
inside the field Qp of p-adic numbers, and the symbol Z(p) is used to indicate
the valuation ring Zp ∩ Q. We fix throughout the book an algebraic closure Q
of Q. A subfield E of Q is called a number field (often assuming [E : Q] :=
dimQ E < ∞ tacitly). For a number field E, OE denotes the integer ring of E,
OE,p = OE ⊗Z Zp ⊂ Ep = E ⊗Q Qp and OE,(p) = OE ⊗Z Z(p) ⊂ E. Often we fix
a base field denoted by F which is usually a totally real field. For the base field
F , we simply write O = OF . A central simple algebra over F of dimension 4 is
called a quaternion algebra over F , which is often denoted by D/F . A quadratic
extension M/F is called a CM field if F is totally real and M is totally imaginary.
For a CM field M , we write R for OM .
The symbol W is exclusively used to indicate a valuation ring inside Q with
residual characteristic p. The ring W could be of infinite rank over Z(p) but with
finite ramification index over Z(p) ; so it is still discrete. The p-adic completion
limn W/pn W is denoted by W , and we write Wm = W/pm W = W/pm W.
←−
The symbol A denotes the adele ring of Q. For a subset Σ of rational primes,
we set A(Σ∞) = {x ∈ A|x∞ = xp = 0 for p ∈ Σ}. If Σ is empty, A(∞) denotes the

ring of finite adeles. We put ZΣ = p∈Σ Zp and define Z(Σ) = ZΣ ∩Q. If Σ = {p}
for a prime p, we write A(p∞) for A(Σ∞) . For a vector space of a number field
E, we write VA = V (A) and VA(Σ∞) = V (A(Σ∞) ) for V ⊗Q A and V ⊗Q A(Σ∞) ,
×
respectively. We identify A(Σ∞) with the group of ideles x ∈ A× with xv = 1
for v ∈ Σ {∞} in an obvious way. The maximal compact subring of A(∞) is
denoted by Z, which is identified with the profinite ring p Zp = limN Z/N Z.
←−
We put Z(Σ) = Z ∩ A(Σ∞) and Z(p) = Z ∩ A(p∞) . For a module L of finite type,
we write L = L ⊗Z Z = limN L/N L, L(Σ) = L ⊗Z Z(Σ) and L(p) = L ⊗Z Z(p) .
←−
An algebraic group T (defined over a subring A of Q) is called a torus if its
scalar extension T/Q = T ⊗R Q is isomorphic to a product Grm of copies of the
multiplicative group Gm . The character group X ∗ (T ) = Homalg-gp (T/Q , Gm/Q )
is simply denoted by X(T ), and elements of X(T ) are often called weights of T .


ACKNOWLEDGEMENTS
The author acknowledges partial support from the National Science Foundation
(through the research grants: DMS 0244401 and DMS 0456252) and from the
Clay Mathematics Institute as a Clay research scholar while he was finishing preparing the manuscript of this book at the Centre de Recherches Math´ematiques
in Montr´eal (Canada) in September 2005.


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CONTENTS

1

Introduction
1.1 Classical Iwasawa theory
1.1.1 Galois theoretic interpretation of the class group
1.1.2 The Iwasawa algebra as a deformation ring
1.1.3 Pseudo-representations
1.1.4 Two-dimensional universal deformations
1.2 Selmer groups
1.2.1 Deligne’s rationality conjecture
1.2.2 Ordinary Galois representations
1.2.3 Greenberg’s Selmer groups
1.2.4 Selmer groups with general coefficients
1.3 Deformation and adjoint square Selmer groups
1.3.1 Nearly ordinary deformation rings
1.3.2 Adjoint square Selmer groups and differentials
1.3.3 Universal deformation rings are noetherian
1.3.4 Elliptic modularity at a glance
1.4 Iwasawa theory for deformation rings
1.4.1 Galois action on deformation rings
1.4.2 Control of adjoint square Selmer groups
1.4.3 Λ-adic forms
1.5 Adjoint square L-invariants
1.5.1 Balanced Selmer groups
1.5.2 Greenberg’s L-invariant
1.5.3 Proof of Theorem 1.80

1
2
9
12

13
17
19
19
26
28
29
31
32
35
41
43
47
47
49
56
59
62
64
67

2

Automorphic forms on inner forms of GL(2)
2.1 Quaternion algebras over a number field
2.1.1 Quaternion algebras
2.1.2 Orders of quaternion algebras
2.2 A short review of algebraic geometry
2.2.1 Affine schemes
2.2.2 Affine algebraic groups

2.2.3 Schemes
2.3 Automorphic forms on quaternion algebras
2.3.1 Arithmetic quotients
2.3.2 Archimedean Hilbert modular forms
2.3.3 Hilbert modular forms with integral coefficients
2.3.4 Duality and Hecke algebras

70
76
76
80
86
87
91
93
95
96
99
104
109


xii

Contents

2.3.5 Quaternionic automorphic forms
2.3.6 The Jacquet–Langlands correspondence
2.3.7 Local representations of GL(2)
2.3.8 Modular Galois representations

The integral Jacquet–Langlands correspondence
2.4.1 Classical Hecke operators
2.4.2 Hecke algebras
2.4.3 Cohomological correspondences
2.4.4 Eichler–Shimura isomorphisms
Theta series
2.5.1 Quaternionic theta series
2.5.2 Siegel’s theta series
2.5.3 Transformation formulas
2.5.4 Theta series of imaginary quadratic fields
The basis problem of Eichler
2.6.1 The elliptic Jacquet–Langlands correspondence
2.6.2 Eichler’s integral correspondence

110
114
117
125
129
129
132
134
138
139
139
141
147
150
153
156

158

Hecke algebras as Galois deformation rings
3.1 Hecke algebras
3.1.1 Automorphic forms on definite quaternions
3.1.2 Hecke operators
3.1.3 Inner products
3.1.4 Ordinary Hecke algebras
3.1.5 Automorphic forms of higher weight
3.2 Galois deformation
3.2.1 Minimal deformation problems
3.2.2 Tangent spaces of local deformation functors
3.2.3 Taylor–Wiles systems
3.2.4 Hecke algebras are universal
3.2.5 Flat deformations
3.2.6 Freeness over the Hecke algebra
3.2.7 Hilbert modular basis problems
3.2.8 Locally cyclotomic deformation
3.2.9 Locally cyclotomic Hecke algebras
3.2.10 Global deformation over a p-adic field
3.3 Base change
3.3.1 p-Ordinary Jacquet–Langlands correspondence
3.3.2 Base fields of odd degree
3.3.3 Automorphic base change
3.3.4 Galois base change
3.4 L-invariants of Hilbert modular forms
3.4.1 Statement of the result
3.4.2 Deformation without monodromy conditions

162

163
163
167
168
174
180
183
183
187
189
200
210
213
217
230
233
243
245
245
246
247
248
251
251
256

2.4

2.5

2.6

3


3.4.3
3.4.4
3.4.5
3.4.6
3.4.7

Contents

xiii

Selmer groups of induced representations
L-invariant of induced representations
Adjoint square Selmer groups and differentials
Proof of Theorem 3.73
Logarithm of the universal norm

262
265
274
279
283

4

Geometric modular forms

4.1 Modular curves
4.1.1 Modular curves and elliptic curves
4.1.2 Arithmetic Weierstrass theory
4.1.3 Moduli of level N
4.1.4 Toric action
4.1.5 Compactification
4.1.6 Action of an adele group
4.2 Hilbert AVRM moduli
4.2.1 Abelian variety with real multiplication
4.2.2 AVRM moduli with level structure
4.2.3 Classical Hilbert modular forms
4.2.4 Toroidal compactification
4.2.5 Tate AVRM
4.2.6 Hasse invariant
4.2.7 Geometric Hilbert modular forms
4.2.8 p-Adic Hilbert modular forms
4.2.9 Hecke operators
4.3 Hilbert modular Shimura varieties
4.3.1 Abelian varieties up to isogenies
4.3.2 Finite level structure
4.3.3 Modular varieties of level Γ0 (N)
4.3.4 Isogeny action
4.3.5 Reciprocity law at CM points
4.3.6 Hilbert modular Igusa towers
4.3.7 Finite level Hecke algebra
4.3.8 q-Expansion
4.3.9 Universal Hecke algebras
4.4 Exceptional zeros and extension
4.4.1 Λ-adic automorphic representations
4.4.2 Extensions of automorphic representations

4.4.3 Extensions of Galois representations

286
286
286
287
289
291
292
294
296
296
300
303
307
311
313
315
317
319
323
324
330
332
332
334
334
336
337
338

341
343
347
351

5

Modular Iwasawa theory
5.1 The cyclotomic tower of deformation rings
5.1.1 Control of deformation rings
5.1.2 Kă
ahler dierentials as Iwasawa modules
5.1.3 Dimension of R

353
353
354
355
363


xiv

Contents

5.2

5.3

Adjoint square exceptional zeros

5.2.1 Order of exceptional zeros
5.2.2 Base change of Selmer groups
Torsion of Iwasawa modules for CM fields
5.3.1 Ordinary CM fields and their Iwasawa modules
5.3.2 Anticyclotomic Iwasawa modules
5.3.3 The L-invariant of CM fields

366
367
375
377
377
379
383

References

387

Symbol Index

397

Statement Index

399

Subject Index

401

1
INTRODUCTION

In this book, we study classical, p-adic and archimedean Hilbert modular and
quaternionic automorphic forms and their Hecke algebras in detail, relating them
to the (corresponding) universal Galois deformation rings and automorphic and
modular L-values. So in this introductory chapter, let us describe some of the
focal points treated in this book. This book is, in some sense, a sequel of [MFG]
and [PAF], where a general introduction to the p-adic deformation theory of
Galois representations and automorphic forms on Shimura varieties is given. In
[MFG], a detailed proof (following Wiles’ fundamental work [Wi2]) of the identification of the (elliptic modular) Hecke algebra and the Galois deformation ring is
given (and this identity of the two naturally given algebras is one of the principal
ingredients of his proof of Fermat’s last theorem). In [PAF], the emphasis is put
more on the automorphic and geometric side of the theory of the p-adic Hecke
algebra (and Shimura varieties). Here, we describe Iwasawa-theoretic aspects of
the theory along with the generalization of Wiles’ identification to the Hilbert
modular case first worked out by Fujiwara in [Fu]. To be more precise, for a
nearly p-ordinary two-dimensional Galois representation ρ associated to a Hilbert
modular Hecke eigenform, we discuss the following four topics:
1. the identification of the local ring of ρ of an appropriate p-adic Hecke algebra
with the universal Galois deformation ring of ρ (the Taylor–Wiles method
worked out by Fujiwara [Fu]; Chapter 3);
2. torsion property (over the Iwasawa algebra) of the adjoint square Selmer
groups of Hilbert modular forms and the anticyclotomic Iwasawa modules
associated with CM fields (Chapter 5);
3. the L-invariant of the adjoint square of ρ (Section 3.4), and its relation to
the tower of Galois deformation rings and Hecke algebras over the cyclotomic Zp -extension (modular Iwasawa theory; Chapter 5) and to nontrivial
extensions of p-adic automorphic representations (Section 4.4);

4. the basis problem of Eichler (Sections 2.6 and 3.2.7).
The second, third, and fourth items are direct applications of the first. The
principal new results besides the first topic in this book are Theorems 3.47,
3.59, 3.73, 4.29, 5.9, 5.27 and 5.33 and Corollaries 3.74, 4.32 and 5.39.
Chapter 4 is a long summary of the results on geometric Hilbert modular forms


2

Introduction

described in [PAF] (which are used throughout this book), though added in Section 4.4 are some new results on the close relation between nontrivial extensions
of automorphic representations and the exceptional zeros of adjoint square p-adic
L-functions. A natural path to follow after describing these topics is
(a) the problem of nonvanishing modulo p of abelian critical L-values as an
application of the theory of Hilbert modular Shimura varieties treated in
[PAF] Chapter 3 (see [H04b] and [H05b]);
(b) the proof of the anticyclotomic main conjecture in Iwasawa theory for CM
fields under appropriate assumptions (see [H05d]).
In the near future, we hope to treat these two topics (a) and (b) in book form.
In this chapter, we shall give a brief outline of the first three topics (1–3).
The fourth topic is to relate quarternionic automorphic forms and elliptic and
Hilbert modular forms through theta series, which will be discussed in Chapter 2.
Since this is an introductory chapter putting forward special cases of the results
described in detail in the later chapters, often we give only a sketch of a proof and
possibly even omit proofs (which will be given later in a more general setting),
or we shall just content ourselves with indicating the place where a proof can be
found. Many open questions are also discussed here.
We fix a prime p > 2, algebraic closures Q of Q and Qp of Qp and the field
embeddings ip : Q → Qp and i∞ : Q → C.

1.1 Classical Iwasawa theory
Before starting with a review of classical Iwasawa theory, we describe the group of
roots of unity as a group scheme. A brief summary of schemes and group schemes
will be given in Section 2.2; so, here we limit ourselves to the group, denoted
by µN , of N -th roots of unity (which is the main subject of research in classical
Iwasawa theory). We regard µN for a positive integer N as a covariant functor
from the category ALG of commutative algebras with identity into the category
AB of abelian groups. See [MFG] Section 4.1 or [GME] Section 1.4 for a brief
description of functors. Thus µN associates with an algebra A the commutative
group àN (A) = { Aì |ζ N = 1} for the identity 1 ∈ A. For two algebras A and
B, the set of homomorphisms Homalg (A, B) is made up of ring homomorphisms
taking the identity of A to the identity of B. Then for φ ∈ Homalg (A, B), the
corresponding group homomorphism φA : µN (A) → µN (B) is given by µN (A)
ζ → φ(ζ) ∈ µN (B), and this gives rise to the covariant functoriality.
For any given algebra R, we can think of a covariant functor ALG → SET S,
written as Spec(R), given by Spec(R)(A) = Homalg (R, A). For φ ∈ Homalg (A, B)
and ϕ ∈ Spec(R)(A), the corresponding map φA : Spec(R)(A) → Spec(R)(B) is
given by φA (ϕ) = φ ◦ ϕ. The functor Spec(R) is called the affine scheme of the
algebra R; we will discuss schemes more fully in Section 2.2.


Classical Iwasawa theory

3

Let W be a complete discrete p-adic valuation ring flat over Zp with residue
field F. If R is a p-profinite W -algebra, we can think of a slightly different covariant functor Spf(R) from the category of p-profinite algebras into
SET S called an affine formal scheme under the p-profinite topology. For any
p-profinite W -algebra R, Spf(R) is called the formal spectrum of R. This is

to impose the p-profinite continuity on morphisms we consider. Thus for any
p-profinite W -algebra A, Spf(R)(A) = HomW -alg (R, A) made up of continuous
W -algebra homomorphisms. The category of p-profinite W -algebras is then sent
into the category of formal affine spectra faithfully by the contravariant functor
R → Spf(R). In particular, we have Hom(Spf(R), Spf(B)) = HomW -alg (B, R),
where W -algebra homomorphisms are assumed to be continuous with respect to
the p-profinite topology.
For two covariant functors Φ, Ψ : ALG → C into a category C, φ : Φ → Ψ is a
morphism (of covariant functors) if we have morphisms φA : Φ(A) → Ψ(A) of C
indexed by A ∈ ALG such that the following diagram is commutative for each
algebra homomorphism ρ : A → B:
φA

Φ(A) −−−−→ Ψ(A)


Ψ(ρ)

Φ(ρ)
φB

Φ(B) −−−−→ Ψ(B),
where Φ(ρ) : Φ(A) → Φ(B) is the functorial map associated with ρ; in particular,
these maps satisfy Φ(ρ ◦ η) = Φ(ρ) ◦ Φ(η) by covariant functoriality. If we have
an inverse morphism ψ : Ψ → Φ such that φA ◦ ψA and ψA ◦ φA are identity
morphisms in C for all A, we call Φ and Ψ isomorphic.
The covariant functor µN has a special property that
ιA : µN (A) ∼
= Homald (Z[X]/((1 + X)N − 1), A))
via µN (A) ζ ↔ ζ ∈ Homald (Z[X]/((1 + X)N − 1), A)) if ζ(1 + X) = ζ. We

verify that ιB ◦ φA (ζ) = φA ◦ ζ for any φ ∈ Homalg (A, B) (so, ζ ↔ ζ gives rise to
an isomorphism of covariant functors). In such a case, we say that the functor µN
is represented by the affine scheme Spec(R) with R = Z[X]/((1 + X)N − 1). We
write µN = Spec(Z[X]/((1 + X)N − 1)) and identify the two functors Spec(R)
and µN . Since µN is actually a functor into the subcategory AB of SET S and
is identified with Spec(Z[X]/((1 + X)N − 1)), we call µN an affine commutative
group scheme, whose A-points µN (A) are made of N -th roots of unity in A
(e.g. [GME] Example 1.6.3). This is a point of view regarding a scheme S/Z as an
association to each commutative algebra A of its A-points S(A). Mathematically,
A → S(A) is a covariant functor from the category ALG of (commutative)
algebras (with identity) into the category of sets. If a scheme G/Z actually has
values in the category GP of groups, G is called a group scheme. Since the ring
Z[X]/((1 + X)N − 1) is free of finite rank, µN is called a locally free (actually
free) group scheme.


4

Introduction

Fixing a base algebra B, we can think of the notion of the B-affine scheme
S = Spec(R)/B for a B-algebra R, which is by definition the covariant functor
A → S/B (A) = HomB-alg (R, A) from the category of B-algebras into SET S. If
S has values in GP , S is called a group B-scheme. Extending the notion of local
freeness, if an affine group B-scheme A → G(A) = HomB-alg (R, A) over a base
ring B is given by a B-algebra R which is locally free of finite rank, we call G
locally free of finite rank over B (or finite flat over B). We call two group schemes
G/B and H/B isomorphic if they are isomorphic as two covariant functors (from
B-algebras) into GP .
We put µp∞ = n µpn . Thus µp∞ (A) is the group of all p-power roots of unity

in A, and if A is sufficiently large, µp∞ (A) is p-divisible; so, it is an example
of a special group scheme called a Barsotti–Tate group. An affine group scheme
G/B is called a Barsotti–Tate group G over B if G/B is p-divisible, is given by
limn G[pn ]/B for its pn -torsion subgroups G[pn ], and G[pn ] is locally free of finite
−→
rank over B (see [T], [CBT] and [ARG] Chapter III for such groups).
If µp∞ (A) is p-divisible, it is not finitely generated, though µN (A) for finite N
is a cyclic group if A is a domain of characteristic prime to N .
Exercise 1.1 An abelian group G is called divisible if x → nx is surjective
for any nonzero integer n. Show that a nonzero divisible group G is not finitely
generated as a group. Hint: use the fundamental theorem of abelian groups.
We can think of the constant functor A → ΦN (A) = Z/N Z for any algebra A,
which is a functor from ALG to GP (with ΦN (ρ) given by the identity map
idZ/N Z of Z/N Z for any ρ ∈ Homalg (A, B)). This functor for N > 1 is not
isomorphic to the group scheme µN , because µp (Fp ) = {1} for a prime p but
Φp (Fp ) = (Z/pZ) = {1}.
Exercise 1.2 Show that ΦN is not of the form Spec(R) for a commutative
ring R (thus, ΦN is not an affine scheme). Propose a modification of ΦN which
is an affine scheme Spec(R) and Spec(R)(A) = ΦN (A) for any indecomposable
commutative algebras A.
If A is a subring of an algebraically closed field B of characteristic different from p (for example, B = Qp or C), we often write A[µpn ] for the
extension of A generated by µpn (B). We have the p-adic cyclotomic character
σ
N (σ)
for all ζ ∈ µp∞ (Q), and for the
N : Gal(Q[µp∞ ]/Q) ∼
= Zì
p given by =
ì
unique subgroup àp1 (Zp ) ⊂ Zp , we write Q∞ for the subfield of Q[µp∞ ] fixed by

N −1 (µp−1 (Zp )). Then Gal(Q∞ /Q) ∼
= Zp
= Γ = 1 + pZp . Note that Γ = (1 + p)Zp ∼
∞
s n
s
p
via Zp
s → (1 + p) =
∈
Γ,
and
Q
is
called
the
cyclotomic
∞
n=0 n
Zp -extension over Q. For a given number field M , the composite M∞ = Q∞ M
is called the cyclotomic Zp -extension of M .
In the late 1950s, Iwasawa started studying the arithmetic of Zp -extensions
(particularly the cyclotomic ones) and created many important profinite modules
X with a continuous action of Γ, which are often called Iwasawa modules, and


Classical Iwasawa theory

5

it turned out to be a success, bringing us new knowledge of the arithmetic of
Zp -extensions (see, for example, [Iw]). The book [ICF] by Washington is a good
introduction to Iwasawa theory. For any complete valuation ring W finite flat
over Zp with residue field F, the completed group algebra
n
n
W [[Γ]] := lim W [Γ/Γp ] = lim W [x]/((1 + x)p − 1) ∼
= W [[x]]
←−
←
−
n
n

acts on X (if X is a W -module in addition to being a Γ-module). Indeed, by
the continuity of the action, we may assume that X = limn Xn for a finite
←−
n
module Xn over the finite group Γ/Γp , and the action is compatible with respect
n
Xn . Passing to the limit, W [[Γ]] =
to the projections Γ
Γ/Γp and X
pn
limn W [Γ/Γ ] acts on X = limn Xn . We write Λ = ΛW for W [[Γ]], which is a
←−
←−
local ring with maximal ideal mΛ = mW + (x) and Λ/mΛ = F.
n
n

∼ W [[x]].
Exercise 1.3 Prove limn W [Γ/Γp ] = limn W [x]/((1 + x)p − 1) =
←−
←−
n
pn
Hint: choose a generator γ ∈ Γ and associate with (γ mod Γ ) ∈ W [Γ/Γp ]
n
n
the element (1 + x mod (1 + x)p − 1) ∈ W [x]/((1 + x)p − 1).
n
n
We will later give a proof of limn W [Γ/Γp ] = limn W [x]/((1 + x)p − 1) ∼
=
←−
←−
W [[x]] as Corollary 1.20 via Galois deformation theory; so, the point of the above
exercise is to give a ring theoretic proof of this fact.
Let us discuss a prototypical example of Iwasawa modules. In this discussion,
we assume W = Zp . For simplicity, we assume only in this section that M/Q is
at most tamely ramified at p. Here a prime is tamely ramified in M/Q if its ramification index is prime to p. Since p fully ramifies in the union of p-extensions
Q∞ /Q, M and Q∞ are linearly disjoint; so, we have Gal(M∞ /M ) ∼
= Γ, canonically, by restricting σ ∈ Gal(M∞ /M ) to Q∞ . The Zp -extension M∞ /M has layers
n
of intermediate fields Mn fixed by Γn = Γp . Let Cn be the p-Sylow subgroup
of the class group of Mn . By Galois conjugation, Gal(Mn /M ) = Γ/Γn ∼
= Z/pn Z
acts on Cn . The norm map Nm,n : Cm → Cn (m > n) is compatible with the
Galois action, and we can form the projective limit C∞ = limn Cn with respect
←−

to the norm map. Since Cn is a finite p-group, C∞ is a p-profinite (compact)
group with a continuous Γ-action. Thus we get an Iwasawa module C∞ , which
is a module over Λ.
By class field theory, Cn is isomorphic to the Galois group Gal(Hn /Mn ) of
p-Hilbert class field Hn by the Artin reciprocity map. Since Hm ⊃ Hn (m > n),
we have the restriction map Resm,n : Gal(Hm /Mm ) → Gal(Hn /Mn ) which is
surjective. By class field theory, we have the following commutative diagram:

∼

Gal(Hm /Mm ) −−−−→ Cm


N

Res
m,n

m,n

∼

Gal(Hn /Mn ) −−−−→ Cn .


6

Introduction

Thus C∞ is canonically isomorphic to X = Gal(H∞ /M∞ ), where H∞ /M∞ is

the maximal unramified p-abelian extension. The action of σ ∈ Gal(Mm /M ) on
Gal(Hm /Mm ) is given by taking an extension σ of σ to Hm /M and conjugating
τ ∈ Gal(Hm /Mm ) by σ: τ → στ σ −1 . Since Gal(Hm /Mm ) is commutative, this
action of σ is independent of the choice of the extension σ. Since Hn Mm is a
maximal abelian extension of Mn inside Hm , the Galois group Gal(Hm /Hn Mm )
is the derived group of Gal(Hm /Mn ). Writing γ = 1 + p and identifying Γ
n
with Gal(M∞ /M ), Gal(Mm /Mn ) is generated by γ p ; so, the derived group
Gal(Hm /Hn Mm ) is generated by the commutators
[γ p , τ ] = γ p τ γ −p τ −1 = (γ p − 1)τ
n

n

n

n

(writing additively on the right-hand side). This shows
Gal(Hm /Hn Mm ) ∼
= (γ p − 1)Cm .
n

Since Mm /Mn fully ramifies at p, we have Gal(Hn Mm /Mm ) ∼
= Gal(Hn /Mn ).
Thus we have the following control theorem (cf. [Iw] Sections 3–4 and [ICF]
Section 13.3):
Theorem 1.4 (Control)

We have C∞ ⊗Λ Λ/(γ p − 1)Λ ∼

= Cn for all n > 0.
n

n
Note that Λ ∼
= Zp [[x]] by γ → 1 + x and (1 + x)p − 1 is in the unique maximal
ideal of Λ, by Nakayama’s lemma, and the finiteness of Cn tells us the following.

Corollary 1.5

The Iwasawa module C∞ is a torsion Λ-module of finite type.

Nakayama’s lemma we referred to is the third assertion of the following lemma,
which is actually due to Krull and Azumaya (according to Nakayama as Nagata
wrote in his books on ring theory):
Lemma 1.6 (Krull–Azumaya, Nakayama)
imal ideal mR and X be an R-module.

Let R be a local ring with max-

1. If X is an R-module of finite type and X = mR X, then X = 0.
2. If mN
R X = 0 for a sufficiently large integer N , then X = mR X implies
X = 0.
3. Suppose that R = limn R/mnR is an mR -adically complete local ring with
←−
finite R/mnR for all n > 0 and that X is a continuous R-module under the
mR -adic topology. Then
(a) X = mR X implies X = 0,
(b) if X ⊗R R/mR is finite dimensional over R/mR , X is an R-module of

finite type,
(c) if further R is a noetherian normal domain, X is an R-module of finite
type and X/xX is a torsion R/xR-module for a prime element x ∈ R,
X is a torsion R-module of finite type.


Classical Iwasawa theory

7

Proof For the proof of the first two assertions, see [MFG] Lemma 1.3. The
assertion (a) of the third assertion follows from (2) applied to quotients X/mnR X
and passing to the limit.
As for (b), take elements x1 , . . . , xr in X whose classes modulo mR X give
rise to a basis of X/mR X. Consider the R-linear map π : Rr → X given by
n
n r
π(a1 , . . . , ar ) =
i ai xi . Applying (2) to Coker((π mod mR ) : (R/mR ) →
n
n
X/mR ), we find that (π mod mR ) is surjective for all n, and passing to the limit,
we find that Im(π) is mR -adically dense in X. Since Rr is compact (because of
finiteness of R/mnR ), Im(π) is closed, and hence π is surjective; so, X is of finite
type.
As for (c), we note that the localization R(x) is a discrete valuation ring
(because a normal noetherian integral domain of dimension 1 is a discrete valuation ring; e.g., [CRT]). Then that X(x) /xX(x) is a torsion R(x) /(x)-module
means X(x) /xX(x) = 0, which implies X(x) = 0 by (2). Then X has to be a
torsion R-module.
✷

Let F be a totally real field of finite degree with integer ring O. We now
assume that M = F [µp ] (note that p is tamely ramified in M/Q). Then we
have Gal(M/F ) → àp1 by the Teichmă
uller character . Again by conjugation,
Gal(M/F ) acts on C∞ . Let Cn [ω i ] = {x ∈ Cn |σ(x) = ω i (σ)x} (n = 0, 1, . . . , ∞).
n
The limit ω = limn→∞ N p exists, giving a character ω : Gal(Q/F ) →
n
n−1
µp−1 (Zp ), because (Z/pn Z) has order pn − pn−1 (so, N p −p
≡ 1 mod pn ).
∞
The limit is the Teichmă
uller character of Gal(M [àp ]/M ) and we regard
it as an ideal character by class field theory so that ω(l) = ω(F robl ) for
the Frobenius element F robl for prime ideal l outside p. We put ω(p) = 0
if p|p (but ω n (p) = 1 for all p including p|p if ω n is the trivial character). Write χ = ω a for a fixed integer a, and regard it as a complex
valued character by composing i∞ ◦ i−1
p . Then we consider the complex
L-function
χ(a)N (a)−s =

L(s, χ) =
a

1−
l

χ(l)
N (l)s

−1

absolutely and locally uniformly convergent on the right half-plane defined
by Re(s) > 1, where l runs over all prime ideals of O and a runs over all
integral ideals of O. This L-function can be continued to the entire complex
plane as a complex meromorphic function having pole at s = 1 only when
χ is the trivial character (e.g., [LFE] Theorem 2.5.1). Since F is totally real,
as was shown by Siegel, Klingen and Shintani (see [LFE] Corollary 2.5.1),
L(1 − n, χ) ∈ Z(p) [µp−1 (C)] for Z(p) = Q ∩ Zp in Qp . Embed Z(p) [µp−1 (C)]
into Zp by ip ◦ i−1
∞ , and regard L(1 − n, χ) ∈ Zp . In the late 1970s, Deligne and
Ribet [DR] (and Barsky and Cassou-Nogu´es; see [LFE] Chapter 3) constructed


8

Introduction

a power series ΦF (x; ω i ) ∈ Λ such that for even integer i with 0 ≤ i < p − 1
ΦF (γ 1−n − 1; ω i )
=

(γ 1−n − 1) p|p (1 − ω −n (p)N (p)n−1 )L(1 − n, ω −n )
i−n
(p)N (p)n−1 )L(1 − n, ω i−n )
p|p (1 − ω

if i = 0
if i > 0

for all n > 0. Since L(s, χ) satisfies a functional equation of the form s ↔ 1 − s
with the Γ-factor having a pole at 1 − n if χ = ω i−n for odd i (e.g., [LFE]
Section 8.6), the value L(1 − n, ω i−n ) vanishes for odd i, and therefore, we only
look into even i.
Here we insert a general fact valid for integral domains of dimension ≤ 2. Over
a principal ideal domain A, any torsion module X of finite type is isomorphic to
e(P )
for finitely many prime elements P . For a regular local ring A of
P A/P
dimension two (including Λ), if X is a torsion A-module of finite type, we can
find a homomorphism ι : X → P A/P e(P ) A for finitely many prime elements P
with small kernel and cokernel (e.g. [ICF] Section 13.2 or [BCM] VII.4.4). When
A = Λ, the word “small” means “finite.” A generator of the ideal P P e(P ) is
called the characteristic element in A and the characteristic power series of X
when A = Λ, which is uniquely determined modulo Λ× .
Here is a theorem of A. Wiles [Wi1] which was originally called the main
conjecture of the Iwasawa theory for F :
Theorem 1.7 (A. Wiles) Let F be a totally real number field of finite degree.
Then we have char(C∞ [ω 1−i ])(x) = ΦF (x; ω i ) for even i up to a unit multiple
in Λ for an odd prime p.
This theorem was originally conjectured by Iwasawa and was proved by Mazur
and Wiles for F = Q in [MzW]. Later Wiles gave a proof valid for general totally
real F in [Wi1]. It appears to have a mismatch because the ω 1−i -part C∞ [ω 1−i ]
is described by the L-function of ω i . However, we may regard L(1 − n, ω i−n ) as
a function of characters N n−1 ω i−n ≡ ω i−1 mod p; so, the ω 1−i -part C∞ [ω 1−i ]
is described by the L-function whose domain is made up of characters congruent
to ω i−1 modulo p.
In [H05d], a theorem similar to this one is given, taking a p-ordinary CM
field to be a base field in place of the totally real field F in the above theorem. For such a generalization, we need to deal with modules over a complete

regular local ring with dimension ≥2. Indeed, we can generalize the construction of the characteristic power series to a more general ring R. We consider a
formal power series ring W [[x1 , . . . , xg ]] and assume that R is a normal integral domain finite and torsion-free over W [[x1 , . . . , xg ]]. Then a prime ideal P
of R is called a prime divisor if dim R/P = dim R − 1 for the Krull dimension
dim R of R. In this situation, the localization RP of R at a prime divisor P
is a discrete valuation ring. Then for a torsion R-module X of finite type, we
have that XP = X ⊗R RP has finite length over RP . We write P (X) for the


Classical Iwasawa theory

9

length. Then char(X) = P P P (X) is an ideal of R, which is called the characteristic ideal of X. When R = W [[x1 , . . . , xg ]], the characteristic ideal of X is
generated by the characteristic power series char(X)(x) ∈ W [[x1 , . . . , xg ]] (see
[BCM] Chapter 7). Basically by definition, we have a morphism of R-modules
i : X → P R/P e(P ) whose cokernel and kernel vanish after localization at every
prime divisor (and this gives a precise definition of the kernel and the cokernel
of i being “small”). An R-module M is said to be pseudo-null if MP = 0 for all
prime divisors P .
Exercise 1.8 Prove that for a Λ-module M of finite type, MP = 0 for all prime
divisors P of Λ if and only if the order of M is finite.
1.1.1 Galois theoretic interpretation of the class group
In order to give a Galois theoretic generalization of the class group, we introduce here briefly Galois cohomology groups. Let Tp = Qp /Zp . For any abelian
p-profinite compact or p-torsion discrete module X, we define the Pontryagin
dual module X ∗ by X ∗ = Homcont (X, Tp ) and give X ∗ the topology of uniform
convergence on every compact subgroup of X. By Pontryagin’s duality theory
(cf. [FAN] or [LFE] Section 8.3), we have (X ∗ )∗ ∼
= X canonically.
Exercise 1.9 Show that X ∗ ∼
= X noncanonically if X is finite.

Exercise 1.10 Prove that X ∗ is a discrete module if X is p-profinite and X ∗
is compact if X is discrete (e.g., [LFE] Lemma 8.3.1).
By this fact, if X ∗ is the dual of a profinite module X = limn Xn for finite
←−
modules Xn with surjections Xm
Xn for m > n, X ∗ = n Xn∗ is a discrete
module which is a union of finite modules Xn∗ .
For the (profinite) Galois group G over a field inside Q or a p-adic field
inside Qp and a continuous G-module X, we denote by H q (G, X) the continuous group cohomology with coefficients in X. If X is finite, H q (G, X)
is as defined in [MFG] 4.3.3. If X = limn Xn for finite G-modules Xn , we
←−
have H q (G, X) = limn H q (G, Xn ), and if X = n Xn is discrete with finite
←−
Xn , we have H q (G, X) = limn H q (G, Xn ). For a general K-vector space V
−→
with a continuous action of G and a G-stable W -lattice L of V , we define
H q (G, V ) = H q (G, L) ⊗W K. For a finite set S of rational primes, let F S /F
be the maximal extension unramified outside S and the archimedean place of F ,
and write GSF for Gal(F S /F ). For a finite set S of rational primes containing
all ramified primes in F/Q, we have QS = F S . By a result of Tate (e.g., [MFG]
4.4.2), GSF has (virtual) cohomological dimension 2; so, only H 0 , H 1 , and H 2 are
important. We have
H 0 (G, X) = X G = {x ∈ X|gx = x for all g ∈ G},


10

Introduction

and if X is finite, the first cohomology is defined by

c

H 1 (G, X) =

→ X : continuous|c(στ ) = σc(τ ) + c(σ) for all σ, τ ∈ G}
{G −
b

{G −
→ X|b(σ) = (σ − 1)x for x ∈ X independent of σ}

.

As for the second cohomology, 2-cocycles c : G×G → X are continuous functions
satisfying the following relation:
c(α, β) + c(αβ, γ) = α · c(β, γ) + c(α, βγ)
for all α, β, γ ∈ G. For any continuous function b : G → X, ∂b(α, β) = b(αβ) −
αb(β) − b(α) is easily checked to be a 2-cocycle by computation. Such 2-cocycles
obtained from b : G → X are called a 2-coboundary. Then
H 2 (G, X) =

{2-cocycles with values in X}
.
{2-coboundaries with values in X}

Exercise 1.11 Check that ∂b as above is a 2-cocycle.
If G = Gal(Qp /K) for a finite extension K/Qp , by Tate duality (see [MFG]
Example 4.42 and Theorem 4.43),
H 2−i (G, X) ∼
= Hom(H i (G, X ∗ (1)), Q/Z)

for finite X.
For each Galois character ψ : Gal(Q/F ) → W × , we write ψ for the F× valued character ψ mod mW for the maximal ideal mW of W . For any W -module
X, we write X(ψ) for the Galois module whose underlying W -module is X
with Galois action given by ψ. We simply write X(i) for X(N i ). In particular
Zp (1) ∼
µ n (Q) as Galois modules. Note that GK = GpK = Gal(F p /K)
= lim
←−n p
for any intermediate field K of F p /F , where F p /F is the maximal extension
unramified outside p and ∞.
Fix an integer i as above, and put ψ = ω 1−i . We consider the Galois
cohomology group H 1 (GF , Tp (ψ)) and define the Selmer group of Tp (ψ) over
K by
Res

SelK (Tp (ψ)) = Ker(H 1 (GK , Tp (ψ)) −−→

H 1 (Il , Tp (ψ))),

(1.1.1)

l

where l runs over all prime ideals of K and Il is a chosen inertia subgroup at l
of GK . By the inflation-restriction sequence (e.g., [MFG] 4.3.4),
0 → H 1 (Gal(M/F ), H 0 (GM , Tp (ψ)))
→ H 1 (GF , Tp (ψ)) → H 0 (Gal(M/F ), H 1 (GM , Tp (ψ)))
→ H 2 (Gal(M/F ), H 0 (GM , Tp (ψ)))
is exact. Since Gal(M/F ) (M = F [µp ]) has order prime to p,
H 1 (Gal(M/F ), H 0 (GM , Tp (ψ))) = H 2 (Gal(M/F ), H 0 (GM , Tp (ψ))) = 0.

Classical Iwasawa theory

11

Thus we have
∼
H 1 (GF , Tp (ψ)) ∼
→ HomGF (GM , Tp (ψ)).
= H 0 (Gal(M/F ), H 1 (GM , Tp (ψ))) −
ι

Since Hm /Mm is the maximal p-abelian extension unramified everywhere and
elements in SelF (Tp (ψ)) are made up of unramified cocycle classes, we find that
SelF (Tp (ψ)) ∼
= Hom(C0 [ψ], Tp ).
Exercise 1.12 Give a detailed proof of the above identity.
This shows that C0 [ψ] ∼
= SelF (Tp (ψ))∗ , which we write as Sel∗F (Tp (ψ)) hereafter. As a consequence of Theorem 1.7, we get the following very precise p-class
number formula:
Corollary 1.13 We have C0 [ψ] ∼
= Sel∗F (Tp (ψ)) canonically, and
|C0 [ψ]| = |SelF (Tp (ψ))| =

|L(0, ψ)|−1
p
|(γ − 1)L(0, ψ)|−1
p

if ψ = ω
if ψ = ω,

where |x|p = |ip (x)|p for the p-adic absolute value | · |p of Qp with |p|p = p−1 .
When F = Q, the direction p |C0 [ψ]| ⇒ p|L(0, ψ) is due to Herbrand (1932), and
the converse p|L(0, ψ) ⇒ p |C0 [ψ]| was later proven by Ribet (1976; see [Ri1]).
The above precise formula follows from Theorem 1.7 because
(a) if X is a torsion Λ-module of finite type and has no nontrivial finite
Λ-submodule, charΛ (X)(α) = charW (X/(x − α)X) by ring theory as long
as X/(x − α)X is finite,
(b) C∞ does not have a nontrivial finite Λ-submodule as proven by
R. Greenberg (cf., [Gr1]).
Exercise 1.14 Let the notation and the assumption be as in (a) above. Using
(b), prove that multiplication by f = x − α on X (f : X
t → f t ∈ X) is
injective. Then using this fact, show (a).
By a similar argument, taking the Pontryagin dual, we find
Corollary 1.15 Let ψ∞ be the restriction of ψ to GM∞ . Then we have a
canonical isomorphism: C∞ [ψ∞ ] ∼
= Sel∗F∞ (Tp (ψ)).
If G is a group with subgroup H, for a given W [H]-module X, we define the
induced G-module IndG
H X = W [G] ⊗W [H] X on which we let g ∈ G act by
g(g ⊗ x) = (gg ⊗ x). Identifying X with 1 ⊗ X ⊂ IndG
H X, the action of H on
1 ⊗ X and X are identical. Thus IndG
H X is an extension of the H-module X to
the bigger group G. Choosing a complete representative set Q of G/H in G, we
∼
∼

find IndG
HX =
q∈Q q ⊗ X =
q∈Q X with q ⊗ X = X as W -modules.
G
We can embed IndH X into HomW [H] (W [G], X) by sending ξ = q q ⊗ aq to
φξ (q) = aq . If we let g ∈ G act on φ ∈ HomW [H] (W [G], X) by (gφ)(x) = φ(g −1 x),