Non-abelian Fundamental Groups and Iwasawa Theory

321 Pages·2011·1.96 MB·English

by  John Coates

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,
UnitedKingdom
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LondonMathematicalSocietyLectureNoteSeries:393
Non-abelian Fundamental Groups
and Iwasawa Theory
Editedby
JOHN COATES
UniversityofCambridge
MINHYONG KIM
UniversityCollegeLondon
FLORIAN POP
UniversityofPennsylvania
MOHAMED SAIDI
UniversityofExeter
PETER SCHNEIDER
UniversitätMünster
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
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PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
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(cid:2)c CambridgeUniversityPress2012
Thispublicationisincopyright.Subjecttostatutoryexception
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Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Non-abelianfundamentalgroupsandIwasawa
theory/editedbyJohnCoates...[etal.].
p. cm.–(LondonMathematicalSocietylecturenoteseries;393)
ISBN978-1-107-64885-2(pbk.)
1. Iwasawatheory. 2. Non-Abeliangroups. I. Coates,J. II. Title. III. Series.
QA247.N56 2011
(cid:3)
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ISBN978-1-107-64885-2Paperback
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Contents
Listofcontributors pagevi
Preface vii
Lecturesonanabelianphenomenaingeometryandarithmetic
FlorianPop 1
OnGaloisrigidityoffundamentalgroupsofalgebraiccurves
HiroakiNakamura 56
AroundtheGrothendieckanabeliansectionconjecture
MohamedSa¨ıdi 72
FromtheclassicaltothenoncommutativeIwasawatheory
(fortotallyrealnumberfields)
MaheshKakde 107
OntheM (G)-conjecture
H
J.CoatesandR.Sujatha 132
GaloistheoryandDiophantinegeometry
MinhyongKim 162
Potentialmodularity–asurvey
KevinBuzzard 188
RemarksonsomelocallyQ -analyticrepresentationsofGL (F)
p 2
inthecrystallinecase
ChristopheBreuil 212
Completedcohomology–asurvey
FrankCalegariandMatthewEmerton 239
Tensorandhomotopycriteriaforfunctionalequationsof(cid:2)-adic
andclassicaliteratedintegrals
HiroakiNakamuraandZdzisławWojtkowiak 258
v
Contributors
ChristopheBreuil Baˆtiment425,C.N.R.S.etUniversite´ Paris-Sud,
91405OrsayCedex,France
KevinBuzzard DepartmentofMathematics,ImperialCollegeLondon,
180Queen’sGate,LondonSW72AZ,UK
FrankCalegari DepartmentofMathematics,NorthwesternUniversity,
2033SheridanRoad,Evanston,IL60208-2730,USA
J. Coates DPMMS, Centre for Mathematical Sciences, Wilberforce Road,
CambridgeCB30WB,UK
Matthew Emerton Department of Mathematics, Northwestern University,
2033SheridanRoad,Evanston,IL60208-2730,USA
Mahesh Kakde Department of Mathematics, University College London,
GowerStreet,LondonWC1E6BT,UK
Minhyong Kim Department of Mathematics, University College London,
GowerStreet,LondonWC1E6BT,UK
HiroakiNakamura DepartmentofMathematics,
OkayamaUniversity,Okayama700-8530,Japan
FlorianPop DepartmentofMathematics,UniversityofPennsylvania,
209South33rdStreet,Philadelphia,PA19104-6395,USA
MohamedSa¨ıdi CollegeofEngineering,Mathematics,andPhysical
Sciences,UniversityofExeter,HarrisonBuilding,NorthParkRoad,
ExeterEX44QF,UK
R.Sujatha SchoolofMathematics,TataInstituteofFundamentalResearch,
HomiBhabhaRoad,Mumbai400005,India
ZdzisławWojtkowiak Universite´ deNice-SophiaAntipolis,De´pt.ofMath.,
LaboratoireJeanAlexandreDieudonne´,U.R.A.auC.N.R.S.,No168,
ParcValrose–B.P.N◦71,06108NiceCedex2,France
vi
Preface
Inhistoricalorigins,serioustopologicalinputtoarithmeticstartsatthelatest
in the early years of the twentieth century with the foundations of class field
theorybyTakagiandArtin.Aroundthemiddleofthecentury,therapiddevel-
opmentofhomologicaltechniquesinalgebraledtoanexplosionofconsequent
activityinnumbertheoryandalgebraicgeometry,includingthetheoryofco-
herentsheaves,thehomologicalinterpretationofArtin’sreciprocitymap,and
Grothendieck’sconstructionofarithmeticcohomologytheoriesbaseduponthe
abstractnotionofGrothendiecktopologies.Then,inthe1980s,Grothendieck
formulated his anabelian conjectures in the celebrated letter to Faltings, and
brought to a hitherto-unexplored depth the interaction between topology and
arithmetic. Therein came into focus the far-reaching vision that the perspec-
tive of non-abelian fundamental groups could lead to a fundamentally new
understandingofdeeparithmeticphenomena,includingthearithmetictheory
ofmoduliandDiophantinefinitenessonhyperboliccurves.Inthe1990s,pro-
found (and perhaps unexpected) progress in Grothendieck’s program was re-
alized through the theorems of Nakamura, Tamagawa, Mochizuki. (See the
contributionsofNakamura,Pop,andSaidi.)
Meanwhile, a certain amount of work in recent years linking fundamental
groups to Diophantine geometry intimates deep and mysterious connections
to the theory of motives and Iwasawa theory, notions usually associated to
Diophantineproblemsinvolvingexactformulaeofwhichthemostcelebrated
example is the conjecture of Birch and Swinnerton-Dyer (cf. Kim’s article).
Infact,theworkthusfarsuggeststhatthestill-unresolvedsectionconjecture
of Grothendieck, whereby maps from Galois groups of number fields to fun-
damentalgroupsofhyperbolicarithmeticcurvesareallproposedtobeofge-
ometric origin, is exactly the sort of key problem that touches the core of all
these areas of number theory and more. Therefore, the time seems right to
vii
viii Preface
encourageamuchbroaderunderstandingofthearithmeticissuessurrounding
anabeliangeometryanditsramifications.
Whiletheoverallimportanceofthetheoremsofanabeliangeometryappears
tobewidelyacknowledged,thereisasyetnotmuchspecificknowledgewithin
thearithmeticgeometrycommunityofitscoherentbodyofconceptsandphi-
losophy,andofthenewtechnologythatyieldsactualresults.Itisourbeliefthat
ahigherlevelofgeneralawarenesswillleadtoagenuinestrengtheningofour
grasp of a wide range of Galois-theoretic and Diophantine phenomena, and
hopefully to significant progress on the section conjecture. The goal then of
thisbookistopresentarticlesthatcontaintheideasandproblemsofanabelian
geometry within the global context of mainstream arithmetic geometry, with
strongemphasisonconnectionstonon-commutativeIwasawatheory.
Broadlyput,anabeliangeometryandnon-commutativeIwasawatheorydeal
indifferentwayswiththeliftingofhomologicalandabelianideasofarithmetic
into the realm of the non-abelian, with connections to homotopy theory. The
complementarynatureofthetwoviewpointsisexpressedinpartbytheexact
sequence
(*) 0→πˆ (X¯,b)→πˆ (X,b)→Gal(Q¯/Q)→0
1 1
associatedtoanalgebraiccurveXdefinedovertherationals,involvingthegeo-
metricfundamentalgroupπˆ (X¯,b)andtheabsoluteGaloisgroupGal(Q¯/Q)of
1
therationals,bothcontributinginintricatelyinter-relatedwaystothestructure
ofthearithmeticfundamentalgroupπˆ (X,b)inthemiddle.Fromthisexactse-
1
quence,oneextractsaninducedouteractionofGal(Q¯/Q)onπˆ (X¯,b).When
1
this actionis viewed atthe level ofthe abelianization H (X¯,Zˆ) of πˆ (X¯,b), it
1 1
extendsnaturallytocompletedgroupalgebrasofquotientgroupsofGal(Q¯/Q),
whichthenbecomethecentralobjectsofstudyinnon-commutativeIwasawa
theory(cf.articlesofCoatesandSujatha,andofKakde).
Here,encodingtheinfinitetowerofnumberfieldsrepresentedbyaninfinite
non-abelianquotientintoasinglegroupalgebraisbothconvenientandcrucial,
and brings with it the intervention of refined tools that combine the structure
theoryofnon-commutativealgebraswithcohomologicalaspectsofGaloisthe-
ory.Inanabeliangeometry,ontheotherhand,themainpointistostudythefirst
group πˆ (X¯,b) in its own right, without abelianizing, even as certain abelian
1
or mildlynon-abelian quotientsor completionsmight mediatethis study.Al-
thoughtheattendantactionofGal(Q¯/Q)isofkeyimportance,itissafetosay
thattheoverwhelminglygeometricnatureofthetechniquesaswellasthenon-
linearity of the action tend to obscure the role of fine invariants of Gal(Q¯/Q)
suchasmightoccurinIwasawatheory.Thereby,withsomeoversimplification,
one could describe both anabelian geometry and non-commutative Iwasawa
Preface ix
theory within the single framework of the sequence (*), the difference lying
onlyinthetechnicaldepthswithwhichattentionisdrawntoeitherthekernel
or the quotient, and in the degree of non-commutativity/non-linearity that is
preservedinitsstudy.
Havingformulatedthusthecommonalityanddifference,theimportanceof
coming up with an approach unified at both the philosophical and the tech-
nologicallevelbecomesquiteobvious.Asdiscussedinanumberofdifferent
chaptersofthisbook,thereisaremarkableoverlapbetweentheanabelianand
Iwasawa-theoreticviewpointsinthestudyofellipticcurves.Ontheotherhand,
when the geometric fundamental group is genuinely non-abelian, as happens
forhyperboliccurves,itremainstodevelopthecorrectanalogueofthisinter-
action.Inadditiontoprogressonthesectionconjectureitself,itishopedthat
an extended program of the Birch and Swinnerton-Dyer type for hyperbolic
curves will result from investigation along these lines. Because these issues
lieatthecrossroadsofimportantresearchinperhapsallofthemainareasof
arithmeticgeometry,ourintentionisthatthecorescientificinteractionofthe
book should be enriched further by surrounding essays (by Breuil, Buzzard,
andCalegariandEmerton)thatdealwithoneotherkeyarenafornon-abelian
ideasandtechniques,namely,theLanglandsprogram.
The articles in this book were written by participants in the special pro-
grammeon‘non-abelianfundamentalgroupsinarithmeticgeometry’thattook
placeattheIsaacNewtonInstituteinthesecondhalfoftheyear2009.Inthe
course of the programme, we benefitted greatly from the advice of director
David Wallace as well as the hospitality of the INI staff, especially Christine
West,EsperanzadeFelipe,andMustaphaAmrani.Wearedeeplyindebtedto
theirefficienthelp.ManythanksareduealsotoDianaGilloolyofCambridge
University Press, whose encouragement was critical to the inception of this
book and whose sterling effort to bring together the disparate contributions
intoasinglemanuscriptwasessentialtoitscompletion.