Geometric Aspects of Dwork Theory. Volumes 1 and 2 PDF
Preview Geometric Aspects of Dwork Theory. Volumes 1 and 2
Geometric Aspects of Dwork Theory, Volumes I and II Editors Alan Adolphson et al. Walter de Gruyter Geometric Aspects of Dwork Theory Volume I Bernie in Erice, October 1994 Bernard M.Dwork 1923(cid:18)1998 Geometric Aspects of Dwork Theory Editors Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, and Franc¸ois Loeser Volume I ≥ Walter de Gruyter · Berlin · New York Editors AlanAdolphson FrancescoBaldassarri PierreBerthelot DepartmentofMathematics DipartimentodiMatematica IRMAR OklahomaStateUniversity Universita`diPadova Universite´deRennes1 Stillwater,OK74078 ViaBelzoni7 CampusdeBeaulieu USA 35131Padova 35042Rennescedex e-mail:[email protected] Italy France e-mail:[email protected] e-mail:[email protected] NicholasKatz Franc¸oisLoeser DepartmentofMathematics E´coleNormaleSupe´rieure PrincetonUniversity De´partementdemathe´matiquesetapplications Princeton,NJ08544-1000 UMR8553duCNRS USA 45rued’Ulm e-mail:[email protected] 75230ParisCedex05 France e-mail:[email protected] MathematicsSubjectClassification2000: 14-06;14Fxx,14Gxx,11Gxx,11Lxx Keywords: p-adiccohomologies,zetafunctions,p-adicmodularforms,D-modules EPPrintedonacid-freepaperwhichfallswithintheguidelinesofthe ANSItoensurepermanenceanddurability. LibraryofCongressCataloging-in-PublicationData Geometric aspects of Dwork theory / edited by Alan Adolph- son…[etal.]. p. cm. Includesbibliographicalreferences. ISBN3-11-017478-2(cloth:alk.paper) 1. Geometry, Algebraic. 2. Number theory. 3. p-adic analysis. I.Adolphson,Alan,1951(cid:18) QA564.G47 2004 516.315(cid:18)dc22 2004011345 ISBN 3-11-017478-2 BibliographicinformationpublishedbyDieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataisavailableintheInternetat(cid:12)http://dnb.ddb.de(cid:14). ”Copyright2004byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany. All rights reserved, including those of translationinto foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, includingphotocopy,recordingoranyinformationstorageandretrievalsystem,withoutpermis- sioninwritingfromthepublisher. PrintedinGermany. Coverdesign:ThomasBonnie,Hamburg. Typesetusingtheauthors’T Xfiles:I.Zimmermann,Freiburg. E Printingandbinding:Hubert&Co.GmbH&Co.KG,Göttingen. Preface Bernard Dwork (New York 5/27/1923–Princeton 5/9/1998) stunned the algebro- geometricworldin1958withhisproofofrationalityofthezetafunctionofanalgebraic varietyoverafinitefieldandwiththep-adicmethodsheintroducedinhisproof. He thenwentontocreateacompletelynewp-adiccohomologytheoryforhypersurfaces of characteristic p > 0, and to study the p-adic variation, in families, of the zeta function,creatingoutofwholeclothap-adictheoryofPicard–Fuchsequationsand theirrelationtozetafunctions. Heestablishedageneraltheoryofp-adicdifferential equationsanddeepresultsontherelationbetweentheHodgeandslopefiltrationsof Picard–Fuchsequations. Inthecourseofthiswork,heintroducedideasthathavebeenfundamentalinthe development of p-adic arithmetic geometry as it presently stands. Among these are thenotionsofFrobeniusstructureforp-adicdifferentialequations, andoftheslope and growth filtrations on their solution spaces. Equally fundamental has been his insistenceontheroleofmonodromyofPicard–Fuchsequationsaroundsupersingular points,andonthestillmysteriousnotionofexcellentliftingofFrobenius. In order to explore and clarify the intrinsic geometric content of Dwork’s arith- meticresultsandhisp-adicanalyticmethodsagroupofDwork’smathematicalheirs organizedanextendedcycleofconferenceswhichwasentitled“TheDworkTrimester inItaly”byreasonofitsdurationandvenue. Moreover,duringthisintensivetrimester newdevelopmentswerepresentedviaaseriesofmini-coursesdedicatedtosomeof themostsalientaspectsofDwork’stheory. The Dwork Trimester took place from May to July of 2001. The principal site wasattheUniversityofPadova. Therewerealsotwoone-weekspecialconferences. The first, on p-adic modular forms, p-adic L-functions and p-adic integration, was organizedbyM.Bertolini,andheldattheVillaMonasteroinVarennaonLakeComo from June 3 to June 9, 2001. The second, which comprised the final week of the DworkTrimester,washeldatthemountainresortBressanonefromJuly1toJuly7, 2001. The Dwork Trimester brought together over a hundred mathematicians from many countries to pay a fitting scientific and personal homage to the remarkable mathematicianandmanwhowasBernardM.Dwork. TheorganizersoftheDworkTrimesterwishtoextendtheirheartfeltthankstothe IstitutoNazionalediAltaMatematica(INdAM),Rome,andtotheEuropeanNetwork “ArithmeticAlgebraic Geometry” which co-sponsored the events, as well as to the Dipartimento di Matematica Pura ed Applicata of the Università di Padova, to the staffsofVillaMonasteroinVarennaandtheScuolaEstivaoftheUniversitàdiPadova atBressanonewhoseunstintingcooperationmadepossibletheDworkTrimester. AsonemightinferfromthefactthattheDworkTrimestertookplaceinnorthern Italy, Bernie had strong and enduring ties with Italy. Already in the fall of 1966 he vi Preface hadacceptedaninvitationfromFrancescoGherardellitobeavisitingprofessoratthe University of Florence. It was an eventful year: early November rains brought the ArnotohistoricallyunprecedentedlevelswhileBernie,Shirleyandtheirthreechildren werevisitingRomeforaverywetAllSaintsholiday. Theheavyrainsandfloodinghad blockedtheroadsbacktoFlorence,thusforcingthefamilytomakeanovernightstop inCortona. FromthereBerniesenthisbrotherLeoamemorabletwo-wordtelegram (“Glug glug”), clearly showing both that the family was safe, and that his sense of humorwasintact. (Hisbrothersentatwowordreply: “UseListerine”.) Thehavoc produced by the flood made a return to Florence by car unthinkable. Following the family’sreturntoRomethethreatoftyphoidfeverinFlorenceruledouteventhetrain triptoFlorencethatBernieandhissonAndrewhadplanned,hopingtoretrievewarm clothingforthefamily. TheplannedsabbaticalinItalyseemedruined. TheDworks were on the verge of returning to the United States, but the timely efforts of Aldo AndreottienabledBernieandhisfamilytomovetotheUniversityofPisafortherest oftheacademicyear. Inthefallof1975FrancescoBaldassarri,recently“laureato”attheUniversityof PadovaunderthedirectionofI.Barsotti,wenttoPrincetonUniversityasapostdoctoral fellow. Dwork soon became his mentor, and a lasting collaboration and friendship ensued. AfewyearslateranotheryoungmathematicianfromPadova,BrunoChiarel- lotto, also went to Princeton to study with Dwork. As a result, both Bernie and ShirleyDworkbecamefrequentvisitorstoPadovaintheyearsthatfollowed. During theirmanyvisitsthroughoutthe1980stheyestablishedfirmfriendshipsbothwithin themathematicalcommunityatPadovaandalsobeyondit. Thus,itwasnatural(al- thoughbynomeansbureaucraticallytrivial)thatwhenBernieretiredfromPrinceton Universityhewas“called”totheUniversityofPadovawhereheservedas“Professore perChiaraFama”from1992untilhisdeathin1998. Duringthisperiodhisinfluence onmathematicsinPadovawasvibrantanddeep. Amonghisotherstudentsandcol- laboratorsfromthisperiodareMaurizioCailotto, LuciaDiVizio, GiovanniGerotto, FrankSullivan,andFrancescaTovena. ButBernie’sinterestinItalywasnotlimited toitsmathematicians. Indeed, BerniehadakeeninterestinItalianlife, culture, and politics. His openness and joie de vivre won him, throughout his life, close friend- shipsinmanyplaces, andItalywasnoexception. HisfriendsinItalywerehonored toorganizeamathematicaltrimesterdedicatedtothedevelopmentswhichcameout ofhisworkandoutofthetoolsheinvented. Theresultofthattrimesteristhepresent publication. April2004 A.Adolphson,F.Baldassarri, P.Berthelot,N.Katz,andF.Loeser Table of Contents of Volume I Preface v TableofContentsofVolumeII ix TheMathematicalPublicationsofBernardDwork xi AlanAdolphson Exponentialsumsandgeneralizedhypergeometricfunctions. I:CohomologyspacesandFrobeniusaction 1 AlanAdolphsonandStevenSperber Exponentialsumsandfreehyperplanearrangements 43 YvesAndré Surlaconjecturedesp-courburesdeGrothendieck–Katz etunproblèmedeDwork 55 FabrizioAndreattaandEyalZ.Goren Hilbertmodularvarietiesoflowdimension 113 FrancescoBaldassarriandPierreBerthelot OnDworkcohomologyforsingularhypersurfaces 177 FrancescoBaldassarriandAndreaD’Agnolo OnDworkcohomologyandalgebraicD-modules 245 LaurentBerger Anintroductiontothetheoryofp-adicrepresentations 255 VladimirG.Berkovich Smoothp-adicanalyticspacesarelocallycontractible. II 293 Jean-BenoîtBost Germsofanalyticvarietiesinalgebraicvarieties: canonicalmetrics andarithmeticalgebraizationtheorems 371 GillesChristol Thirtyyearslater 419 RobertF.ColemanandWilliamA.Stein Approximationofeigenformsofinfiniteslopebyeigenformsoffiniteslope 437 viii TableofContentsofVolumeI RichardCrew Crystallinecohomologyofsingularvarieties 451 AndreaD’AgnoloandPietroPolesello Stacksoftwistedmodulesandintegraltransforms 463 JanDenefandFrançoisLoeser Onsomerationalgeneratingseriesoccuringinarithmeticgeometry 509 MladenDimitrov CompactificationsarithmétiquesdesvariétésdeHilbert etformesmodulairesdeHilbertpour(cid:1) (c,n) 527 1 Table of Contents of Volume II MladenDimitrovandJacquesTilouine VariétésetformesmodulairesdeHilbertarithmétiquespour(cid:1) (c,n) 555 1 LuciaDiVizio Introductiontop-adicq-differenceequations 615 MatthewEmertonandMarkKisin AnintroductiontotheRiemann–HilbertcorrespondenceforunitF-crystals 677 Jean-YvesEtesse IntroductiontoL-functionsofF-isocrystals 701 OferGabber Notesonsomet-structures 711 HaruzoHida Non-vanishingmodulopofHeckeL-values 735 LucIllusie Onsemistablereductionandthecalculationofnearbycycles 785 NicholasM.KatzandRahulPandharipande InequalitiesrelatedtoLefschetzpencilsandintegralsofChernclasses 805 KiranS.Kedlaya FullfaithfulnessforoverconvergentF-isocrystals 819 BernardLeStum Frobeniusaction,F-isocrystalsandslopefiltration 837 ShigekiMatsuda ConjectureonAbbes–SaitofiltrationandChristol–Mebkhoutfiltration 845 ChristineNoot-Huyghe TransformationdeFourierdesD-modulesarithmétiquesI 857 TomohideTerasoma BoyarskyprincipleforD-modulesandLoeser’sconjecture 909 NobuoTsuzuki Cohomologicaldescentinrigidcohomology 931
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