Arithmetic and Geometry over Local Fields: VIASM 2018 PDF
Preview Arithmetic and Geometry over Local Fields: VIASM 2018
Lecture Notes in Mathematics 2275 Bruno Anglès Tuan Ngo Dac Editors Arithmetic and Geometry over Local Fields VIASM 2018 Lecture Notes in Mathematics Volume 2275 Editors-in-Chief Jean-MichelMorel,CMLA,ENS,Cachan,France BernardTeissier,IMJ-PRG,Paris,France SeriesEditors KarinBaur,UniversityofLeeds,Leeds,UK MichelBrion,UGA,Grenoble,France CamilloDeLellis,IAS,Princeton,NJ,USA AlessioFigalli,ETHZurich,Zurich,Switzerland AnnetteHuber,AlbertLudwigUniversity,Freiburg,Germany DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK AngelaKunoth,UniversityofCologne,Cologne,Germany ArianeMézard,IMJ-PRG,Paris,France MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg SylviaSerfaty,NYUCourant,NewYork,NY,USA GabrieleVezzosi,UniFI,Florence,Italy AnnaWienhard,RuprechtKarlUniversity,Heidelberg,Germany Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Bruno Anglès (cid:129) Tuan Ngo Dac Editors Arithmetic and Geometry over Local Fields VIASM 2018 Editors BrunoAnglès TuanNgoDac LaboratoiredeMathématiquesNicolas InstitutCamilleJordan,UMR5208 Oresme,CNRSUMR6139 ClaudeBernardUniversityLyon1 UniversitédeCaenBasse-Normandie Villeurbanne,France Caen,France ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-030-66248-6 ISBN978-3-030-66249-3 (eBook) https://doi.org/10.1007/978-3-030-66249-3 MathematicsSubjectClassification:Primary:11Fxx,11Gxx,11Lxx,14Fxx,14Gxx;Secondary:12Hxx ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Arithmetic geometry is a very active branch of mathematics, with important and deepconnectionstovariousareassuchas algebraicgeometry,numbertheory,and Lietheory. Thegoalofthisvolumeistointroducegraduatestudentsandyoungresearchers tosomerecentresearchtopicsinarithmeticgeometryoverlocalfields.Thelectures arecenteredaroundtwocommonthemes:thestudyofDrinfeldmodulesandnon- Archimedeananalyticgeometry. The notes of this volume grew out from the lectures given during the research program“Arithmeticandgeometryoflocalandglobalfields,”whichtookplaceat the Vietnam Institute of Advanced Study in Mathematics (VIASM) from June to August 2018. Two of them were given at the VIASM School on Number Theory (see https://hanoi-nt18.sciencesconf.org/).The others were presented as advanced coursesduringtheresearchseminar. Theauthors,allleadingexpertsinthesubject,havemadeagreatefforttomake thenotesasself-containedaspossible.Inadditiontointroducingthebasictools,the lectures aim to present an overview of recent developments in the arithmetic and geometry of local fields and related topics. The included examples and suggested concrete research problems will enable young researchers to quickly reach the frontiersofthisfascinatingbranchofmathematics. Contents ofThisVolume Thevolumeconsistsofsevenlectures: 1. SomeElementsonBerthelot’sArithmetic D-ModulesbyDanielCaro(Univer- sityofCaenNormandy,France), 2. Difference Galois Theory for the “Applied”Mathematicianby Lucia Di Vizio (CNRSandUniversityofVersaillesSaintQuentin,France), v vi Preface 3. Igusa’sConjectureonExponentialSumsModulopmandLocal-GlobalPrinciple by NguyenHuuKien(KULeuven,Belgium), 4. From theCarlitzExponentialtoDrinfeldModularFormsbyFedericoPellarin (InstituteCamilleJordanandUniversityofSaint-Etienne,France), 5. BerkovichCurvesandSchottkyUniformizationI:The BerkovichAffineLineby JérômePoineau(UniversityofCaenNormandy,France)andDanieleTurchetti (DalhousieUniversity,Canada), 6. Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves by Jérôme Poineau (University of Caen Normandy, France) andDanieleTurchetti(DalhousieUniversity,Canada), 7. OntheStarkUnitsofDrinfeldModulesbyFloricTavaresRibeiro(University ofCaenNormandy,France). The first lecture by D. Caro offers an introduction to p-differential methods in arithmetic geometry. First, he reviews Berthelot’s ring of p-adic differential operators, which plays an important role in the theory of arithmetic D-modules. Next, he extends it to some finite level on p-adic formal affine smooth schemes. Finally, concrete examples and a guide to further reading are also provided. The materialassumesabasicknowledgeofringtheoryandalgebraicgeometry. The second lecture by L. Di Vizio gives an overview of the Galois theory of difference equations. The first part presents a guide to the key definitions and results of difference Galois theory. In the second part, interesting applications to transcendence and differential transcendence are treated in detail. Note that the framework is the same as that of Papanikolas’ theory in the setting of Drinfeld modules. The curious reader may wish to refer to the lectures of F. Pellarin and F.TavaresRibeiroformoredetails. The third lecture by K. Nguyen is a survey on Igusa’s conjecture around exponential sums motivated by the study of local-global principles for forms of higherdegree.Afterintroducingthenotionofexponentialsumsandthosemodulo pm with some examples, he formulates a coarse form of Igusa’s conjecture on a uniform bound of those exponential sums and explains its relations with Igusa’s localzetafunctions,themonodromyconjecture,andfiberintegrals.Hethenstates Igusa’s conjecture on exponential sums and gives an overview of recent progress onthisconjecture,inparticularthemostrecentbreakthroughofCluckers,Musta¸ta˘, andtheauthor.Thelectureendswithageneralpictureofthelocal-globalprinciple forformsandthecontributionoftheaforementionedconjectureinthisdirection. The fourth lecture by F. Pellarin presents a friendly introduction to the theory of Drinfeld modular forms attached to the affine line over a finite field. Drinfeld modular forms in positive characteristic are defined as analogues of classical modularformsbythe pioneeringworksofGossandGekeler. Thenotesgradually introducetheveryfirstbasicelementsofthearithmetictheoryofDrinfeldmodules, thentheDrinfeldupper-halfplaneanditstopology,andendwithDrinfeldmodular forms.Thekeynotionsareillustratedwithmanyexamples.Thelecturealsocontains several advanced parts such as Drinfeld modular forms with values in Banach Preface vii algebras. These course notes will enable the reader to gently enter into this rich andstilldevelopingtheory. Theself-containedsurveybyJ.PoineauandD.Turchetticonsistsoftwolectures onnon-ArchimedeancurvesandSchottkyuniformizationfromthepointofviewof Berkovichgeometry.Thefirstpart,thefifthlecture,couldbereadasanelementary course on the theory of Berkovich spaces with emphasis on the affine line. The authorsintroducebasicdefinitionsandpropertieswithfulldetailsandproofs.The second part, the sixth lecture, is more advanced and deals with the theory of uniformization of curves under Berkovich’s theory. After introducing the notion of Mumfordcurves and Schottky groups, the authorspresent an analytic proofof Schottkyuniformization.Many examples,explicitresearchproblems,and a guide for further reading are also provided. The reader who is interested in Schottky groupsinthe languageofrigidanalyticspacesis invitedto readthe relevantparts ofthelectureofF.Pellarin. The last lecture by F. Tavares Ribeiro presents some recent developments in the arithmetic theory of the special values of Goss zeta functions. This lecture is an exposition on Stark units of Drinfeld modules over the ring of polynomials overa finitefield.Thenotesarealsotheoccasiontointroducebasicdefinitionsof Drinfeldmodulesand morerecentfundamentalobjectslinkedto an analyticclass numberformulaobtainedbyL.Taelman:L-values,unitmodules,andclassmodules attachedtoaDrinfeldmodule.TheauthorthenpresentsthenotionofStarkunitsand givestheirbasicproperties.Finally,hegivesseveralapplicationsofStarkunits,in particular,tothestudyofcongruencepropertiesofBernoulli–Carlitznumbers.He alsogiveshintsforageneralbasering. Caen,France BrunoAnglès Lyon,France TuanNgoDac October2020 Acknowledgments We wouldlike to acknowledgethe manypeoplewhosehelp was essentialfor this volume. We would like to thank the speakers and the authors, Daniel Caro, Lucia Di Vizio, Nguyen Huu Kien, Federico Pellarin, Jérôme Poineau, Floric Tavares Ribeiro, Lenny Taelman, and Daniele Turchetti, for their enthusiastic investment tothissemesterandthisvolume.We also thankalltherefereesformanyvaluable commentsandsuggestionsthathelpedinimprovingtheexpositionofeachlecture aswellasthewholevolume. We would like to express our gratitude to the directors, the researchers, and administrative staff of the VIASM for the hospitality and excellent working conditionsduringthespecialsemester.SpecialthanksareduetoLêMinhHa,Ngô Bao-Châu,PhungHôHai,LêThiLanAnh,andNguyênHoangAnh. Our semester was partially supported by the VIASM, the Institute of Math- ematics in Hanoi (IMH), the European Research Council (ERC Starting Grant TOSSIBERG),theANRGrantPerCoLaTor(ANR-14-CE25-0002-01),theInstitut Henri Poincaré (IHP), the Foundation Compositio Mathematica, and the GDR Structurationdelathéoriedesnombres(CNRS). ix Contents 1 SomeElementsonBerthelot’sArithmeticD-Modules .................. 1 DanielCaro 1.1 Introduction............................................................ 1 Introduction.................................................................... 1 1.2 RingofDifferentialOperators........................................ 2 1.2.1 KählerDifferential ............................................ 2 1.2.2 RingofDifferentialOperators................................ 4 1.2.3 SmoothDifferentialCase ..................................... 6 1.2.4 LeftD -Modules........................................... 8 A/R 1.3 Berthelot’sRingofDifferentialOperatorsofFiniteLeveland InfiniteOrder........................................................... 12 1.3.1 FormalAffineCase,StandardRingofDifferential Operators....................................................... 12 1.3.2 FormalAffineCase,Berthelot’sRingofDifferential OperatorsofLevelm.......................................... 14 (m) 1.3.3 LeftD -modules ........................................... 19 A/R 1.3.4 Weakp-AdicCompletion..................................... 23 1.3.5 Sheafification,Coherence..................................... 25 Appendix:FurtherReading................................................... 27 References..................................................................... 28 2 DifferenceGaloisTheoryforthe“Applied”Mathematician............ 29 LuciaDiVizio 2.1 Introduction............................................................ 29 2.2 GlossaryofDifferenceAlgebra....................................... 31 2.3 Picard-VessiotRings................................................... 33 2.4 TheGaloisGroup...................................................... 37 2.4.1 AShortDigressiononGroupSchemes ...................... 38 2.4.2 TheGaloisGroupofaLinearDifferenceSystem ........... 40 2.4.3 TranscendenceDegreeofthePicard-VessiotExtension..... 42 2.5 TheGaloisCorrespondence(FirstPart).............................. 44 xi
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