Rational points and arithmetic of fundamental groups : evidence for the section conjecture

253 Pages·2013·1.57 MB·English

by  Jakob Stix

Lecture Notes in Mathematics 2054
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
•
Jakob Stix
Rational Points
and Arithmetic of
Fundamental Groups
Evidence for the Section Conjecture
123
JakobStix
MathematicsCenterHeidelberg(MATCH)
UniversityofHeidelberg
Heidelberg,Germany
ISBN978-3-642-30673-0 ISBN978-3-642-30674-7(eBook)
DOI10.1007/978-3-642-30674-7
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012945519
MathematicsSubjectClassification(2010):14H30,14G05,14H25,11G20,14G32,14F35
(cid:2)c Springer-VerlagBerlinHeidelberg2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
To Antonia,Jadenand Lucie
andtoSabine
•
Preface
The section conjecture,as stated by Grothendieck[Gr83] in a letter to Faltings in
1983,speculatesaboutarepresentationofrationalpointsintherealmofanabelian
geometry.Every k-rational point of a geometrically connected variety X=k gives
risetoaconjugacyclassofsections
s W(cid:2) .Spec.k//!(cid:2) .X/
1 1
ofthe naturalprojection(cid:2) .X/ ! (cid:2) .Spec.k// ofe´tale fundamentalgroups.The
1 1
section conjecture suggests that the converse also holds for smooth, projective
curvesofgenusatleast2overfieldskthatarefinitelygeneratedoverQ.
If the section conjecture turns out to be true, then it would shed new Galois
theoreticlightontheoldDiophantineproblemofdescribingrationalpoints.
This volume of Lecture Notes consists of the author’s Habilitationsschrift and
aimstodevelopthefoundationsoftheanabeliangeometryofsectionsandtopresent
ourknowledgeaboutthesectionconjecturewithanaturalbiastowardstheworkof
theauthor.Inaddition,wediscussthesectionconjectureovernumberfieldsfroma
localtoaglobalpointofviewandprovidedetaileddiscussionsofvariousanalogues
ofthesectionconjecture,whichmightserveassupportingevidenceinfavourofthe
conjecture.
Acknowledgments
Asthispresentbookbuildsonseveralyearsofwork,itisquitenaturalthatoverthis
periodI have been influencedby many people and institutionsand owe to them a
certaindebtofgratitudethatIwillnowtrytobalance.
I have worked at several places: the Mathematische Institut Bonn, the Institute
forAdvancedStudyinPrincetonfortheacademicyear2006/2007,theUniversityof
PennsylvaniainPhiladelphiain2007/2008,theIsaacNewtonInstituteinCambridge
during the summer of 2009, and most recently at the Mathematisches Institut
Heidelberg and in particular the MAThematics Center Heidelberg (MATCH).
vii
viii Preface
I thank all these institutes, particularly the people there, I thank for the excellent
working conditions and the supportive environment that they provided, for the
coffeeandforthediscussionsonmathematics,life,theuniverse,andalltherest.
IamgratefultoFlorianPopforhisongoinginterestinmywork,forthenumerous
discussions, and for becoming, beyond an academic teacher, a collaborator and
friend.I thankHiroakiNakamuraand Akio Tamagawaformy severalstimulating
visits to Okayama and Kyoto. To Kirsten Wickelgren and Gereon Quick, I owe
thanks for the enlightening discussions on the point of view of homotopy fixed
points towards the section conjecture. My thanks also go to Jochen Ga¨rtner for
the discussions on Galois groups of number fields with restricted ramification.
I thank David Harari for the pleasant collaboration on the descent obstruction,
and I am grateful to He´le`ne Esnault and Olivier Wittenberg for the wonderful,
stimulating, and very intense meetings in Essen, Paris, and Heidelberg, and also
fortheirenthusiasticapproachtowardsmathematics.
Thereare certainlyaspectsofthe sectionconjecturebeyondwhatis coveredin
thisbook,andalso someaspectsthatwere originallyintendedto beincluded.But
then,amongotherthings,AntoniaandJadenwouldnotsleepintheirbelovedbunk
bed.So, bykeepingme busyotherwise,my kidsearnedthe honourof forcingme
to focuson finding an end pointfor this piece of work.I am thankfulto Antonia,
Jaden,andLucieforthejoy thattheyareandgiveeveryday,andforbringingme
coffeeandbreakfastwhilethesefinallineswerebeingwritteninJanuary2011.
Tomyparents,GiselaandMichaelStix,Iamextremelygratefulforbeingthere
wheneverahelpinghandwasneeded.
Writing my Habilitationsschrift as a culmination of research conducted over
several years while being part of a growing family would have been simply
impossiblewithoutthewarm,enthusiastic,andinexhaustiblesupportofmybeloved
wife,Sabine.Ithankherfromthebottomofmyheartwithwholeheartedloveand
admiration,althoughthere are no wordsthat capture the uncountablegratefulness
shedeserves.
Heidelberg,Germany JakobStix
Contents
PartI FoundationsofSections
1 ContinuousNon-abelianH1withProfiniteCoefficients................ 3
1.1 Torsors,SectionsandNon-abelianH1 .............................. 3
1.2 Twisting,DifferencesandComparison.............................. 4
1.3 LongExactCohomologySequence................................. 7
1.4 Non-abelianHochschild–SerreSpectralSequence................. 9
2 TheFundamentalGroupoid............................................... 13
2.1 FibreFunctorsandPathSpaces ..................................... 13
2.2 TheFundamentalGroupoid.......................................... 14
2.3 TheFundamentalGroupoidintheRelativeCase................... 16
2.4 GaloisInvariantBasePoints......................................... 18
2.5 AbstractSections..................................................... 20
2.6 HomotopyFixedPointsandtheSectionConjecture ............... 21
3 BasicGeometricOperationsinTermsofSections...................... 25
3.1 FunctorialityintheSpaceandAbelianization...................... 25
3.2 BaseChange.......................................................... 26
3.3 CentralisersandGaloisDescentforSections....................... 27
3.4 FibresAboveSections ............................................... 28
3.5 WeilRestrictionofScalarsandSections............................ 34
4 TheSpaceofSectionsasaTopologicalSpace ........................... 37
4.1 SectionsandClosedSubgroups ..................................... 37
4.2 TopologyontheSpaceofSections.................................. 39
4.3 LimitsofSections.................................................... 40
4.4 TheDecompositionTowerofaSection............................. 42
5 EvaluationofUnits......................................................... 45
5.1 KummerTheorywithFiniteCoefficients........................... 45
5.2 KummerTheorywithProfiniteCoefficients........................ 47
ix