Introduction to Non-Abelian Class Field Theory, An: Automorphic Forms of Weight 1 and 2-Dimensional Galois Representations

177 Pages·2017·5.88 MB·English

by  Toyokazu Hiramatsu

m
o
c
c.
ntifi
e
ci
s
d
orl
w
w.
m wwonly.
oe
ed fral us
dn
ao
os
nler
wp
DoFor
ory 17.
Field Theon 08/21/
s 9
Clas53.5
n 1.
a1
n-Abeliby 78.1
o
N
o
n t
o
cti
u
d
o
ntr
n I
A
Series on Number Theory and Its Applications* ISSN 1793-3161
Series Editor: Shigeru Kanemitsu (Kindai University, Japan)
Editorial Board Members:
V. N. Chubarikov (Moscow State University, Russian Federation)
Christopher Deninger (Universität Münster, Germany)
Chaohua Jia (Chinese Academy of Sciences, PR China)
Jianya Liu (Shangdong University, PR China)
H. Niederreiter (RICAM, Austria)
m

o
c
c. Advisory Board:
ntifi A. Schinzel (Polish Academy of Sciences, Poland)
e
sci M. Waldschmidt (Université Pierre et Marie Curie, France)
d
orl
w
w.
m wwonly. Published
oe
ed fral us Vol. 7 Geometry and Analysis of Automorphic Forms of Several Variables
dn Proceedings of the International Symposium in Honor of Takayuki Oda
ao
os
nler on the Occasion of His 60th Birthday
wp
DoFor edited by Yoshinori Hamahata, Takashi Ichikawa, Atsushi Murase &
ory 17. Takashi Sugano
Field Theon 08/21/ V ol. 8 NPruomcebeedri nTghse oorfy t:h Ae r6itthh mCehtiinca i–nJ Saphaann gSrei-mLianar
Class 53.59 edited by S. Kanemitsu, H.-Z. Li & J.-Y. Liu
n 1. Vol. 9 Neurons: A Mathematical Ignition
a1
n-Abeliby 78.1 by Masayoshi Hata
o Vol. 10 Contributions to the Theory of Zeta-Functions: The Modular
N
o Relation Supremacy
n t
o by S. Kanemitsu & H. Tsukada
cti
u
od Vol. 11 Number Theory: Plowing and Starring Through High Wave Forms
n Intr Proceedings of the 7th China–Japan Seminar
A edited by Masanobu Kaneko, Shigeru Kanemitsu & Jianya Liu

Vol. 12 Multiple Zeta Functions, Multiple Polylogarithms and
Their Special Values
by Jianqiang Zhao
Vol. 13 An Introduction to Non-Abelian Class Field Theory: Automorphic
Forms of Weight 1 and 2-Dimensional Galois Representations
by Toyokazu Hiramatsu & Seiken Saito
*For the complete list of the published titles in this series, please visit:
www.worldscientific.com/series/sntia
LaiFun - An Introduction to Non-Abelian Class Field Theory.indd 1 25-08-16 9:11:34 AM
m
o
c
c.
ntifi
e
ci
s
d
orl
w
w.
m wwonly.
oe
ed fral us
dn
ao
os
nler
wp
DoFor
ory 17.
Field Theon 08/21/
s 9
Clas53.5
n 1.
a1
n-Abeliby 78.1
o
N
o
n t
o
cti
u
d
o
ntr
n I
A
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
m
o
c
c.
ntifi
e
ci
s
d
worl British Library Cataloguing-in-Publication Data
w. A catalogue record for this book is available from the British Library.
m wwonly.
oe
ed fral us Book cover: The artwork “Circle, Triangle and Square (The Universe)” by Sengai, Edo period.
dn
oaso Courtesy of Idemitsu Museum of Arts.
nler
wp
DoFor
ory 17.
Field Theon 08/21/ SAAeuNrti oe ImsN ooTnrR pNhOuicDm FUboeCrrmT TIsh OoefoN rW y T eaOign hd Nt I O1ts aN An-pdAp B2l-iEcDaLitmiIoAennNss i— oCn LVaAlo GlS. a1Sl3 o FisI ERLepDr e TseHnEtaOtiRonYs
Class 53.59 Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
n 1. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
a1
n-Abeliby 78.1 esylesctetrmo nniocw o rk nmoewchna onri ctoa lb, ein icnlvuednintegd p, hwoitthoocoupt ywinrigtt,e rne cpoerrdminisgs ioorn a fnryo min ftohrem pautbiolins hsetor.rage and retrieval
o
N
o
n t
o
cti
u
d For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
o
ntr Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
n I is not required from the publisher.
A

ISBN 978-981-3142-26-8
Printed in Singapore
LaiFun - An Introduction to Non-Abelian Class Field Theory.indd 2 19-08-16 5:18:20 PM
August16,2016 15:15 ws-book9x6 AnIntroductiontoNon-AbelianClassFieldTheory ws-book9x6 pagev
m
o
c
c.
ntifi
e
ci
s
d
orl Dedicated to Tomio Kubota
w
w.
m wwonly.
oe
ed fral us
dn
ao
os
nler
wp
DoFor
ory 17.
Field Theon 08/21/
s 9
Clas53.5
n 1.
a1
n-Abeliby 78.1
o
N
o
n t
o
cti
u
d
o
ntr
n I
A

v
May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws
m TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
o
c
c.
ntifi
e
ci
s
d
orl
w
w.
m wwonly.
oe
ed fral us
dn
ao
os
nler
wp
DoFor
ory 17.
Field Theon 08/21/
s 9
Clas53.5
n 1.
a1
n-Abeliby 78.1
o
N
o
n t
o
cti
u
d
o
ntr
n I
A
August19,2016 17:15 ws-book9x6 AnIntroductiontoNon-AbelianClassFieldTheory ws-book9x6 pagevii
Preface
m
o
c
c.
ntifi
e
ci
s
d
orl
w This monograph is intended to provide a brief exposition of the theory of
w.
m wwonly. athuetoomutosrtpahnidcifnogrmprsoobflewmesigihnta1riatnhdmtehtiecirisaptoplgiceanteiroanlisziencalarsitshfimeeldticth.eOornyetoof
oe
ded frnal us nporonp-aobseedliabnyGTaakloaigsieinxtheinsstiaolnksaotftnhuemStbrearsbfioeuldrsg.CTohnigsrpesrso,b1l9em20.wSaisgnalirfiecaadnyt
ao
os
nler progress has been made in recent years by Langlands and others. In the
wp
DoFor monograph,wediscusssomeoftherelationsbetweenthisproblemandcusp
ory 17. forms of weight 1.
Field Theon 08/21/ tersT1heanmdo2n,owgreadpihsccuosnsshisitgsheorfrneicnieprcohcaitpytelraswasnadndanaraitphpmenetdiicx.coInngrCuhenapce-
s 9 relations for (non-abelian) dihedral polynomials. In addition, Chapter 1
Clas53.5 contains an overview of modular forms and Hecke operators. Chapter 3
n 1.
a1 will be devoted to the study of Hecke’s indefinite modular forms of weight
on-Abeliby 78.1 1se,raiensd. a relation between positive definite theta series and indefinite theta
N
o LetΓ beafuchsiangroupofthefirstkindandletd bethedimensionof
n t 1
o thelinearspaceofcuspformsofweight1onthegroupΓ. Itisnoteffective
cti
odu tocomputethenumberd1 bymeansoftheRiemann-Rochtheorem. Inthe
An Intr fitrnaaclecfhoarmptuelra,ofanPdaratlsIo,dwiescguisvseds1ommoedf2o.rmula for d1 by using the Selberg

The first chapter of Part II contains a very remarkable account of var-
ious aspects of the theory including Galois representations of odd type,
the Artin conjecture, the Langlands program, base change and icosahedral
representations. In Chapter 6, we discuss some relations between Maass
cusp forms and Galois representations of even type. We also introduce
some aspects of automorphic hyperfunctions of weight 1 related to Hecke’s
theta series. Chapter 7 begins with reviewing some basic facts concerning
vii
August16,2016 15:15 ws-book9x6 AnIntroductiontoNon-AbelianClassFieldTheory ws-book9x6 pageviii
viii An Introduction to Non-Abelian Class Field Theory
five conjectures in arithmetic, and we discuss some relations between these
conjectures. In Chapter 8, we describe a family of modular series associ-
ated with indefinite binary quadratic forms. These series introduced by
Polishchuk generate the same space of weight 1 modular forms as Hecke’s
indefinite theta series.
The dimension of the space of Hilbert modular type cusp forms has
been calculated in most of cases, but not yet for the case of weight 1. In
m Chapter 9, we shall discuss the dimension for this remaining case. Let K
o
c.c be a real quadratic field and OK be the ring of integers in K. By the
ntifi technical reason, we assume that the class number of K is equal to 1. The
e
ci purpose of this chapter is more precisely to study the dimension for the
s
d
orl Hilbert modular type cusp forms of weight 1 with respect to the Hilbert
w
w. modular group SL2(OK) through Selberg’s trace formula.
m wwonly. The articles in the Appendix are the reproduction of a manuscript re-
oe lated to the first author’s G¨ottingen talk.
ed fral us
adon Kobe Toyokazu Hiramatsu
os
nler Tokyo Seiken Saito
wp
DoFor June, 2015
ory 17.
Field Theon 08/21/
s 9
Clas53.5
n 1.
a1
n-Abeliby 78.1
o
N
o
n t
o
cti
u
d
o
ntr
n I
A
August16,2016 15:15 ws-book9x6 AnIntroductiontoNon-AbelianClassFieldTheory ws-book9x6 pageix
Contents
m
o
c
c.
ntifi
e
ci
s
d
orl
w
w. Preface vii
m wwonly.
oe
ed fral us Part I 1
dn
ao
os
nler 1. Higher reciprocity laws 3
wp
DoFor 1.1 Some examples of non-abelian case . . . . . . . . . . . . . 4
ory 17. 1.1.1 f(x)=x3−d . . . . . . . . . . . . . . . . . . . . 4
Field Theon 08/21/ 11..11..23 ff((xx))==x4x43−−24xx22++21. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 193
n Class 1.53.59 1.2 M1.2o.d1ularSLfo2r(mZs)aannddHitescckoenogpruereantcoerssub.gr.o.up.s. .. .. .. .. .. .. .. 1155
a1
n-Abeliby 78.1 11..22..23 TMhoeduulpaprefrorhmalsf-apnladnceus.p.fo.rm. s. .. .. .. .. .. .. .. .. .. .. .. 1167
o
N 1.2.4 Hecke operators . . . . . . . . . . . . . . . . . . . 18
o
n t
ctio 2. Hilbert class fields over imaginary quadratic fields 21
u
d
o 2.1 The classical theory of complex multiplication . . . . . . . 21
ntr
n I 2.2 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . 24
A
2.3 Schl¨afli’s modular equation . . . . . . . . . . . . . . . . . 29
2.4 The case of q =47 . . . . . . . . . . . . . . . . . . . . . . 30
3. Indefinite modular forms 37
3.1 Hecke’s indefinite modular forms of weight 1 . . . . . . . . 38
3.2 Ray class fields over real quadratic fields . . . . . . . . . . 38
3.3 Positive definite and indefinite modular forms of weight 1 40
3.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . 44
ix
August16,2016 15:15 ws-book9x6 AnIntroductiontoNon-AbelianClassFieldTheory ws-book9x6 pagex
x An Introduction to Non-Abelian Class Field Theory
3.5 Higher reciprocity laws for some real quadratic fields . . . 51
3.6 Cusp forms of weight 1 related to quartic residuacity . . . 53
3.7 Fundamental lemmas . . . . . . . . . . . . . . . . . . . . . 57
3.8 Three expressions of Θ(τ; K) . . . . . . . . . . . . . . . . 60
4. Dimension formulas in the case of weight 1 67
4.1 The Selberg eigenspace M(k,λ) . . . . . . . . . . . . . . . 67
m 4.2 The compact case . . . . . . . . . . . . . . . . . . . . . . . 71
o
c
ntific. 4.3 T4.h3e.1ArfTinhveaAriarfntinavnadriadn1tmoofdq2uad.r.at.ic.fo.r.m.s.m.o.d2. .. .. .. .. 7766
e
sci 4.3.2 The Atiyah invariant on spin structures . . . . . . 78
d
orl 4.3.3 The Arf invariant and d mod2 . . . . . . . . . . . 80
w 1
w. 4.4 The finite case 1 (: Γ (cid:54)(cid:51)−I) . . . . . . . . . . . . . . . . 82
m wwonly. 4.5 The finite case 2 (: Γ (cid:51)−I) . . . . . . . . . . . . . . . . 86
ed froal use 4.6 The case of Γ0(p) . . . . . . . . . . . . . . . . . . . . . . . 88
dn
ao
os
nler
DowFor p Part II 91
s Field Theory 9 on 08/21/17. 5. 2n5-.o1dni-mdeiGhneasdlioorinaslarlceupGsrpaelsfoeoinsrtmarsteipoornfeswseoenfigtoahdttido1ntsypoef o.d.d.ty.p.e.a.n.d. . . . . 9933
Clas53.5 5.1.1 Artin L-functions and the Artin conjecture . . . . 93
an 11. 5.1.2 2-dimensional Galois representations of odd type
n-Abeliby 78.1 5.2 The caseanodf ttyhpeeLsaAnglaannddsSp:roBgraasme c.ha.n.ge. t.he.o.ry. .. .. .. .. .. 9984
o 4 4
N
o 5.2.1 Results of Serre-Tate . . . . . . . . . . . . . . . . 98
ction t 5.2.2 Base change for GL2 . . . . . . . . . . . . . . . . . 98
u 5.2.3 The case of types A and S . . . . . . . . . . . . 99
d 4 4
o
ntr 5.3 The case of type A5 . . . . . . . . . . . . . . . . . . . . . 101
n I 5.3.1 The first example due to Buhler . . . . . . . . . . 101
A

5.3.2 Icosahedral Artin representations . . . . . . . . . 102
5.4 The Serre conjecture . . . . . . . . . . . . . . . . . . . . . 103
5.5 The Stark conjecture in the case of weight 1 . . . . . . . . 104
5.5.1 The Stark conjecture . . . . . . . . . . . . . . . . 104
(cid:18) (cid:19)
1
5.5.2 The value of L , ε . . . . . . . . . . . . . . . 105
2
6. Maass cusp forms of eigenvalue 1/4 107