Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10-15, 2007 PDF
Preview Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10-15, 2007
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 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotim- ply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:SPiPublisherServices Printedonacid-freepaper springer.com 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
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