Hilbert Modular Forms and Iwasawa Theory

416 Pages·2006·2.407 MB·English

by  Haruzo Hida

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Hida:Hilbert modular forms and Iwasawa theory
Hilbert Modular Forms and
Iwasawa Theory
HARUZO HIDA
Department of Mathematics, UCLA,
Los Angeles, CA 90095-1555, USA
·
CLARENDON PRESS OXFORD
2006
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PREFACE
When I was a toddler, my parents brought me to an esoteric Buddhist temple
(Kongobu-ji“temple”oftheShingonBuddhistsect)inthesouthernhillypartof
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 pic-
ture; afterwards, I was often bothered by nightmarish dreams somehow finding
myself in one of the ghostly mandalas. This is something like placing oneself in
betweentwomirrors,andthenfindinginfinitelymanycopiesofoneself,andthen
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 par-
tial 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
undergraduateeducationincludingmathematics).Ifoundinthebooks,asortof
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 introduc-
tion 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
andautomorphicforms.Fromthattimeon,Ihavebeendeterminedtocreatemy
owntwinmandalasdepictingmyownmathematicaltwinworlds.Ihaverevealed
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.
WhenIarrivedatPrinceton(InstituteforAdvancedStudy)asapostdoctoral
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).Iwassolvingsmallproblemsandgivinganswersashadbeenpredicted.
Small-problemsolvinggivesmesomepleasurebutnotmuch.Afterhavingspent
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
relatingcongruencesamongHeckeeigenformsto(nowcalled)theadjointsquare
L-value at s = 1. Trying to prove that mod p congruence of a Hecke eigen-
form 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
oftheL2-spaceoffunctionsonthehyperbolicPoincar´eupperhalf-planekillsall
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),asnonholomorphicautomorphicformsareasimportantasholomorphicones.
The point is that this p-ordinary projector creates a world where p-adic deform-
ation theory can be built in the neatest way. In this book, I try to describe the
worldofp-ordinaryHilbertmodularformsandtheirdeformationforwhichmany
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 identi-
fication of an appropriate Hecke algebra and the corresponding universal Galois
deformationring(ofMazur).Asapreparationtothis,Igiveadetailedexposition
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 argu-
ments which might still be premature in book form. Thus the identification of
theHeckealgebraandtheGaloisdeformationringtreatedinthisbookislimited
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.
Someearlychaptersarefrommygraduatecoursesin2002–2005atUCLAand
also from my lectures in Peking University in February 2004 and at the morn-
ing 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
hasbeenusedinmyearlierbooks[MFG],[GME]and[PAF],andtheabbreviation
is (basically) common to all of the above three books.
Asforthenotationandtheterminology,wedescribeheresomestandardones
usedatmanyplacesinthisbook.ThesymbolZ denotesthep-adicintegerring
p
inside the field Q of p-adic numbers, and the symbol Z is used to indicate
p (p)
the valuation ring Z ∩Q. We fix throughout the book an algebraic closure Q
p
of Q. A subfield E of Q is called a number field (often assuming [E : Q] :=
dimQE < ∞ tacitly). For a number field E, OE denotes the integer ring of E,
OE,p =OE ⊗ZZp ⊂Ep =E⊗QQp and OE,(p) =OE ⊗ZZ(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 = O . A central simple algebra over F of dimension 4 is
F
called a quaternion algebra over F, which is often denoted by D . A quadratic
/F
extensionM/F iscalledaCMfieldifF istotallyrealandM istotallyimaginary.
For a CM field M, we write R for O .
M
The symbol W is exclusively used to indicate a valuation ring inside Q with
residualcharacteristicp.TheringW couldbeofinfiniterankoverZ butwith
(p)
finite ramification index over Z ; so it is still discrete. The p-adic completion
(p)
lim W/pnW is denoted by W, and we write W =W/pmW =W/pmW.
←− m
n
The symbol A denotes the adele ring of Q. For a subset Σ of rational primes,
wesetA(Σ∞) ={x∈A|x∞ =xp =(cid:1)0 for p∈Σ}.IfΣisempty,A(∞) denotesthe
ringoffiniteadeles.WeputZ = Z anddefineZ =Z ∩Q.IfΣ={p}
Σ p∈Σ p (Σ) Σ
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 ⊗QA and V ⊗QA(Σ∞),
respectively. We identify A(Σ∞)× with the group of ideles x ∈ A× with x = 1
v
for v ∈ Σ(cid:7){∞} in an obvious way. The maximal compact subring of A(∞) is
(cid:1)
denoted by Z(cid:2), which is identified with the profinite ring Z = lim Z/NZ.
p p ←−N
We put Z(Σ) =Z(cid:2)∩A(Σ∞) and Z(p) =Z(cid:2)∩A(p∞). For a module L of finite type,
we write L(cid:2) =L⊗ZZ(cid:2) =←lim− L/NL, L(cid:2)(Σ) =L⊗ZZ(cid:2)(Σ) and L(cid:2)(p) =L⊗ZZ(cid:2)(p).
N
An algebraic group T (defined over a subring A of Q) is called a torus if its
scalar extension T = T ⊗ Q is isomorphic to a product Gr of copies of the
/Q R m
multiplicative group G . The character group X∗(T) = Hom (T ,G )
m alg-gp /Q m/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
ClayMathematicsInstituteasaClayresearchscholarwhilehewasfinishingpre-
paring the manuscript of this book at the Centre de Recherches Math´ematiques
in Montr´eal (Canada) in September 2005.
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