Noncommutative Iwasawa Main Conjectures over Totally Real Fields: Münster, April 2011

215 Pages·2013·1.61 MB·English

by  John Coates

Springer Proceedings in Mathematics
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Volume 29
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John Coates Peter Schneider R. Sujatha
(cid:2) (cid:2) (cid:2)
Otmar Venjakob
Editors
Noncommutative Iwasawa
Main Conjectures over
Totally Real Fields
Mu¨nster, April 2011
123
Editors
JohnCoates PeterSchneider
DepartmentofPureMathematics InstituteofMathematics
andMathematicalStatistics(DPMMS) UniversityofMu¨nster
UniversityCambridge Mu¨nster
Cambridge Germany
UnitedKingdom
R. Sujatha OtmarVenjakob
DepartmentofMathematics InstituteofMathematics
UniversityofBritishColumbia UniversityofHeidelberg
Vancouver Heidelberg
BritishColumbia Germany
Canada
ISSN2194-1009 ISSN2194-1017(electronic)
ISBN978-3-642-32198-6 ISBN978-3-642-32199-3(eBook)
DOI10.1007/978-3-642-32199-3
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012950783
MathematicalSubjectClassification(2010):11R23,11S40,14H52,14K22,19B28
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Preface
Hinohikari
Yabushiwakaneba
Isonokami-
Furinishisatoni
Hanamosakikeri
Forthelightofthesun
Shunsnotthewildthickets,
EveninIsonokami-
Thisvillagegrownancient,
Theflowersareinbloom.
-FuruImamichi(Kokinshu).
The mysterious link between special values of complex zeta and L-functions and
purelyarithmetic problemswas discoveredby Dirichlet and Kummer in the nine-
teenthcentury,andspectacularlygeneralizedinthetwentiethcenturybyBirchand
Swinnerton-Dyerwith the formulation of their celebrated conjecture on the arith-
meticofellipticcurves.WeowetoIwasawathegreatdiscoverythattheseproblems
canbeattackedbyp-adicmethods,wherep isanyprimenumber,providedoneis
preparedtoworkwithaclassofinfiniteGaloisextensionsofthebasefieldF (which
isalwayssupposedtobeafiniteextensionofQ).Iwasawahimselfonlyconsidered
thoseGaloisextensionswhoseGaloisgroupisisomorphictotheadditivegroupof
theringofp-adicintegersZ andthetrivialTatemotive.However,itsoonbecame
p
apparentthathismethodsoughttoapplytoamuchwiderclassofinfiniteextensions,
namelythosewhoseGaloisgroupoverF isap-adicLiegroupofdimension(cid:3) 1,
and to a largeclass of motivesdefinedover F. While it is still notknownhow to
formulateitincompletegenerality,itisnowwidelybelievedthat,inthisgeneralset-
ting,thelinkbetweenspecialvaluesofcomplexL-functionsandarithmeticshould
beexpressedbywhatisknownasamainconjecture.Veryroughlyspeaking,such
v
vi Preface
amainconjectureshouldassertthatanappropriatep-adicL-function,interpolating
specialvaluesofthe relevantcomplexL-functions,shouldcoincidewith a certain
algebraicallydefinedinvariant,usuallycalledacharacteristicelement,whicharises
naturallyfromthearithmeticofthemotiveoverthep-adicLieextension.
ThisbookarosefromaworkshopheldattheUniversityofMu¨nsterfromApril
25–30, 2011. The principal aim of this Workshop was to present the proof of the
first key example of these general ideas, namely, the case when the motive is the
trivialTatemotiveandthep-adicLieextensionF ofF istotallyreal(inaddition,
1
we always assume that F containsthe cyclotomicZ -extensionof Q). The first
1 p
importantprogressonthisproblemgoesbacktoIwasawahimself,althoughweowe
toMazurandWilesthefirstcompleteproofofthemostclassicalcaseofthismain
conjecture(when F is a the compositum of a real abelian base field F with the
1
cyclotomicZ -extensionofQ).Subsequently,Wilesdiscoveredadeepnewmethod,
p
relyingheavilyon the theoryof automorphicforms,for attackingthese problems.
Inthisway,hesucceededinprovingthemainconjecturewhenthebasefieldF is
anytotally realnumberfield, and the Galois groupof F overF is abelian.This
1
bookisconcernedwiththeproblemofhowonecanextendWiles’worktoestablish
thegeneralnon-abeliantotallyrealmainconjectureforthetrivialTatemotive.Two
approaches for doing this were discovered independently and simultaneously, by
Kakde on the one hand, generalizing ideas of Kato, and by Ritter and Weiss on
the other. Both methodsdo in factrequireone to assume a standard conjectureof
Iwasawaaboutthevanishingofhiscyclotomic(cid:2)-invariant,andsofarthishasonly
been proven when the base field F is an abelian extension of Q and the Galois
group of F over F is assumed to be pro-p. Both approaches are discussed in
1
thisbook,but,followingthelecturesattheWorkshop,itislargelyKakde’smethod
which is treated in detail here. One reason for doing this is that the remarkable
setofcongruencesestablishedbyKakdetodescribetheK -groupoftheIwasawa
1
algebra of any compact p-adic Lie group should also apply to attacking the non-
commutativemain conjecture for other motives. Finally, for reasons of space, the
book only contains a written version of the lectures at the workshop which were
closelyrelatedtotheproofofthemainconjecture.
The Scientific Committee for the Mu¨nster Workshop consisted of J. Coates,
P.Schneider(Chairman),R.Sujatha,andO.Venjakob.Itwasmadepossiblebythe
generous financial support of Project A2 within the DFG Collaborative Research
Center 878 “Groups, Geometry, and Actions” at Mu¨nster and by some additional
fundingfromtheERCStartingGrantIWASAWAatHeidelberg.
Cambridge,UK JohnCoates
Mu¨nster,Germany PeterSchneider
Vancouver,Canada R.Sujatha
Heidelberg,Germany OtmarVenjakob
Contents
IntroductiontotheWorkofM.KakdeontheNon-commutative
MainConjecturesforTotallyRealFields ..................................... 1
JohnCoatesandDohyeongKim
ReductionsoftheMainConjecture............................................ 23
R.Sujatha
TheGroupLogarithmPastandPresent ...................................... 51
TedChinburg,GeorgiosPappas,andM.J.Taylor
K ofCertainIwasawaAlgebras,AfterKakde............................... 79
1
PeterSchneiderandOtmarVenjakob
CongruencesBetweenAbelianp-AdicZetaFunctions...................... 125
MaheshKakde
OntheWorkofRitterandWeissinComparisonwithKakde’s
Approach.......................................................................... 159
OtmarVenjakob
NoncommutativeMainConjecturesofGeometricIwasawaTheory....... 183
MalteWitte
vii
•
Contributors
TedChinburg DepartmentofMathematics,UniversityofPennsylvania,Philadel-
phia,USA
John Coates Department of Pure Mathematics and Mathematical Statistics,
UniversityofCambridge,Cambridge,UK
MaheshKakde DepartmentofMathematics,King’sCollegeLondon,London,UK
Dohyeong Kim Department of Mathematics, POSTECH, Pohang, Gyeongbuk,
RepublicofKorea
Georgios Pappas Department of Mathematics, Michigan State University, East
Lansing,USA
PeterSchneider Universita¨tMu¨nster,MathematischesInstitut,Mu¨nster,Germany
R. Sujatha School of Mathematics, Tata Institute of Fundamental Research,
Mumbai,India
DepartmentofMathematics,UniversityofBritishColumbia,Vancouver,Canada
MartinJ.Taylor MertonCollege,OxfordUniversity,Oxford,UK
Otmar Venjakob Universita¨t Heidelberg, Mathematisches Institut, Heidelberg,
Germany
Malte Witte Universita¨t Heidelberg, Mathematisches Institut, Heidelberg,
Germany
ix
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