Class Field Theory– Its Centenary and Prospect

ADVANCED STUDIESIN PURE MATHEMATICS
Chief Editor:30

Eiichi Bannai (Kyushu University)

Class Field Theory– Its Centenary and Prospect

Edited byKatsuya Miyake (Tokyo Metropolitan Univ.)

Mathematical Society of Japan

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ISBN 4-931469-11-6
2000 Mathematics Subject Classification.
Primary 11R37; Secondary 11F70, 11H99, 11M06, 11R23, 11R39, 11R42,11R56, 11S37, 01A60.

Advanced Studies in Pure Mathematics 30
Chief Editor
Eiichi Bannai(Kyushu University)

Editorial Board of the Series
Hitoshi Arai

Eiichi Bannai

Yoshikazu Giga

(Univ. of Tokyo)(Kyushu Univ.)(Hokkaido Univ.)

Ki-ichiro Hashimoto

Kazuya KatoToshitake Kohno(Waseda Univ.)(Univ. of Tokyo)(Univ. of Tokyo)

Yoichi Miyaoka

Seiki Nishikawa

Junjiro Noguchi(Kyoto Univ.)(Tohoku Univ.)(Univ. of Tokyo)

Toshiyuki Tanisaki(Hiroshima Univ.)
PRINTED IN JAPANby Tokyo Shoseki Printing Co., Ltd.

Preface
The Seventh MSJ International Research Institute of the Mathematical Society of Japan was held in Tokyo for ten days from June 3rd
to 12th, 1998. The theme was ‘Class Field Theory – its Centenary andProspect’, which is taken as the title of the book. The program of this international conference is attached at the end of the preface. This volume
is a collection of articles contributed by the speakers of the conference.
All but a few of them are full scale papers. Some of them are expositoryon those subjects which are of central issues of algebraic number theory,
and are prepared by the leading experts; they contains important andinteresting problems with extensive references. Some of them are historical, and vividly explain how number theorists were motivated andexchanged their mathematical ideas.
In 1920 Takagi published the complete version of his class ﬁeld theory as ‘Ueber eine Theorie des relativ Abel’schen Zahlkörpers’ in J.
Coll. Sci. Tokyo, vol. 41. Chapter V of it is devoted to an aﬃrmative solution to ‘Kronecker’s youth-dream’. This problem asks, roughly
speaking, whether all abelian extensions of an imaginary quadratic number ﬁeld could be obtained by the singular moduli and special values ofelliptic functions which have complex multiplication by the elements of
the quadratic ﬁeld; and it was reformulated in a general frame-work byHilbert as the twelfth problem of his celebrated 23 problems. Thereis another problem behind class ﬁeld theory: that is, the principal
ideal theorem which was ﬁnally proved by Furtwängler in 1930 basedon Artin’s general reciprocity law. Artin established this signiﬁcant result in his short paper, ‘Beweis des allgemeinen Reziprozitätsgesetzes’,in Abh. Math. Sem. Univ. Hamburg, vol. 5 (1927). Like the former
problem, the origin of the latter goes back to Kronecker. We may say,however, that the Takagi-Artin class ﬁeld theory has its direct origins inseveral papers by Weber and by Hilbert published in 1897–1898:
H. Weber, Ueber Zahlengruppen in algebraischen Körpern I, Math. Ann.
48 (1897), 433–473; II, 49 (1897), 83–100; III, 50 (1898), 1–26;
D. Hilbert, Die Theorie der algebraischen Zahlkörper, Jber. Deutschen
Math.-Ver.4 (1897), 175–546; Über die Theorie der relativ-Abelschen
Zahlkörper, Nachr. Akad.Wiss. Göttingen 5 (1898), 377–399;(cf. e.g. K. Miyake, The Establishment of the Takagi-Artin Class Field
Theory, in The Intersection of History and Mathematics (ed. J. W.

Dauben et al), Birkhäuser Verlag, Basel-Boston-Berlin, 1994, pp.109–128). This is one of the reasons why the editor of this volume chose theabove stated theme of the Seventh MSJ International Research Institutein 1998 when he was appointed as organizer.
After the marvelous results of Takagi and Artin, the next act of thedrama of class ﬁeld theory was played by H. Hasse. In this energetic
performance, he established his reciprocity law in the form which beautifully embodied the Local–Global Principle in class ﬁeld theory (cf. e.g.
G. Frei’s article in this volume). Under his inﬂuence F. K. Schmidt could
demonstrate local class ﬁeld theory in 1930 based on the global theory.
Then Chevalley gave an ‘arithmetic proof’ to class ﬁeld theory, and also
introduced idèles (cf. e.g. C. Chevalley, La théorie du corps de classes,
Ann. Math. 41 (1940), 394–418). And ﬁnally in 1950, K. Iwasawa andJ. Tate independently developed functional analysis over idele groups ofalgebraic number ﬁelds to present zeta- and L-functions with functional
equations in an elegant manner. Here ideles became a natural and basic
concept of algebraic number theory. It should also be mentioned thatA. Weil gave an important impetus by introducing the ring of adèles in
his lecture notes, Adèles and algebraic groups, at Princeton in 1959.
In 1951 and 1952, G. Hochschild, T. Nakayama and J. Tate successfully gave a cohomological description to class ﬁeld theory. In hislectures at Princeton in 1951–52, E. Artin treated class ﬁeld theory withthis method. It enabled him to give an axiomatic presentation, ‘classformation’. Then extensive studies were carried out by Y. Kawada withthe partial help of I. Satake.
It would be too much for me to give even a brief sketch of the development of the theory of automorphic functions and forms. I wouldlike, however, to point out just two signiﬁcant works. Mainly motivated by Hilbert’s twelfth problem, E. Hecke developed his works on‘Hilbert modular functions’ of two variables in his two papers publishedin Math. Ann. 71 (1912) and ibid. 74 (1913); the title of the second
paper is ‘Über die Konstruktionen relativ-Abelscher Zahlkörper durch
Modulfunktionen von zwei Variabeln’. Here he deals with SL2 over thering of integers of a real quadratic number ﬁeld which discontinuously
acts on a direct product of two copies of the complex upper half plane.
The other work I point out here is one of C. L. Siegel’s works, Symplectic Geometry, Amer. J. Math. 65 (1943). In this paper he worked on
Spn over the rational number ﬁeld and established the analytic theory ofmoduli of (polarized) Abelian varieties. There followed extensive works
on automorphic functions and forms in many variables with respect tosemi-simple and reductive algebraic groups over algebraic number ﬁelds.
Weil’s lecture notes referred above should also be mentioned here.

Towards the end of the 1960’s there appeared three important works
by K. Iwasawa, by G. Shimura and by R. P. Langlands. Concerning theﬁrst one and the third, we have splendid articles by R. Greenberg andby B. Casselman, respectively, in this book. Let me just point out oneof Shimura’s works here:
G. Shimura, On canonical models of arithmetic quotients of boundedsymmetric domains, Ann. Math. 91 (1970), 144–222; II, ibid. 92 (1970),
528–558.
This paper represented a modern summit in the language of adele geometry of which Kronecker once dreamed and then Hilbert anticipatedwith his twelfth problem.
On the history of Hilbert’s twelfth problem, N. Schappacher gave
two interesting lectures in our conference as you will see from the program at the end of the preface. However, we could not include his article
in this book. For those who are interested in it, I refer to his article, On
the history of Hilbert’s twelfth problem: a comedy of errors, Matériaux
pour l’histoire des mathématiques au XXe siècle (Nice, 1996), 243–273,
Sémin. Congr., 3, Soc. Math. France, Paris, 1998. In our conferenceJohn Coates gave two inspiring lectures on Iwasawa theory of elliptic
curves for graduate students and young mathematicians at the requestof the organizers. Those who are interested in the subject are recommended to see his article, Fragments of the GL2 Iwasawa theory of elliptic curves without complex multiplication, Arithmetic theory of elliptic
curves (Cetraro, 1997), 1–50, Lecture Notes in Math., 1716, Springer,Berlin, 1999.
I have mentioned only a strictly limited number of important works
on those topics which are closely related with class ﬁeld theory. It is,of course, apparent that this list is far from suﬃcient. I certainly failed
to mention such important topics as class ﬁeld theory in positive characteristic, higher dimensional class ﬁeld theory, and so on. (As for theﬁrst theme, we are happy to include an article of P. Roquette in this volume.) I just hope this humble review of mine may help younger number
theorists who might be interested in this volume.

Acknowledgement
The chief organizer of the international conference and the editor ofthis volume would like to express his gratitude to those who ﬁnancially
supported these scientiﬁc activities:

Mathematical Society of Japan,
Japan Association for Mathematical Sciences,
Waseda University Advanced Research Institute for Science and Engineering,Tokyo Metropolitan University,
IBM Japan, Ltd.,
Nissan Science Foundation,
Mr. Kenshiro Koto, Director and General Manager, Tokyo Gas Co.,Ltd.,
TAMAT Number Theory Association,and several anonymous personae.
We were also partially supported by
the Grant-in-Aid for Scientiﬁc Research (A) No. 08304004, Ministryof Education, Science, Sports and Culture,
the Grant-in-Aid for Scientiﬁc Research (A) No. 10304004, Japan
Society for the Promotion of Science, andthe Grant-in-Aid for Scientiﬁc Research (B) No. 11440013, Japan
Society for the Promotion of Science.

December 25, 2000
Katsuya Miyake,
EditorAll papers in this volume have been refereed and are in final form.
No version of any of them will be submitted for publication elsewhere.

The Seventh International Research Institute of theMathematical Society of Japan– Class Field Theory – its Centenary and Prospect
DATE: ARRIVAL DAY Tuesday, June 2, 1998
TALKS BEGIN Wednesday, June 3, 1998
TALKS END Friday, June 12, 1998
(There will be no talks in the afternoon of Saturday, June 6, and on
Sunday, June 7.)
PLACE: Building 55, School of Science and Engineering, Waseda
University Okubo, Shinjuku, Tokyo 169, JapanPROGRAM:June 3
10:10 Opening Address
10:20 - 10:50 Shokichi Iyanaga (Japan Acad.)
Memories of Professor Teiji TAKAGI
11:00 - 12:00 Günther Frei (Univ. Laval)
How H. Hasse was led to the Local-Global Principle,
the Reciprocity Laws and Local Class Field Theory.
13:30 - 14:20 Noriyuki Otsubo (Univ. Tokyo)
Recent Progress on the Finiteness of Torsion Algebraic
Cycles14:30 - 15:20 Kanetomo Sato (Tokyo Inst. Tech.)
Finiteness of a certain Motivic Cohomology of Varieties
over Local and Global Fields
15:50 - 16:40 Moulay Chrif Ismaili (Univ. Oujda)
The Capitulation Problem for Certain Number Fields
16:50 - 17:50 Hiroshi Suzuki (Nagoya Univ.)
On the Capitulation Problem

June 4
11:15 - 12:15 Peter Stevenhagen (Univ. Amsterdam)
Generating Class Fields using Shimura Reciprocity
13:45 - 14:45 Takeshi Saito (Univ. Tokyo)
Modular Forms and p-adic Hodge Theory
15:40 - 16:40 Spencer Bloch (Univ. Chicago)Higher Theta Functions
16:50 - 17:50 Kazuya Kato (Univ. Tokyo)
Log Hodge Structures and Classifying Spaces

June 5
10:00 - 11:00 Shuji Saito (Tokyo Inst. Tech.)
Generalization of the Theorem of Brauer-Hasse-Noether
11:15 - 12:15 Ivan Fesenko (Univ. Nottingham)
Neukirch Construction of the Reciprocity Map
13:45 - 14:45 Masakazu Yamagishi (Nagoya Inst. Tech.)
A Survey of p-Extensions
15:15 - 16:15 Helmut Koch (Humboldt-Univ., Berlin)
History of the Theorem of Shafarevich in the Theory ofClass Formations
16:30 - 17:30 Thong Nguyen Quang Do (Univ. Franche-Comté)Galois Module Structure of Class-Formations and free
pro-p-Extensions
18:30 - 20:30 RECEPTION at Avaco

June 6
10:00 - 11:00 Peter Roquette (Univ. Heidelberg)
Class Field Theory in Characteristic p: its Origin andHistory
11:15 - 12:15 Tsuneo Tamagawa (Yale Univ.)
Class ﬁeld theory is easy: Hazewinkel-Iwasawa Theoryof Class Fields of Function Fields over Finite Fields

June 810:00 - 10:50 Winfried Kohnen (Univ. Heidelberg)
Class Numbers of Imaginary Quadratic Fields
11:00 - 11:50 Ryotaro Okazaki (Doshisha Univ.)
On Parities of Relative Class Numbers of Certain
CM-Extensions
13:20 - 14:10 Akito Nomura (Kanazawa Univ.)
On Embedding Problems with restricted Ramiﬁcations
14:20 - 15:10 Sang Geun Hahn (KAIST)
Some Congruences for Binomial Coeﬃcients
15:40 - 16:40 Florian Pop (Univ. Bonn)On the Absolute Galois Group of the Rationals

June 910:00 - 11:00 Ralph Greenberg (Univ. Washington)
Iwasawa Theory – Past and Present
11:10 - 12:00 Manabu Ozaki (Waseda Univ.)
Iwasawa Invariants of Zp -Extensions over an imaginary
Quadratic Field
13:30 - 14:20 Masato Kurihara (Tokyo Metropolitan Univ.)
Iwasawa Theory and Ideal Class Groups of large NumberFields
14:30 - 15:20 Hiroaki Nakamura (Tokyo Metropolitan Univ.)
Artin Braid Groups and Families of Elliptic Curves
15:50 - 16:40 Hisao Taya (Tohoku Univ.)
On p-adic Zeta Functions and Class Groups of Zp Extensions of certain totally real Fields
16:50 - 17:50 John H. Coates (Univ. Cambridge)
Iwasawa Theory of Elliptic Curves without Complex
Multiplication I

June 10
10:00 - 11:00 Norbert Schappacher (Univ. Strasbourg)
On the History of Hilbert’s 12th Problem, 1900–1932
11:15 - 12:15 William Casselman (Univ. British Columbia)
Early Years of the L-group
13:45 - 14:45 Norbert Schappacher (Univ. Strasbourg)
Complex Multiplication in Arithmetic Algebraic Geometry:
Hecke Characters, Abelian Varieties, and Motives
15:15 - 16:15 Roland Gillard (Institut Fourier)
Obstructions in Deformation Problems
16:30 - 17:30 Thomas Zink (Univ. Bielefeld)
P -adic Periods and Displays

June 1110:00 - 11:00 René Schoof (Univ. Roma II)
Abelian Varieties over Number Fields with good Reduction
everywhere
11:15 - 12:15 Hiromichi Yanai (Aichi Inst. Tech.)
Exceptional Hodge Cycles and unramiﬁed Class Fields
13:45 - 14:45 David R. Kohel (Univ. Singapore)
Hecke Module Structure of Quaternions
15:15 - 16:15 Ram Murty (Queen’s Univ. at Kingston)
What is known about Artin L-Functions ?
16:30 - 17:30 Ichiro Satake (Univ. Chuo)
Classiﬁcation Theory of Semisimple Algebraic Groups

June 1210:00 - 11:00 John H. Coates (Univ. Cambridge)
Iwasawa Theory of Elliptic Curves without Complex
Multiplication II
11:15 - 12:15 Jean Cougnard (Univ. Caen)
On Rings of Integers which are stably free and non free
H8 × C2 Modules
13:45 - 14:45 M. Morishita (Kanazawa Univ.) andT. Watanabe (Osaka Univ.)
Adele Geometry of Numbers
15:05 - 16:05 Takashi Ono (Johns Hopkins Univ.)
On Shafarevich-Tate Sets
16:15 - 17:15 Yoshihiko Yamamoto (Osaka Univ.)
The Canonical Power Series associated with EllipticCurves over Q
17:30 - 18:30 Farewell Party

CONTENTS
Shokichi Iyanaga — Memories of Professor Teiji Takagi

1Maruti Ram Murty — On Artin L-Functions

13

Günther Frei — How Hasse was led to the Theory of QuadraticForms, the Local-Global Principle, the Theory of the Norm
Residue Symbol, the Reciprocity Laws, and to Class Field Theory
31
Ivan Fesenko — Nonabelian Local Reciprocity Maps

63

Akito Nomura — Embedding Problems with restricted Ramiﬁcations and the Class Number of Hilbert Class Fields
79
Helmut Koch — The History of the Theorem of Shafarevich in theTheory of Class Formations
87
Masakazu Yamagishi — A Survey of p-Extensions

107

Thong Nguyen Quang Do — Galois Module Stucture of p-Class
Formations
123
Thomas Zink — A Dieudonné Theory for p-Divisible Groups

139

Peter Stevenhagen — Hilbert’s 12th Problem, Complex Multiplication and Shimura Reciprocity
161David R. Kohel — Hecke Module Structure of Quaternions

177

Ichiro Satake — On Classiﬁcation of Semisimple Algebraic Groups 197
Bill Casselman — The L-Group217

Roland Gillard — Groupe des Obstructions pour les Déformations
de Représentations Galoisiennes
259√René Schoof — Abelian Varieties over Q( 6) with Good Reduction Everywhere287
Hiromichi Yanai — Hodge Cycles and Unramiﬁed Class Fields

307

Noriyuki Otsubo — Recent Progress on the Finiteness of Torsion
Algebraic Cycles
313
Kanetomo Sato — Finiteness of a certain Motivic Cohomology
Group of Varieties over Local and Global Fields
325

Ralph Greenberg — Iwasawa Theory – Past and Present

335

Manabu Ozaki — Iwasawa Invariants of Zp -Extensions over anImaginary Quadratic Field
387
Hisao Taya — On p-Adic Zeta Functions and Class Groups of Zp Extensions of certain Totally Real Fields
401
Winfried Kohnen — Class Numbers of Imaginary Quadratic Fields 415
Ryotaro Okazaki — On Parities of Relative Class Numbers ofcertain CM-Extensions
419
Sang Geun Hahn and Dong Hoon Lee — Some Congruences for
Binomial Coeﬃcients
445
Jean Cougnard — Stably Free and Not Free Rings of Integers

463

Mohammed Ayadi, Abdelmalek Azizi and Moulay Chrif Ismaili— The Capitulation Problem for certain Number Fields
467
Hiroshi Suzuki — On the Capitulation Problem

483

Masanori Morishita and Takao Watanabe — Adele Geometry ofNumbers
509
Takashi Ono — On Shafarevich–Tate Sets

537

Peter Roquette — Class Field Theory in Characteristic p, itsOrigin and Development549