Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday PDF
Preview Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday
Progress in Mathematics Volume253 SeriesEditors HymanBass JosephOesterle´ AlanWeinstein Algebraic Geometry and Number Theory In Honor of Vladimir Drinfeld’s 50th Birthday Victor Ginzburg Editor Birkha¨user Boston • Basel • Berlin VictorGinzburg UniversityofChicago DepartmentofMathematics Chicago,IL60637 U.S.A. [email protected] MathematicsSubjectClassification(2000):03C60,11F67,11M41,11R42,11S20,11S80, 14C99,14D20,14G20,14H70,14N10,14N30,17B67,20G42,22E46(primary); 05E15,11F23,11G45,11G55,11R39,11R47,11R58,14F20,14F30,14H40,14H42,14K05, 14K30,14N35,22E67,37K20,53D17(secondary) LibraryofCongressControlNumber:2006931530 ISBN-10:0-8176-4471-7 eISBN-10:0-8176-4532-2 ISBN-13:978-0-8176-4471-0 eISBN-13:978-0-8176-4532-8 Printedonacid-freepaper. (cid:1)c2006Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMediaLLC,RightsandPermissions,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. (JLS) 987654321 www.birkhauser.com VladimirDrinfeld Preface VladimirDrinfeld’smanyprofoundcontributionstomathematicsreflectbreadthand greatoriginality. Thetenresearcharticlesinthisvolume,coveringadiversityoftopics predominantlyinalgebraandnumbertheory,reflectDrinfeld’svisioninsignificant areasofmathematics,andarededicatedtohimontheoccasionofhis50thbirthday. ThepaperbyGoncharovandFockisdevotedtothestudyofclustervarietiesand theirquantizations. ThissubjecthasitsoriginsintheworkofFominandZelevinsky on cluster algebras and total positivity on the one hand, and, on the other hand, on various attempts to understand Kashiwara’s theory of crystals and quantizations of modulispacesofcurves. StartingwithasplitsemisimplerealLiegroupGwithtrivialcenter,Goncharov and Fock define a family of varieties with additional structures called cluster X- varieties. These varieties have a natural Poisson structure. The authors define a PoissonmapfromaclustervarietytothegroupGequippedwiththestandardPoisson– LiestructureasdefinedbyV.Drinfeld. ThemapisbirationalandthusprovidesGwith canonicalrationalcoordinates. Further,GoncharovandFockshowhowtoconstruct complicatedclusterX-varietiesfrommoreelementaryonesusinganamalgamation procedure. This is used, in particular, to produce canonical (Darboux) coordinates forthePoissonstructureonaZariskiopensubsetofthegroupG. SomeoftheclustervarietiesareverycloselyrelatedtothedoubleBruhatcells studiedbyA.Berenshtein,S.Fomin,andA.Zelevinisky. Ontheotherhand,theresults ofthepaperplayakeyroleindescribingtheclusterstructureofthemodulispaces oflocalsystemsonsurfaces,asstudiedbyGoncharovandFockinanearlierwork. TheimportantroleofDrinfeld’sideas—indeed,oneofthecentralthemesofhis research—isevidentinthepaperbyFrenkelandGaitsgory,whichisdevotedtothe (local) geometric Langlands correspondence from the point of view of D-modules andtherepresentationtheoryofaffineKac–Moodyalgebras. LetgbeasimplecomplexLiealgebraandGaconnectedalgebraicgroupwith Lie algebra g. The affine Kac–Moody algebra(cid:1)g is the universal central extension oftheformalloopagebrag((t)). Representationsof (cid:1)g haveaparameter,aninvar- iant bilinear form on g, which is called the level. Representations corresponding to the bilinear form that is equal to minus one-half of the Killing form are called viii Preface representations of critical level. Such representations can be realized in spaces of globalsectionsoftwistedD-modulesonthequotientoftheloopgroupG((t))byits “compact’’subgroupK equaltoG[[t]],ortotheIwahorisubgroupI. ThisworkbyFrenkelandGaitsgoryisthefirstinaseriesofpapersdevotedto thestudyofthecategoriesofrepresentationsoftheaffineKac–Moodyalgebra(cid:1)gof thecriticallevelandD-modulesonG((t))/K fromthepointofviewofageometric versionofthelocalLanglandscorrespondence. ThelocalLanglandscorrespondence sets up a relation between two different types of data. Roughly speaking, the first data consist of the equivalence classes of homomorphisms from the Galois group ofalocalnon-archimedeanfieldKtoG(C)∨,theLanglandsdualgroupofG. The seconddataconsistoftheisomorphismclassesofirreduciblesmoothrepresentations, denotedbyπ,ofthelocallycompactgroupG(K). Anaiveanalogueofthiscorrespondenceinthegeometricsituationseekstoassign toaG(C)∨-localsystemontheformalpunctureddiscarepresentationoftheformal loopgroupG((t)). However,theauthorsshowthatincontrasttotheclassicalsetting, thisrepresentationofG((t))shouldbedefinednotonavectorspace,butonacategory, asexplainedinthepaper. InthecontributionbyIhara,andinthecloselyrelatedappendixbyTsfasman,the authorsstudytheζ-functionζK(s)ofaglobalfieldK. Specifically,theyareinterested intheso-calledEuler–KroneckerconstantγK, arealnumberattachedtothepower seriesexpansionoftheζ-functionatthepoints =1. Inthespecialcaseofthefield K=Qofrationalnumbers,thisconstantreducestotheEulerconstant (cid:2) (cid:3) 1 1 1 γQ = lim 1+ + +···+ −logn . n→∞ 2 3 n TheconstantγKplaysanimportantroleinanalyticnumbertheory. Ontheother hand,forK=F (X),thefieldofrationalfunctionsonacompletealgebraiccurveX q overafinitefield,thecorrespondingEuler–Kroneckerconstantiscloselyrelatedto thenumberofF -rationalpointsofX. q IharaaddressesthequestionofhownegativetheconstantγKmaybe,depending on the field K. In the number field case, this happens when K has many primes withsmallnorm. Inthefunctionfieldcase, thereareknowntowersofcurveswith manyFq-rationalpoints;theauthorstudiesthebehaviorofγKusingthegeneralized Riemannhypothesis. Inthisway,heobtainsveryinterestingexplicitestimatesofγK. Forinstance,inthecaseK=F (X)Iharaestablishesanupperbound q γK ≤2log((g−1)logq)+logq, whereg denotesthegenusofthecurveX. Healsoobtainssimilarestimatesforthe lowerbound. HrushovskiandKazhdanintheirpaperlaythefoundationsofintegrationtheory over, not necessarily locally compact, valued fields of residue characteristic zero. A valued field is a field K equipped with a “ring of integers’’O ⊂ K, satisfying the property that K = O∪(O\{0})−1. In particular, the authors obtain new and base-fieldindependentfoundationsforintegrationoverlocalfieldsoflargeresidue characteristic,extendingresultsofDenef,Loeser,andCluckers. Preface ix The work of Hrushovski and Kazhdan is on the border of logic and algebraic geometry. Their methods involve an analysis of definable sets. Specifically, they obtainaprecisedescriptionoftheGrothendiecksemigroupofdefinablesetsinterms ofrelatedgroupsovertheresiduefieldandvaluegroup. Thisyieldsnewinvariantsof alldefinablebijections,aswellasinvariantsofmeasure-preservingbijections. Their results are intended to be applied to the construction of Hecke algebras associated withreductivegroupsoveranotnecessarilylocallycompactvaluedfield. Inthecase ofatwo-dimensionallocalfield,thecorrespondingHeckealgebraisexpectedtobe closelyrelatedtothedoubleaffineHeckealgebraintroducedbyCherednik. Kisin’spaperisdevotedtop-adicalgebraicgeometryandnumbertheory. This subjectisrapidlydevelopingatthispointintime. FollowingtheideasofBergerand Breuil, Kisin gives a new classification of crystalline representations. The objects involvedmaybeviewedaslocal,characteristic0analoguesofthe“shtukas’’intro- ducedbyDrinfeld. Kisinalsogivesaclassificationofp-divisiblegroupsandfinite flatgroupschemes,conjecturedbyBreuil. Furthermore,heshowsthatacrystalline representationwithHodge–Tateweights0,1arisesfromap-divisiblegroup—aresult conjecturedbyFontaine. Letkbeaperfectfieldofcharacteristicp >0,W =W(k)itsringofWittvectors, K = W(k)[1], and K : K a finite totally ramified extension. Breuil proposed a 0 p 0 new classification of p-divisible groups and finite flat group schemes over the ring of integers OK of K. For p-divisible groups and p > 2, this classification was established in an earlier paper by Kisin, who also used a variant of Breuil’s theory todescribeflatdeformationrings,andtherebyestablishamodularityliftingtheorem forBarsotti–TateGaloisrepresentations. Inthepresentpaper,theauthorgeneralizesBreuil’stheorytodescribecrystalline representationsofhigherweightor,equivalently,theirassociatedweaklyadmissible modules. Krichever’spaperanalyzesdeepandimportantrelationsbetweenthetheoryofin- tegrablesystemsandtheRiemann–Schottkyproblem. TheRiemann–Schottkyprob- lemonthecharacterizationoftheJacobiansofcurvesamongabelianvarietiesismore than120yearsold. QuiteafewgeometricalcharacterizationsoftheJacobianshave beenfound. Noneofthem,however,providesanexplicitsystemofequationsforthe imageoftheJacobianlocusintheprojectivespaceunderthelevel-2thetaimbedding. Thelinkofthisproblemtointegrablesystemswasfirstdiscoveredinthe1980s. Specifically,T.ShiotaestablishedthefirsteffectivesolutionoftheRiemann–Schottky problem, known as Novikov’s conjecture. The conjecture says the following: An indecomposable principally polarized abelian variety (X,θ) is the Jacobian of a curveofagenusgifandonlyifthereexistg-dimensionalvectorsU (cid:7)=0,V,W such thatthefunction u(x,y,t)=−2℘2lnθ(Ux+Vy+Wt +Z) x isasolutionoftheKadomtsev–Petviashvilii(KP)equation 3u =(4u +6uu −u ) . yy t x xxx x x Preface (Hereθ(Z)=θ(Z|B)istheRiemannthetafunction.) Inthepresentpaper,Kricheverprovesthatanindecomposableprincipallypolar- izedabelianvarietyXistheJacobianofacurveifandonlyifthereexistvectorsU (cid:7)= 0,V suchthattherootsx (y)ofthethetafunctionalequationθ(Ux+Vy+Z)=0 i satisfy the equations of motion of the formal infinite-dimensional Calogero–Moser system. ThemaingoalofLaumon’spaperistoidentifythefibersoftheaffineSpringer resolutionforthegroupGL withcoveringsofcompactifiedJacobiansofprojective n singular curves. This work is part of a more general project of obtaining a geo- metric version of the “Fundamental Lemma’’that appears in Langlands’works on automorphicforms. LetF bealocalnon-archimedeanfieldofequalcharacteristic,letO beitsring F of integers, and let k be the residue field. Let E be a finite-dimensional F-vector space. TheauthorconsiderstheaffineGrassmannianformedbyO -latticesM inE. F Given a regular semisimple and topologically nilpotent endomorphism γ of E, one defines the affine Springer fiber, X , as the closed reduced subscheme of the γ affineGrassmannianformedbytheγ-stablelatticesM ⊂ E. KazhdanandLusztig haveshownthatX isascheme,locallyoffinitetypeoverk. Moreover,thisscheme γ comesequippedwithanaturalfreeactionofanabelianalgebraicgroup(cid:6) suchthat γ thequotientZ =X /(cid:6) isaprojectivek-scheme. γ γ γ Inhispaper,theauthorattachestoγ aprojectivealgebraiccurveC overkwith γ asinglesingularpointsuchthatthecompletedlocalringatthispointisisomorphic toO [γ]⊂F[γ]⊂Aut (E). Furthermore,theauthorrelatesthevarietiesX and F F γ Z withthecompactifiedJacobianofthecurveC . Thisallowshimtoreprovesome γ γ irreducibility results about compactified Jacobians due toAltman and Kleiman. In addition,thetechniquesdevelopedinthepaperprovideanapproachtoanimportant “purityconjecture’’concerningthecohomologyofcertainaffineSpringerfibers,due toGoresky,Kottwitz,andMacPherson. ThegoaloftheworkofManinpresentedinthisvolumeistostudypropertiesof theiteratedintegralsofmodularformsintheupperhalf-plane. Thissettinggeneral- izessimultaneouslythetheoryofmodularsymbolsandthatofmultiplezetavalues. Multiplezetavaluesarethenumbersgivenbythek-multipleDirichletseries (cid:4) 1 ζ(m ,...,m )= (0.1) 1 k nm1···nmk 0<n1<···<nk 1 k or,equivalently,bythem-multipleiteratedintegralsm=m +···+m , 1 k (cid:5) (cid:5) (cid:5) (cid:5) ζ(m ,...,m )= 1 dz1 z1 dz2 z2··· zmk−1 dzmk ··· . (0.2) 1 k z z 1−z 0 1 0 2 0 0 mk Multiplezetavaluesareinterestingbecausetheyandtheirgeneralizationsappear inmanydifferentcontextsinvolvingmixedTatemotives,deformationquantization (Kontsevich),knotinvariants,etc. Multiplezetavaluessatisfycertaincombinatorialrelations,calleddouble-shuffle relations. The relations in question can be succinctly written in terms of formal Preface xi generating series for (regularized) iterated integrals (0.2). Such integrals appeared more than 15 years ago in the celebrated work by Drinfeld on what is nowadays knownastheDrinfeldassociator. However,thequestionaboutinterdependenceof (double-)shuffleandassociatorrelationsdoesnotseemtobesettledatthemoment. Inthepaper, theauthordefines1-formsofmodularandcuspmodulartypeand studies iterated integrals and the total Mellin transform for families of such forms. ThefunctionalequationforthetotalMellintransformisdeduced. Thisresultextends theclassicalfunctionalequationforLseries. Theauthoralsointroducesaniterated modular symbol as a certain noncommutative 1-cohomology class of the relevant subgroupofthemodulargroup. Thepaperestablishessomeanaloguesoftheclassical identity (0.1) = (0.2) but different from it in two essential respects. First, iterated integrals are only linear combinations of certain multiple Dirichlet series. Second, theidentitiesobtainedinthepaperinvolveintegralswhicharenotoftheusualtype, (cid:4) a ···a 1,n1 n,nk; nm1···nmk 0<n1<···<nk 1 k infact,theircoefficientsdependonpairwisedifferencesn −n . j i In the paper by Eskin and Okounkov, the authors prove that natural generating functionsforenumerationofbranchedcoveringsofthepillowcaseorbifoldarelevel- 2quasimodularforms. Thisgivesusawaytocomputethevolumesofthestrataof themodulispaceofquadraticdifferentials. ConsideracomplextorusT2 =C/L,whereL⊂Cisalattice. Itsquotient P =T2/±1 by the automorphism z (cid:8)→ −z is a sphere with four (Z/2)-orbifold points, which is sometimes called the pillowcase orbifold. The map T2 → P is essentially the Weierstraß℘-function. Thequadraticdifferential(dz)2onT2descendstoaquadratic differentialonP. ViewedasaquadraticdifferentialontheRiemannsphere, (dz)2 hassimplepolesatcornerpoints. Let µ be a partition and ν a partition of an even number into odd parts. The authorsareinterestedinenumerationofdegree2d maps π :C →P (0.3) withthefollowingramificationdata. Viewedasamaptothesphere, π hasprofile (ν,2d−|ν|/2)over0∈P andprofile(2d)overtheotherthreecornersofP.Addition- ally,π hastheprofile(µi,12d−µi)oversome(cid:8)(µ)givenpointsofP andunramified elsewhere. Here(cid:8)(µ)isthenumberofpartsinµ. ThepaperbySchechtmanmaybeviewedasacontinuationoftheworkbyGor- bunov,Malikov,andSchechtmanonthechiraldeRhamcomplex. Specifically,the paperinthevolumeintroducesacertainchiralanalogueofthethirdChern–Simons classofavectorbundle. · LetXbeasmoothvarietyoverafieldkofcharacteristiczero,andwrite(cid:9) for X thedeRhamcomplexofX. AssociatedwithanyvectorbundleE onX,onehasthe correspondingChernclasses
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