Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10-15, 2007

250 Pages·2010·2.19 MB·English

by  Jean-Louis Colliot-Thélène

Lecture Notes in Mathematics 2009
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
C.I.M.E.meansCentroInternazionaleMatematicoEstivo,thatis,InternationalMathematicalSummer
Center.Conceivedintheearlyfifties,itwasbornin1954andmadewelcomebytheworldmathemat-
icalcommunitywhereitremainsingoodhealthandspirit.Manymathematicians fromalloverthe
worldhavebeeninvolvedinawayoranotherinC.I.M.E.’sactivitiesduringthepastyears.
So they already know what the C.I.M.E.is all about. For the benefit of future potential users and
co-operators the main purposes and the functioning of the Centre may be summarized as follows:
everyyear,duringthesummer,Sessions(threeorfourasarule)ondifferentthemesfrompureand
appliedmathematicsareofferedbyapplicationtomathematiciansfromallcountries.Eachsessionis
generallybasedonthreeorfourmaincourses(24−30hoursoveraperiodof6−8workingdays)
heldfromspecialistsofinternationalrenown,plusacertainnumberofseminars.
AC.I.M.E.Session,therefore,isneitheraSymposium,norjustaSchool,butmaybeablendofboth.
Theaimisthatofbringingtotheattentionofyoungerresearcherstheorigins,laterdevelopments,and
perspectivesofsomebranchoflivemathematics.
Thetopicsofthecoursesaregenerallyofinternationalresonanceandtheparticipationofthecourses
cover the expertise of different countries and continents. Such combination, gave an excellent
opportunitytoyoungparticipants tobeacquaintedwiththemostadvancedresearchinthetopicsof
thecoursesandthepossibilityofaninterchangewiththeworldfamousspecialists.Thefullimmersion
atmosphereofthecoursesandthedailyexchangeamongparticipantsareafirstbuildingbrickinthe
edificeofinternationalcollaborationinmathematicalresearch.
C.I.M.E.Director C.I.M.E.Secretary
PietroZECCA ElviraMASCOLO
DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini”
Universita`diFirenze Universita`diFirenze
ViaS.Marta,3 vialeG.B.Morgagni67/A
50139Florence 50134Florence
Italy Italy
e-mail:zecca@unifi.it e-mail:[email protected]fi.it
FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
– INdAM(IstitutoNazionalediAltaMatematica)
Jean-Louis Colliot-The´le`ne
Peter Swinnerton-Dyer
Paul Vojta
Arithmetic Geometry
Lectures given at the C.I.M.E. Summer School
held in Cetraro, Italy, September 10–15, 2007
Editors:
Pietro Corvaja
Carlo Gasbarri
ABC
Authors
Prof.Jean-LouisColliot-The´le`ne Prof.PeterSwinnerton-Dyer
Universite´Paris-SudXI UniversityofCambridge
CNRS Dept.ofPureMath.&Math.Statistics
Labo.Mathe´matiques WilberforceRoad
Orsay91405CX CB30WBCambridge
Baˆtiment425 UnitedKingdom
France [email protected]
[email protected]
Prof.PaulVojta
UniversityofCalifornia,Berkeley
DepartmentofMathematics
970,EvansHall
Berkeley,CA94720-3840
USA
[email protected]
Editors
Prof.PietroCorvaja Prof.CarloGasbarri
Universita`diUdine Universite´deStrasbourg
Dipto.diMatematicaeInformatica InstitutdeRecherche
ViadelleScienze206 Mathe´matiqueAvance´e
33100Udine 7,rueRene´Descartes
Italy 67084Strasbourg
[email protected] France
[email protected]
ISBN:978-3-642-15944-2 e-ISBN:978-3-642-15945-9
DOI:10.1007/978-3-642-15945-9
SpringerHeidelbergDordrechtLondonNewYork
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2010938613
MathematicsSubjectClassification(2010):11G35,11G25,11D45,14G05,14G10,14G40,14M22
(cid:2)c Springer-VerlagBerlinHeidelberg2011
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Preface
ArithmeticGeometrycanbedefinedasthepartofAlgebraicGeometryconnected
with the study of algebraic varieties over arbitrary rings, in particular over non-
algebraically closed fields. It lies at the intersection between classical algebraic
geometryandnumbertheory.
In recent years, significant progress has been achieved in this field, in several
directions. More importantly, new links between arithmetic geometry and other
branchesofmathematicshavebeendeveloped,andnewpowerfultoolsfromgeom-
etry, complex analysis, differential equationsand representation theory have been
importedintonumbertheory,thusputtingarithmeticgeometryatthecrossroadsof
mostofcontemporarymathematics.
Somelinksbetweenarithmeticgeometryandclassicalalgebraicgeometrycome
fromthe classification ofalgebraicvarieties,an old subjectinitiatedby the Italian
schoolinthecaseofsurfacesanddevelopedatarapidpaceinrecenttime.
AsdiscoveredbyOsgoodandVojta about20yearsago,thereisa formalanal-
ogybetweencomplexanalysisandbothdiophantineapproximationandarithmetic
geometry.Suchanalogyhasrevealeditselfasafertilesourceofideasandproblems
in both complex analysis and arithmetic geometry, and it has recently led to new
achievements.
The algebraic theory of differential equations is also connected to arithmetic
geometry, especially with algebraic geometry in positive characteristic; many au-
thors,startingwiththefoundersoftranscendentalnumbertheory,stressedtherole
of differentialequations in transcendence. Recently, the theory of algebraic folia-
tionsshowednewrelationsbetweenthesetopicsanddiophantineapproximation.
The C.I.M.E. Summer School Arithmetic Geometry, held in Cetraro (Cosenza,
Italy),September10–15,aimedatpresentingsomeofthemostinterestingnewde-
velopmentsofarithmeticgeometry.Itconsistedoffourcourses,givenbysome of
themosteminentcontributorstothefield.
Hereisanoverviewofthethreecourseswhichhavebeenwrittenup.
Section 1 Varie´te´s presque rationnelles, leurs points rationnels et leurs
de´ge´ne´rescences,byJean-LouisColliot-The´le`ne.
This surveyaddresses the generalquestion:Over a given type of field, is there
anaturalclassofvarietieswhichautomaticallyhavearationalpoint?Fieldsunder
v
vi Preface
considerationhereinclude:finitefields,p-adicfields,functionfieldsinoneortwo
variablesoveranalgebraicallyclosedfield,C-fields.Classicalanswersaregivenby
i
theChevalley-WarningtheoremandbyTsen’stheorem.Moregeneralanswerswere
providedbyatheoremofGraber,HarrisandStarrandbyatheoremofEsnault.The
latterresultsapplytorationallyconnectedvarieties.
Colliot-The´le`nediscussesthesevarietiesfromvariousangles:weakapproxima-
tion(seealsoSwinnerton-Dyer’scontribution),R-equivalenceonthesetofrational
points,Chowgroupofzero-cycles.
Looselyspeaking,R-equivalenceonthesetofrationalpointsofavarietydefined
overagivenfieldisgeneratedbytheelementaryrelation:tobeconnectedbyara-
tionalcurvedefinedoverthegivenfield.Rationallyconnectedvarietiesarevarieties
for which R-equivalence becomestrivial when one extendsthe groundfield to an
arbitraryalgebraicallyclosedfield.Rationallyconnectedvarietiesplayanimportant
roˆleintheclassificationofalgebraicvarieties.
Ongoingworkon“rationallysimplyconnected”varietiesoverfunctionfieldsin
twovariablesisalsomentioned.Acommonthreadinthisreportisthestudyofthe
special fibre of a scheme over a discrete valuation ring: if the generic fibre has a
simplegeometry,whatdoesitimplyforthespecialfibre?
Manyexamplesare presentedin the courseshowingthat, despite importantre-
centadvancements,stillmanyquestionsremainopen,keepingthesubjectstrongly
alive.
Section2Topicsindiophantineequations,bySirPeterSwinnerton-Dyer.
The notes by Swinnerton-Dyeraddress the main problem in the theory of dio-
phantine equations: to decide whether a given algebraic equation has solutions in
integerorrationalnumbers.
Anobviousnecessaryconditionforthe existenceof rationalsolutionsto a dio-
phantine equation is its solubility over the reals, and more generally over p-adic
completionsofQ.Sinceaneffectiveproceduretodecideaboutsolubilityoverlocal
fieldsisknown,suchconditionisveryusefulinmanycases.Henceitisnaturalto
askforwhichclassofdiophantineequationstheconversealsoholds:
1. Iftheequationissolubleovereverylocalcompletionoftherationalnumberfield,
isitsolubleovertherationals?
This is called the Hasse principle. It is known that its does not hold for an ar-
bitrary equation. An obstruction for its validity was discovered by Manin in the
seventies and is nowadayscalled the Brauer-Manin obstruction. The notes briefly
describethisobstruction,andthenaddressthesecondnaturalquestion:
2. IstheBrauer-ManinobstructiontheonlyobstructiontotheHasseprinciple?
Inthecasewhenagivenequationisknowntobesoluble,onemaybeinterested
in the distribution of its solutions, i.e., of rational points on the algebraic variety
V definedbythatequation.WhensuchpointsareZariski-dense,onewouldliketo
“measure”theirdensity.Thereareatleasttwoverydistinctnotionsofdensity.First:
foreverypositiveintegerH,weletN(H)bethenumberofrationalpointsofheight
lessthanH.Weask:
Preface vii
3. CanoneestimatethegrowthofN(H),forH→∞,intermsofthegeometryofV?
Secondly:embedV(Q)intheproduct∏ V(Q )andconsiderthecorresponding
p p
producttopology.
4. (Weak approximation) Is the image of V(Q) dense in every finite product as
above?
Theseproblemsandquestionsarerelatedwithmanyotheraspectsofarithmetic
andgeometry,andthe authorillustratestheselinksinthefirstchaptersofhistext,
which can be viewed as an introduction to most of twentieth century Arithmetic
Geometry.
In the second part of the notes, answers are given to the above mentioned
questions in many concrete nontrivialcases, especially for surfaces. The methods
employedhave been pioneeredby Swinnerton-Dyerhimself and his collaborators
inthelasttenyears;hereapanoramicviewofthesemethodologiesisgiven.Also,
severalnewexamplesarepresentedforthefirsttime,inparticularforthemostim-
portantcaseofellipticandrationalsurfaces.
Section3DiophantineapproximationandNevanlinnatheory,byPaulVojta.
Intheeighties,P.VojtadiscoveredstrikinganalogiesbetweenNevanlinnatheory
incomplexanalysis,diophantineapproximation,someresultsonentirecurvesand
thedistributionofintegralandrationalpointsonalgebraicvarieties.
SupposethatX isaprojectivevarietydefinedoverafieldKofcharacteristiczero.
IfKisanumberfieldweareinterestedinthestructureofthesetX(K)ofitsrational
points.IfK=Cweareinterestedintheimageofanalyticmaps f :C→X.
Wemayaskthefollowingquestionsinthetwocases:
(1 ) IsthesetX(K)Zariskidense?
ar
(1 ) Maywefindmaps f :C→X withZariskidenseimage?
an
(2 ) IsthereafiniteextensionL/KsuchthatX(L)isZariskidense?
ar
(2 ) Isthereafinitecoveringh:Y →Cwithamap f :Y →X withZariskidense
an
image?
(3 ) Maywe controlthe size of therationalpointsin X(K)outsideof a proper
ar
Zariskiclosedset?
(3 ) Isitpossibletocontroltheorderofgrowthofamap f :C→X withZariski
an
denseimage,intermsofthegeometryofX?
AnalogousquestioncanbeaskedforopensubsetsY ⊂X ofalgebraicvarieties,
namely:
(4 ) LetO betheringofintegersofK.IsthesetY(O )Zariskidense?
ar K K
(4 ) Doesthereexistamap f :C→Y withZariskidenseimage?
an
Manyothersimilarquestionsmaybeasked.
ThenotesbyP. Vojtabeginbyformalizingthelanguageneededtoattackthese
questions:Inthearithmeticcontext,thetheoryofheightandWeilfunctionsisde-
scribed, while in the analytic context, the appropriate Nevanlinna theory is used.
viii Preface
Vojta shows how, using an appropriate dictionary, the two theories have striking
similarities.Alsoheshowshowhis“dictionary”canbeusedasasourceofproblems
inboththeories.Inparticular,theanalogiesbetweenRoth’stheoremindiophantine
approximation and Nevanlinna’s Second Main Theorem, between Schmidt’s sub-
space theoremin diophantineapproximationandCartan’sTheoremin Nevanlinna
theoryare presented,and thisleads to the naturalanalogybetweenGriffiths’ con-
jectureincomplexanalysisandhisownconjectureonrationalpoints.
After showing the classical results on the distribution of rational and integral
pointsin their historicalperspective,he presentssome of the recentdevelopments
obtainedfromSchmidt’ssubspacetheorem(andfromCartantheoremintheNevan-
linnasetting),togivenontrivialanswerstoquestions(4 )and(4 )incertaincases.
ar an
Inthelastpartofthecourse,heexplainstherelationsofthesetheorieswithdiffer-
entversionsofthefamousabcconjectureofMasserandOesterle´,andgivessome
ideasonrecentdevelopmentsobtainedbyMcQuillanandYamanoiontheso-called
1+εconjecture,in the function field case. Finally, he formulatessome new con-
jecturesinarithmetic,whicharestronglyinspiredbytheworkofMcQuillanonthe
abcconjectureoverfunctionfields.
PietroCorvaja
CarloGasbarri
Contents
Varie´te´spresquerationnelles,leurspointsrationnelsetleurs
de´ge´ne´rescences .................................................................... 1
Jean-LouisColliot-The´le`ne
1 Introduction .................................................................... 1
2 Notations,rappelsetpre´liminaires............................................ 2
3 Sche´masau-dessusd’unanneaudevaluationdiscre`te....................... 4
3.1 A-sche´mas de type (R), croisements normaux,
croisementsnormauxstricts ........................................... 4
3.2 Quand la fibre spe´ciale a une composante
demultiplicite´1 ........................................................ 5
3.3 Quandla fibre spe´ciale contientunesous-varie´te´
ge´ome´triquementinte`gre............................................... 6
3.4 Quand la fibre spe´ciale a une composante
ge´ome´triquementinte`gredemultiplicite´1............................ 7
3.5 Unexemple:quadriques............................................... 9
4 Groupede Brauer des sche´mas au-dessusd’un anneau
devaluationdiscre`te........................................................... 10
5 CorpsC ........................................................................ 12
i
6 R-e´quivalenceete´quivalencerationnellesurlesze´ro-cycles................ 14
7 Autourduthe´ore`medeTsen:varie´te´srationnellementconnexes........... 14
8 Autourduthe´ore`medeChevalley-Warning:varie´te´sdontle
groupedeChowge´ome´triqueesttrivial...................................... 21
9 Approximationfaiblepourlesvarie´te´srationnellementconnexes .......... 22
10 R-e´quivalencesurlesvarie´te´srationnellementconnexes.................... 23
11 E´quivalencerationnellesur les ze´ro-cyclesdesvarie´te´s
rationnellementconnexes...................................................... 27
12 Verslesvarie´te´ssupe´rieurementrationnellementconnexes................. 29
12.1 Deuxexemples ......................................................... 29
12.2 Fibres spe´ciales avec une composante
ge´ome´triquementinte`gredemultiplicite´1............................ 30
12.3 Varie´te´srationnellementsimplementconnexes....................... 32
12.4 Existence d’un point rationnel sur un corps de
fonctionsdedeuxvariables............................................ 34
ix
x Contents
12.5 Approximationfaibleentouteslesplacesd’uncorps
defonctionsd’unevariable............................................ 35
12.6 R-e´quivalenceete´quivalencerationnelle.............................. 36
13 Surjectivite´arithme´tiqueetsurjectivite´ge´ome´trique ........................ 37
13.1 Morphismes de´finis sur un corps de nombres
etapplicationsinduitessurlespointslocaux ......................... 38
13.2 Quelquesautresquestions ............................................. 40
Bibliographie ........................................................................ 41
TopicsinDiophantineEquations................................................. 45
SirPeterSwinnerton-Dyer
1 Introduction .................................................................... 45
2 TheHassePrincipleandtheBrauer-ManinObstruction..................... 47
3 Zeta-FunctionsandL-Series .................................................. 52
4 Curves.......................................................................... 55
5 VarietiesofHigherDimensionandtheHardy-LittlewoodMethod ......... 58
6 Manin’sConjecture............................................................ 60
7 Schinzel’sHypothesisandSalberger’sDevice............................... 65
8 TheLegendre-JacobiFunction................................................ 69
9 PencilsofConics............................................................... 75
10 2-DescentonEllipticCurves.................................................. 80
11 PencilsofCurvesofGenus1.................................................. 86
12 SomeExamples................................................................ 93
12.1 DiagonalQuarticSurfaces............................................. 93
12.2 SomeKummerSurfaces ............................................... 98
12.3 DiagonalCubicSurfaces............................................... 98
13 TheCaseofOneRational2-DivisionPoint..................................101
14 DelPezzoSurfacesofDegree4...............................................105
References...........................................................................108
DiophantineApproximationandNevanlinnaTheory..........................111
PaulVojta
1 Introduction ....................................................................111
2 NotationandBasicResults: NumberTheory ................................113
3 Heights .........................................................................115
4 Roth’sTheorem................................................................117
5 BasicsofNevanlinnaTheory..................................................120
6 Roth’sTheoremandNevanlinnaTheory .....................................123
7 TheDictionary(Non-GeometricCase).......................................127
8 Cartan’sTheoremandSchmidt’sSubspaceTheorem........................130
9 VarietiesandWeilFunctions..................................................134
10 HeightFunctionsonVarietiesinNumberTheory............................140
11 ProximityandCountingFunctionsonVarietiesinNumberTheory ........145
12 Height,Proximity,andCountingFunctionsinNevanlinnaTheory .........147
13 IntegralPoints..................................................................151