Sequences of enumerative geometry congruences and

arXiv:math/0610286v1[math.NT]9Oct2006Sequencesofenumerativegeometry:congruencesandasymptoticsDanielB.Gr¨unberg,PieterMoreeAppendixbyDonZagierAbstractWestudytheintegersequencevnofnumbersoflinesinhypersurfacesofdegree2n−3ofPn,n1.Weproveanumberofcongruencepropertiesofthesenumbersofseveraldifferenttypes.Furthermore,theasymptoticsofthevnaredescribed(inanappendixbyDonZagier).Anattemptismadeatasimilaranalysisoftwootherenumerativesequences:thenumbersofrationalplanecurvesandthenumbersofinstantonsinthequinticthreefold.WestudythesequenceofnumbersoflinesinahypersurfaceofdegreeD=2n−3ofPn,n1.Thesequenceisdefinedby(seee.g.[8])vn:=ZG(2,n+1)c2n−2(SymDQ),(1)whereG(2,n+1)istheGrassmannianofC2subspacesofCn+1(i.e.projectivelinesinPn)ofdimension2(n+1−2)=2n−2,Qisthebundleoflinearformsontheline(ofrankr=2,correspondingtoaparticularpointoftheGrassmannian),andSymDisitsDthsymmetricproduct–ofrankD+r−1r−1
=D−1=2n−2.ThetopChernclass(Eulerclass)c2n−2istheclassdualtothe0-chain(i.e.points)correspondingtothezerosofthebundleSymD(Q),i.e.tothevanishingofadegreeDequationinPn;thisisthegeometricrequirementthatthelineslieinahypersurface.Theintegral(1)canactuallybewrittenasasum:vn=X0≤ij≤nQDa=0(awi+(D−a)wj)Q0≤k≤n,k6=i,j(wi−wk)(wj−wk),(2)wherew0,...,wnarearbitrarycomplexvariables.ThisisaconsequenceofalocalisationformuladuetoAtiyahandBottfromequivariantcohomology,whichsaysthatonlythe(isolated)fixedpointsofthe(C∗)n+1actioncontributetothedefiningintegralofvn.Hencethesum.Forthefirstfewvaluesofncomputationyieldsv2=1,v3=27,v4=2875,v5=698005,v6=305093061.D.Zagiergaveasimpleproofthattherighthandsideof(2)isindependentofMathematicsSubjectClassification(2000).14N10,11A07,41A601w0,...,wn(asitmustbefor(2)tohold),andthatinfactitcanbereplacedbythemuchsimplerformulavn=h(1−x)2n−3Yj=0(2n−3−j+jx)ixn−1(3)wherethenotation[...]xnmeansthecoefficientofxn.Infact,formula(2)wasprovedinaverydifferentwayusingmethodsfromSchubertcalculusbyB.L.vanderWaerden,whoestablisheditinpart2ofhiscelebrated20part‘ZuralgebraischenGeometrie’seriesofpapers[19,20].ThenumberoflinearsubspacesofdimensionkcontainedinagenerichypersurfaceofdegreedinPn,whenitisfinite,canbelikewiseexpressedasthecoefficientofamonomialinacertainpolynomialinseveralvariables,seee.g.[15,Theorem3.5.18].Zagieralsogavetheformulavn∼r27π(2n−3)2n−7/21−98n−111640n2−999925600n3+···,(4)wheretherighthandsideisanasymptoticexpansioninpowersofn−1withrationalcoefficientsthatcanbeexplicitlycomputed.Theproofofthisformula,aswellasthederivationof(3)from(2),canbefoundintheappendix.Theremainingresults,summarizedinTheorem1andTheorem2,areconcernedwithcongruencepropertiesofthenumbersvn.Inthiscontextitturnsouttobeconvenienttodefinev1=1(eventhoughthereisnosuchthingasahypersurfaceinP1ofdegree−1)andevenmoreremarkablyv0=−1.Wedonotdoubtthatthecongruenceresultspresentedhereformonlythetipofaniceberg.Afirstversionofthispaperwassingleauthoredbythefirstauthor.Indeed,thepresentversionofthispaperissimilartothefirstone,exceptforsections2and3whichhavebeengreatlyrevisedandexpandedbythesecondauthor.Theconjecturesoutsidethesetwosectionsareduetothefirstauthoralone.Sections4and5wererevisedbyboththesecondauthorandDonZagier.1IntroductionThemotivatingideabehindthispaperistheexpectationthatcertainproblemsinenu-merativegeometryarecoupledtomodularity.Thisisarecurrentthemeinstringtheory,wherepartitionfunctionshaveoftenanenumerativeinterpretationascountingobjects(instantons,etc)andmustsatisfytheconditionofmodularitycovarianceinordertoobtainthesameamplitudewhentwoworldsheetshavethesameintrinsicgeometry.Modularforms,asiswellknown,haveFouriercoefficientssatisfyingmanyinterest-ingcongruences(thinkofRamanujan’scongruencesforpartitions,orforhisfunctionτ(n)).Thesamecanhappenforthecoefficientsofexpansionsrelatedtomodularforms,e.g.theexpansionsy=PAnxnobtainedbywritingamodularformy(locally)asapowerseriesinamodularfunctionx.Forinstance,thefamousAp´erynumbersrelatedtoAp´ery’sproofoftheirrationalityofζ(3)areobtainableinthisway[3]andsatisfymanyinterestingcongruences[18].Thenumbersappearinginthecontextofmirrorsymmetry,Picard-FuchsequationsforCalabi-Yaumanifolds,Gromov-Witteninvariants2andsimilarproblemsofenumerativegeometryaresometimesrelatedtomodularformsandsometimesnot,sowecanreasonablyhopeforinterestingcongruencepropertiesinthesecontextsalso.InSection2weshallfindastonishinglymanycongruencesforoursequencevn.Weshallfirstdrawafewtablesforcongruencesmod2,3,4,5or11,andthensummarizetheobservedcongruences.InSection3weprovethosecongruencesbyelementarymeansstartingfrom(3),andafewconjectureswillbeformulated.Sequencesofnumberscomingfrommodularformsalsooftenhaveinterestingasymptoticpropertiesandwethereforewishtostudythis,too.InSection4wefindtheasymptoticpropertiesofthevnnumericallybyusingacleverempiricaltrickshowntousbyDonZagierwhichwecalltheasympktrick.(Arigorousproofoftheseasymptotics,asalreadymentionedabove,wasalsoprovidedbyhimandisreproducedintheappendix.)Section5presentscongruencesandasymptoticsfortwofurtherexamplesofenumerativesequences,with-outproofs.Thesequenceofrationalcurvesontheplaneandthesequenceofinstantonso