Congruence between modular forms - PDF document

forexample,thefollowingcongruenceofRamanujan[Ram16]:
:=q1Yn=1(1�qn)241Xn=111(n)qnmod691:Toprovethiscongruencerequiresknowingonlythreefacts:thatboth
andE12areclassicalmodularformsofweight12,thattheringofclassicalmodularformsisgivenbyZ[E4;E6] C,andthatthenumeratorofB12isdivisibleby691.Atthesametime,thiscongruencealsopointstowardsadeeperstrucure;itrepresentstherstincarnationofthemainconjectureofIwasawatheory|atheoremrelatingthespecialvaluesofDirichletL-functionstocorrespondingeigenspacesofclassgroupsofabelianextensionsofQ.Thetheoryofcongruencesofmodularformscanbe(roughly)distinguishedintotwotypes:(1)congruencesbetweenHeckeeigenforms,(2)congruencesbetweenclassicalholomorphicormeromorphicmodularforms.Therstsubjectisveryrichindeedandencompasses(broadlyconstrued)theentiretheoryoftwodimensionaloddGaloisrepresentationsofGQ.Weshallnotconcernourselveswithsuchcongruencesinthesenotes(excepttotheextentthattheyarerequiredtounderstandcongruencesofthesecondkind).Instead,weshallgrapplewiththesecondclassofcongruences,whichhasasitsgenesisvariouscon-jecturesofRamanujanconcerningthepartitionfunctionprovedbyWatson[Wat38](seeTheorem1.1.2below).Throughoutthistext,weshallconsiderthefollowingtwoexamples,which,althoughenjoyingsomespecialpropertieswhichdistinguishthemslightlyfromthegeneralcase,exhibitthetypicalbehaviorwithrespecttothetypeofcongruencestreatedinthesenotes.Letj=1+240X3(n)qn3 q1Yn=1(1�qn)24=1 q+744+196884q+:::=:Xc(n)qn:ThisisthefunctionknownasKlein'smodularinvariant(or,inaslightlydierentcontext,assimplythej-invariant).Itisameromorphicmodularfunctionofweightzero,andistheuniquesuchfunctionwhichisholomorphicawayfromasimplepoleatthecuspsuchthatj()=0andj(i)=1728.Theq-expansionjhascoecientsinZwhichgrowsub-exponentiallybutfasterthanpolynomially.Weshallbeinterestedinthecongruencepropertiesofthecoecientsc(n).1.1.1.Exercise.TheWikipediaentryonthej-invariantisembarrassing|makeitbetter.Oursecondexample(whichweconsidermorebrie y)willconcerntheinverseofDedekind'setafunction,whichis(essentially)thegeneratingfunctionforthepartitions,namely:�1=1 q1=24Q1n=1(1�qn)=Xpn+1 24qn:(Followingthestandardconvention,p(m)=0ifmisnotaninteger.)ThefollowingtheoremsareduetoLehner[Leh49]andWatson[Wat38]respec-tively.2 1.1.2.Theorem.Thefollowingcongruencesaresatised:(1)Ifn0mod2mandn�0,thenc(n)0mod23m+8.(2)If24n1mod5m,thenp(n)0mod5m.Theproofofthesecongruencesreliedonajudicioususeofmodularequations,thatis,theexplicitfunctionalrelationshipsbetweenmodularfunctionsofcertainsmalllevels.Weshalltherebydubthistechniquethemodularequationmethod;itwasalsoappliedbyAtkinandO'Brientoprovesimilarcongruencesforhighermoduli[AO67].Weshallarguethatthesystematicuseofoverconvergentmodularformsisadirectdescendantofthemodularequationmethod.1.1.3.Thescopeofthisdocument.Thesenotesarenotintendedtobeanintro-ductiontothetheoryofmodularforms,althoughweshallsummarizesomeofthesalientdetails.Rather,itisdirectedtowardsthreespecicaudiences,namely:(1)Graduatestudentsinnumbertheorywithabasicunderstandingofclassicalmodularformsandtheirq-expansions.(2)Thosewhoareinterestedincongruencesconcerningspecicmodularforms,forexampleinvolvingpartitions,butwhomarenotfullyconversantwiththemoderngeometricandrigidanalyticviewpointofDwork,Katz,andColeman.(3)Thosewhounderstandthetheoryofoverconvergentmodularforms,andarecuriousabouttheapplicationstoconcretecongruences.Sincetheseaudiencesbydenitionhavesomewhatdierentbackgrounds,Iwillhavetoapologizeinadvanceforsayingthingsthatyou,dearreader,willndobvious.Iwillalsoapologizeforelidingtechnicaldetailswhoseabsencemaypushthemorecarefulreaderintoanapoplectict.However,thetheoryofellipticcurvesandmodularformsencompassesquitealotofmathematics,andsoIwillnecessarilybecursoryonseveralimportantpoints(mostimportantly,thetechnicaldetailsconcerningtheconstructionofmodularcurves[DR73,KM85],aswellasanyrigorousdetailsatallconcerningrigidanalyticspaces).Inparticular,Iwillconcentrateontheissuesthataremostrelevanttomypurpose,andleavethesecondarymattertotheliterature,whichisextensiveand(quitefrequently)verywellwritten,e.g.[Sil86,Sil94,Kat73,DS05,Buz03].Indeed,aswithanylecturenotes,thekeychoiceistodecidewhichpointstoelide,whichpointstoskip,andwhichpointstoemphasize.SincemuchofwhatIsayinthersthalfofthesenotesoverlapswithwhatisin[Kat73],IleaveoutseveralargumentsthatKatzgivesindetail,andinsteadconcentrateongivingexamplesandemphasizingthepointsthatsomemightndconfusingifapproaching[Kat73]withlimitedbackground.Letmeincludeatthispointthefollowingtable,whosecontent1isself-explanatory. Themodernmethod Theclassicalantecedent ThecompactnessoftheUoperator Themodularequationsmethod SerreweightsandAsh{Stevens Theweightltrationinlowweights p-adicLanglandsforGL2(Qp) Theweightltrationinhigherweights Holomorphicsectionsovertheordinarylocus Serre'sp-adicmodularforms 1Forreasonsoftime,Iwillnotdiscussinanydetailthesecondandthirdrowsofthistable.Fortheconnectionbetweenthe-operatorandSerreweights,oneshouldconsult[AS86].Theonlytimetheseideasariseinanyformwithinthesenotesissecretly|viaanappealtoaresultofBuzzard{Gee[BG09]concerningGaloisrepresentationsassociatedwithsmallslopeforms.However,Iwillsuppressallofthedetailsofthatpaper,togetherwiththeirconcomitantdicultiesrelatingto,interalia,p-adiclocalLanglands.3 Explicitly,themapisgivenbyx=}(z;)=1 z2+Xn01 (z�)2�1 2;y=dx dz=}0(z;)=X�2 (z�)3:Moreover,everyellipticcurveoverCadmitssuchauniformization.Inparticular,givenalattice,oneobtainsanellipticcurveE.(AnellipticcurveEoverCis,bydenition,asmoothgenusonecurvewithamarkedpoint,whichinthiscaseisthepoint\atinnity",e.g.z=0.ShowingthatellipticcurvesoverCadmitaWeierstrassequationisanelementaryexerciseusingtheRiemann{RochTheorem.)Itisimportanttonote,however,thismapisnotabijection.Itiseasytoseethatifonescalesthelatticebyahomothety,sayreplacingby,thenG4isreplacedby�4G4andG6isreplacedby�6G6.Thecorrespondingellipticcurvesareisomorphicunderascalinginxandy.Inparticular,thismapisabijectionbetweenlatticesinCuptohomothetyandellipticcurvesEoverC.Modularforms,however,arenotfunctionsonlatticesuptohomothetyunlessk=0.Itisnaturaltoask,therefore,whetherWeierstrass'theoremgivesanaturalbijectionbetweenlatticesandellipticcurvesenrichedwithsomeextrastructure.1.2.3.Lemma.ThespaceofholomorphicdierentialsonanellipticcurveEoverCisonedimensional,thatis,H0(E; 1)=C.Proof.Ifonedenesanellipticcurvetobeasmoothprojectivecurveofgenusone,thenthislemmaisatautology.IfoneimaginesanellipticcurvetobegivenbyaquotientE=C=,thenonecanargueasfollows.Anyholomorphicdierentialpullsbacktoadierential!=f(z)dzonCwhichisinvariantundertranslation.Sincethedierentialdzhasnopolesandnozeroes,itfollowsthatf(z)mustbeholomorphiconCanddoublyperiodic,andthus(byLiouville'stheorem)constant.Hencetheonlysuchdierential(uptoscalar)isdz.Thisdenitionallowsustounderstandwhatextrastructurealatticecontainsbeyondtheisomorphismclassofthecorrespondingellipticcurve.1.2.4.Lemma.ThereisabijectionbetweenlatticesCandellipticcurvesEtogetherwithanon-zerodierential!2H0(E;C).ThebijectionisgivenbytakingalatticetothecorrespondingWeierstrassequation,andthentaking!:=dz=dx dx=dz=dx y:Thisbijectioniscanonical,butitisnotcanonicallycanonical,sinceonecouldformsuchabijectionusinganyxedmultipleofdX=Y.Thischoice,however,alsomakessenseintegrally(atleastuptofactorsof2).Underthisbijection,wecancomputeexplicitlywhathappensifwereplaceby.Wedothisingorydetail.Explicitly:7!:y2=4x3�60G4x�140G6;!=dx y;7!:v2=4u3�60G4�4u�140G6�6;!=du v:5 Nowwemaywritethelattercurveas(v3)2=4(u2)3�60G4(u2)�140G6,andhence!=du v=dx�2 y�3=
!:Thisleadstoanewdenitionofamodularform.1.2.5.Denition(Version2).AmodularformfofweightkoverCisafunctiononpairs(E;!)consistingofanellipticcurveEandanon-zeroelement!2H0(E; 1E)suchthatf(E;!)=�kf(E;!);andsuchthatf(C=(Z+Z);dz)isboundedas!i1.Thiscoincideswiththepreviousdenitionifweletf()=f(C=;dz),since(aswesawabove)f()=f(C=;dz).1.2.6.ThefundamentaldomainandX(1).TheactionofPSL2(Z)ontheupperhalfplaneHhasafundamentaldomaingivenbytheshadedregioninFigure1.2.6: Figure1.ThisisapictureyouhaveseenbeforeThequotientY(1)=H=PSL2(Z)hasanaturalstructureasacomplexorbifoldwithconepointsofangles2=2atz=i,2=3andz=,and2=1atz=i1.ThereisanaturalcompacticationX(1)obtainedbyllinginthecuspatinnity.Thefunctionj:X(1)!P16 Figure2.Themonodromyof!Xfor 2�thesheafofrelativedierentials 1E=Y(�)onY(�).If:E!Y(�)denotesthenaturalprojection,thenwelet:!Y:= 1E=Y:Intuitiontellsusthatthebreof!YatapointE2YcorrespondingtoanellipticcurveEshouldbeexactlywhatwearelookingfor,i.e., 1E=H0(E; 1E)(asheafataclosedpointissimplythemoduleofglobalsections).Thisintuitioniscorrect|itrequiresonlythatthemapisproper.1.2.18.Denition(Version3a).AmeromorphicmodularformfofweightkoverRandlevel�isasectionofH0(Y(�)R;! k).Notethatwealsowouldliketounderstandwhathappensatthecusps.For-tunately,theconstructionof[DR73]providesuswithageneralizedellipticcurveE=X(�),andacorrespondinglocalsystem!XonX(�),andwemayset:1.2.19.Denition(Version3b).AmodularformfofweightkoverRandlevel�isasectionofH0(X(�)R;! k).It'susuallysensibletoassumethatRisaZ[1=N]-algebrawhereNisthelevelof�.DenotetheR-moduleH0(X(�)R;! k)ofmodularformsbyMk(�;R).1.2.20.Warning.Inordertodene!Yor!X,oneneedstheexistenceofauniver-salgeneralizedellipticcurveE,whichrequiresthemoduliproblemtobene,which11 requiresworkingwithX1(N)ratherthanX0(N).Thisisnotanarticialproblem|thereisnoappropriatesheaf!onX0(N),andmodularformsofoddweightonX0(N)areautomaticallyzeroforparityreasons.1.2.21.Exercise.Showthat|whenitmakessensetocomparethem|allthedenitionsofmeromorphicmodularformscoincide.1.2.22.Kodaira{Spencer:Anotherdescriptionwhenk=2.Anotherdescriptionofmodularformsofweight2overCarisesfromthefactthat,forsuchforms,f()disinvariantunder�.Thismightleadonetosuspectthat 1X'! 2X;butthisisonlycorrectalongY(�).Theproblemisthatthedierentialdisnotsmoothatthecusp.Thenatural(analytic)parameterat1isq=e2i,andd=1 2idq q:Inparticular,asectionofH0(X; 1)willbe(locally)amultipleofdq,andsothecorrespondingfunctionf()dq=2iqf()dwillautomaticallyvanishatthecusp.Inparticular,thecorrectisomorphismis 1X(1)'!2X;whereD(1)indicatesthatdierentialsareallowedtohavepolesofordersatmostoneatthecusps.TheseisomorphismsgobythenameoftheKodaira{SpencerIsomorphism|overYitcanbededucedmoredirectlyusingdeformationtheory(seex3Bof[HM98]forageometricdiscussion).1.2.23.Changeofcoecients.MostlythecoecientsRjustcomealongfortheride.Inparticular,Mk(�;R)denotestheformsofweightkandlevel�overR,thenonemighthopethatMk(�;S)=Mk(�;R) RSforanS-algebraR.ThisiscertainlytrueifSisa atR-algebra,butitisnotalwaystrue.Theexceptions,however,aremainlyconnedtovaryparticularcir-cumstances:1.2.24.Proposition.LetSbeanR-algebra,andsupposethatNisinvertibleinR.Thenthereisanisomorphism:Mk(�(N);S)'Mk(�(N);R) RSprovidedthatN3andk2.TheonlyinterestingcaseisreallywhenR=ZpandS=FpforaprimepnotdividingN.Thereisamap:0!!!!!!=p!0oflocalsystemsonX(N)=Zp.SincejX(N)=Fp!X(N)=Zpisaclosedimmer-sion,thereisanisomorphismHi(X(N)=Fp;! k)'Hi(X(N)=Zp;j! k)=Hi(X(N)=Zp;! k=p):ItsucestoshowthatH1(X(N)=Zp;! k)=0.Sincethisisnitelygenerated,itsucestoshowthatH1(X(N)=Zp;! k)=piszero,andhenceitsucestoshowthatH1(X(N)=Fp;! k)iszero.Thereisaverynaturalwaytoshowthatthe12 isthenaprioriameromorphicmodularform,however,bylookingattheq-expansion,weseethatitisregularatthecuspsandthusholomorphicbytheq-expansionprinciple.(ThisniceargumentisduetoGross[Gro90]x4.)Ontheotherhand,fordiscussionofHeckeoperators\withoutthecrutchofq-expansions"see[Con07].1.5.TheFrobeniusmorphism.SupposethatSisaringwithpS=0,andsupposethatX=Sisascheme.Inthiscontext,thereareapairofmapswhichgoviathenameFrobenius.First,onehastheabsoluteFrobenius,whichinducesmapsFabs:Spec(S)!Spec(S)andX!X.Thismapisgiven,locallyonrings,bythemapx7!xp.Thereisacommutativediagramasfollows:X Fabs-XSpec(S)? Fabs-Spec(S)? Inparticular,FabsisnotamapofschemesoverS,unlessFabsonShappenstobeconstant(soS=Fp).TherelativeFrobeniusisawayofmodifyingthistogiveamorphismofschemesoverS.Namely,letX(p)=XSS,whereSisthoughtofoverSnotbythetrivialmapbutviaFabs.Then,byconstruction,thereisamap,relativeFrobenius,givenbyF:X!X(p),suchthatthecompositionwiththenaturalmapX(p)!XisFabs.1.5.1.Exercise.LetXbethesmoothcurve:ax3+by3+cz3=0overk,where3abc6=0.ProvethatX(p)isthecurveapx3+bpy3+cpz3=0:NotethatthemapFabsonH(X;OX)isnotS-linear,althoughitisFp-linear.1.6.TheHasseinvariant.LetSbeasintheprevioussection.SupposethatE=Sisanellipticcurvetogetherwithadierential!Sgenerating 1E=S.BySerreduality,wemayassociateto!Sadualbasiselement2H1(E;OE).TheFrobeniusmapinducesamap:FabsH1(E;OE)!H1(E;OE);andwemaywriteFabs()=A(E;!)
forsomeA(E;!)2S.1.6.1.Lemma.Aisameromorphicmodularformofleveloneandweightp�1overS.Proof.Ifonereplaces!by!0=
!for2S,thenbecomes0=�1,andFabs(0)=Fabs(�1)=�pA(E;!)
=1�pA(E;!)
0;andthusA(E;
!)=1�pA(E;!):18 IfSisaeld,thenA(E;!)iseitherzerooraunit|itiszeropreciselywhenEissupersingular(thisisessentiallyadenitionofwhatitmeanstobesupersingular,see[Sil86]).WecallAtheHasseinvariantofE.Tocomputetheq-expansionofA,weneedtoevaluateAonthepair(T(q);!can).Inordertodothis,weneedtounderstandtheoperatorFabsmoreexplicitlyoncurves.1.7.TheCartieroperatoroncurves.SupposethatX=Sisasmoothcurveofgenusg,andthatSisperfect.TheCartieroperatorCdenesamapfromthemeromorphicdierentialsonXtoitselfsatisfyingthefollowingproperties:(1)CpreservestheholomorphicdierentialsH0(X=S; 1).(2)C(fp!)=f
C(!)foranymeromorphicfunctionf.(3)C(fn�1df)=(df;n=p0;0np:.(4)If2H1(X;OX),then,underthepairingofSerreduality,hC(!);i=h!;Fabsi:Toimaginewhysuchanoperatormightexist,considerthecompletion[OX;xofalocalringatx.BecauseXissmooth,thecorrespondingringisS[[x]],themero-morphicdierentialsareoftheformS[[x]]dx.ThenCmaybedenedasfollows,writing!=f(x)dx:C(!):=pr �dp�1f dxp�1dx:ClearlyCpreservesholomorphicityatapoint.Bythechainrule,onehas:dp�1fpg dxp�1=fpdp�1g dxp�1;dp�1fn�1df dxp�11 n
dp dxp(fn):TheusageofismeanttoindicatethatthisequalityisonlyanequalityofformalexpressionsoverZ[[x]]|overS[[x]]itmakesperfectsenseif(n;p)=1andimpliesthatC(fn�1df)=0,becausedp=dxpisclearlythezerooperator.ForpjnitisnottoohardtomakeformalsenseofwhatthismeansandcomputethatC(fp�1df)=df.Tomakethisrigorous,oneneedstoshowthatCmayalsobedenedalgebraically,andthatitdoesnotdependonthechoiceofauniformizer.Itwouldtakeusalittletoofaraeldtoprovethesestatements,however.1.7.1.Deligne'sComputationofA.Nowletuscomputetheq-expansionofA,namely,tocomputeA(T(q);!can)overR=Fp((q)).1.7.2.Theorem(Deligne).A(T(q);!can)=1.Theideaissimplytoshowthatthecorresponding1-formispreservedunderFabs.FromthecharacterizingpropertiesoftheCartieroperator,itthussucestoshowthatC(!can)=!can.Letx=t�1beauniformizingparameterofGmattheorigin.Thena(Gm-invariant)dierentialisgivenbydx 1+x=dt t=!can:YetaneasycomputationshowsthatC(dt=t)=t�1C(tp�1dt)=dt=t.Wecan'tquitearguethisway,becauseRisnotperfect(eventhelocaldenitionofCaboveinvolvestakingpthrootsandthusrequiresthattheunderlyingringSbeperfect).19 Ontheotherhand,ifRperistheperfectionofR,then,sinceR!Rperis at,bytheq-expansionprinciple(Corr.1.3.2),itsucestoworkoverRper,wheretheargumentabovegoesthrough.1.8.LiftingtheHasseinvariant.(cf.[Buz03]).Ifp5,thenAliftstoamodularformincharacteristiczero.Fromthecomputationabove,wenotethattheq-expansionofanysuchliftiscongruentto1modp.FromtheKummercongruences,wededuce:1.8.1.Theorem.Supposethatp5.ThemodularformEp�1isaliftofAsuchthatEp�1Amodp.Ifp=2orp=3,themodularformsE4andE6areliftsofA4mod8andA3mod9respectively.Proof.Forp=2and3thisisacomputation;forp5itisanimmediatecon-sequenceofthevonStaudt{ClausentheoremonBernoullinumbers,aswellastheidenticationoftheconstanttermoftheclassicalholomorphicEisensteinseries.Justtobeclear,hereby\mod"pwereallymeanmodulotheideal(p),sothatifEp�1isthoughtofasamodularformoversomeringR,thiscongruenceidentiesthevalueofAinR=p,evenifthelatterisnotreduced.Inparticular,givenaWeierstrassequationforE(andhenceacanonicaldierential!)onemaycomputeAmodpbycomputingthecorrespondingvalueofEp�1,withappropriatemodicationsifp=2or3.2.p-adicmodularformsLetusxacongruencesubgroup�oflevelprimetop.Thedenitionofp-adicmodularformandoverconvergentp-adicmodularformatlevelonearevirtuallythesameasthecorrespondingdenitionatlevel�|oneneedonlyaddthenaturalcompatibilitywiththelevelstructureawayfromp.Inthesequel,therefore,weshallessentiallyignorethisdistinctionandworkatlevelone,makingremarksaboutthelevelstructureawayfromp(the\tame"levelstructure)whenappropriate.2.1.p-adicmodularforms:TheSerreapproach.Serrewroteabeautiful3andelementarypaperonp-adicmodularforms[Ser73b].Thebasicidea(translatedintosomewhatdierentlanguage)isasfollows.Inordertocapturethenotionofcongruencesbetweenmodularformsinsometopologicalway,thenwewouldliketosaythattwoq-expanionsaandbarecloseifabmodpnforlargen.Recallthat(uptonormalization)thespaceofmodularformshasabasiswithcoecientsinZ.ThereisanaturaltopologyonZp[[q]] Qp(note,thisisdierentfromQp[[q]]) 3Asonewouldexpect,Serreeortlesslyexplainseverythingseeminglystartingfromrstprin-ciplesandgivesabeautifulexplanationoftheconstructionofp-adicL-functions.Followingtheelementaryargumentsdowntothesource,thekeyfactistoshowthat=1Xn=1p�2(n)qndoesnotlieintheeldoffractionsoftheringofmodularformsmoduloponthecomplementXnSofthesupersingularlocus([Ser73b],p.199Ser-9).Anelementaryargumentusingweightsshowsthatitselfisnotamod-pmodularform.Serrenotesthat�p= forsomeexplicit ,andthenusesthefactthatH0(XnS;OX)isintegrallyclosed(becauseXandthusXnSissmooth)toobtainacontradiction.YettogettheidenticationofgM0withH0(XnS;OX),oneusesthefactthattheHasseinvariantAiscongruentto1modp,whichisofanorderofdicultyhigherthantherestoftheargumentsinthepaper.Thusitisbettertoread[Ser73b]inconjuctionwithSerre'sBourbakiseminaronthesubject[Ser73a],whichgivesalittlemoredetailconcerningthisargument.20 whichexactlyrecordsthisnotionofcongruence,andwemaydenep-adicmodularformstobetheclosureofthesetofmodularforms(see[Ser73b]).LetAbe(any)liftoftheHasseinvariant.SinceA1modp,itfollowsthatthepowersofA(atleastthepnthpowers)arebecomingmoreandmorecongruentto1modp.Hencetheyconvergeto1.ItfollowsthatthepowersApn�1areconvergingtoA�1,orinparticularthatanyliftoftheHasseinvariantisinvertible.Thespaceofp-adicmodularfunctionsthendenesitself:2.1.1.Denition.Thep-adicmodularfunctionsonX(�)arethefunctionswhicharewelldenedatallpointsofordinaryreduction.Naturallyenough,onemightbealittlesuspiciousofthisdenition,sinceoneisallowingpolesataninnitenumberofsupersingularpoints(thereareonlynitelymanysupersingularpointsmodulop,butthereareinnitelymanyliftstocharac-teristiczero).Aninitialsteptorepairingthisisgivearule{baseddenition.2.1.2.Denition(Version2).Ap-adicmodularformfofweightkandleveloneoverap-adicallycompletealgebraAisafunctiononpairs(E=R;!)forap-adicallycompleteA-algebraRsatisfying(1)!isanowherevanishingsectionof 1E=A,(2)A(E=B;!B)isinvertible,whereB=A=p,suchthat:(1)f(E=A;!)dependsonlyontheA-isomorphismclassof(E=A;!).(2)f(E;!)=�kf(E;!)forany2A.(3)If:A!Bisanymapofrings,thenf(E=B;!B)=(f(E=A;!)).(4)f(T(q);!can)2A[[q]].Asexpected,thereisananalogousdenitionatlevel�primetop,whereoneconsidersfunctionsf(E=A;!;)forsomelevelstructureawayfrompcorrespond-ingtothegroup�.DenotethisspacebyMk(�;R;0).If�=SL2(Z)(whichisanaturalchoiceforwhichallthephenomenacanalreadybeseen)wesimplywriteMk(1;R;0).ClearlyanyclassicalmodularformofweightkoverRgivesap-adicmodularform.Moreover,ap-adicmodularformoveraniteeldk(necessarilyofcharacteristicp)consistsofsectionsofH0(XnS;!k),whereSisthesupersingularlocus(sinceinvertibleoveraeldisthesameasnon-vanishing).IfAistheHasseinvariant,thenAisamodularformofweightp�1overFp,andA�1isamodularformofweight1�poverFp.Moreover,Apn�1denesaninvertiblep-adicmodularformofweightpn�1(p�1)overZ=pnZ,andthus:2.1.3.Lemma.Supposethatpn=0inR.Thenthemap;Apn�1:Mk(�;R;0)!Mk+pn�1(p�1)(�;R;0)isanisomorphism.Fromthiswededucetheq-expansionprinciple.2.1.4.Lemma.ThereisaninjectivemapMk(�;R;0)!R[[q]].Proof.Byconstruction,Mk(�;R;0)=Mk(�;lim R=pn;0)=lim Mk(�;R=pn;0):21 2.2.3.Remark.Sincenewformsoflevel�0(p)haveUp-eigenvalueequaltooneofthenumbersp(k�2)=2,whenk�2,theordinaryprojectionisgeneratedbyoldformsfromMk(�;R).2.3.Whyp-adicmodularformsarenotgoodenough.Letusexplainwhyonemightasktogobeyondthetheoryofp-adicmodularforms.Wedothiswithanexample,rstobservedbyAtkinandO'Brien[AO67].Thej-invariantdenesameromorphicmodularformofweightzeroandlevel1foranyp.ItfollowsthatUpjalsoap-adicmeromorphicmodularformofweightzeroandlevel1.Ontheotherhand,sinceUpj2Zp[[q]],thefunctionUpjextendstothecuspanddenesanhonestp-adicmodularform.ByHida'stheorem,itfollowsthatepUpjliesintheordinaryspaceepM0(1;Zp;0),andcanthusbewrittenasanitesumof(generalized)ordinaryeigenforms.AtkinandO'Brienconsiderthespecialcaseofp=13.Inthiscase,thedimensionofe13M0(1;Z13;0)isthesamedimensionase13M12(1;Z13;0),whichistheimageoftheclassicalspaceM12(1;Z13)undere13.Thespaceofmodularformsofweight12andlevel1overZ13isgeneratedbyE121mod13and
.TheeigenvalueofT13on
is(13)=�577738;andthustheordinaryprojectione13M12(�0(13);Z13;0)consistsoftheforms:E12(q)�1311E12(q13);
(q)�
(q13);withU13eigenvalues1andrespectively,where=�288869�p �1708715094876=5
1311+3
1312+9
1313+:::=�288869+p �1708715094876=8+5
13+10
133+5
134+:::aretheroots(inZ13)ofx2+577738x+1792160394037=0.Inparticular,theordinaryspacehasdimensiontwo.Aprioriitiseasytoobtainanupperboundof2,butthelowerboundrequiresthecomputation(13)860mod13.Itfollowsthate13M0(�;Z13;0)hasdimensiontwo.Thefunction1isaclassicalmodularformofleveloneandweight0,anditisalsoanordinaryformwitheigenvalue1.Theotherordinaryformisthusanormalizedcuspidaleigenformh2Zp[[q]].Indeed,h
mod13.Itfollowsthatthereisanidentitye13U13j=744+hforsome2Z13.Inparticular,wededucethat,foranyxedd,Un13j�744mod13d=1Xk=1c(13nk)qkmod13disaHeckeeigenformforsucientlylargendependingond.Theseargumentsdonotmakeclear,however,howlargenhastobeforanyparticulard.Ontheotherhand,AtkinandO'Brienconjecturedsomethingmuchmoreprecise,namely,theyconjecturethat1Xk=1c(13nk)qkmod13nisalreadyaHeckeeigenform.Thatis,theconvergenceofUn13jtotheordinaryprojectionislinear.Wemayask:isthisageneralphenomenonforallp-adicmodularforms?Inthisgenerality,itturnsoutthattheanswerisno.25 aslongasweinterpretXrig0(p)[0]tobethecomponentoftheordinarylocuscon-taining1.Yetthissectionextendsasfarasthecanonicalsubgroupcanbedened,namely,toXrig[r]foranyrp p+1:Toseethisexplicitlyforp=2,weneedtorecallsomeofthegeometryofX0(2).Ithasgenuszero,andthusitisgivenbytheprojectivelineforsomemodularfunctionf.Therearevariouschoicesofftomake,butoneclassicaloneistheinverseoftheHauptmodul:f=
(2)
()=q1Yn=1(1+qn)24=q+24q2+:::Onehastheclassicalmodularequation:f (1+28f)3=j�1=1 q+744+:::�1=q�744q2+356652q3:::Thefunctionsfandj�1arebothuniformizingparametersatthecusp1.Letustrytocomputeasectionbysolvingthecorrespondingcubicequation:(1+28f)3�jf=0inaneighbourhoodofj�1=0.Theslopes(valuationsofthecoecients)ofthispolynomial(asapolynomialinf)are[24;16;v(3
28�j);0]:Inparticular,aslongas:kj�1k2k2�8k=28;thereisauniquerootfofvaluation&#x]TJ/;༔ ; .96;& T; 10;&#x.516;&#x 0 T; [0;�8.HowdoesthisrelatetoourpreviouscomputationandLemma3.0.4?NotethatE4=1�240X3(n)qnisaliftofA4.Moreover,wehaveE34
=j:IfweareclosetothecuspofX(1),then
isclosetozeroandthereisacanon-icalsubgroupcorrespondingtopinT(q)(Thecorrspondingellipticcurveshavemultiplicativereduction).SupposeinsteadthatEhasgoodreductionat2.Then,choosing!soastoobtainaminimalmodelforE,wendthat
(E;!)isaunit,andhencev(E34)=v(j):Inparticular,theregionv(j)8corresponds(forcurvesofgoodreduction)totheregionv(A)=1 4v(E4)=1 12v(E34)8 12=2=3;whichisexactlytheboundrequiredtoadmitacongruencesubgroup.3.2.3.Exercise.Showthat,fortheellipticcurveEofexercise3.0.5,onehasj=26+29+O(211),andf(E;C)=2�6+2�4+24+O(25).30 ofoperatorshavethisproperty.Imagine,forexample,weletC(r)denotethecom-plexanalyticfunctionsontheclosedballjzjr.SupposewehadancontinuousoperatorU:C(1)!C(2):ThenwecouldcomposeUwiththerestrictionmapC(2)!C(1).Whatisamazingaboutthislastmapisthatitiscompact.3.4.Overconvergentp-adicmodularforms.3.4.1.Denition.Let0rp=(p+1)berational.Thespaceofoverconvergentmodularformsofweightk,level�,andradiusr,isdenedtobe:Myk(�;r)=H0(Xrig[r];!k):3.4.2.Remark.Thereisaninclusion:Myk(�;r)!Mk(�;Cp;0)!Cp[[q]]:Henceoverconvergentmodularformssatisfytheq-expansionprinciple.3.4.3.Example.SupposethatN=1andp=2andk=0.ThenMy0(�;r)=C2h2rfiisaballofradius2�r.3.4.4.Lemma.My0(�;r)isaBanachspacewithrespecttothesupremumnormonXrig[r].Denotethenormbyk
kr.Therestrictionmaps;:My0(�;s)!My0(�;r)arecontinuous,sincekgksk(g)kr:Thenormalsomakessensewhenr=0.Inthiscasetheformsarenolongeroverconvergentandthuswedropthey.3.4.5.Exercise.Showthatk
k0co-incideswiththeq-expansionnorm.DeducethatanysequenceofoverconvergentmodularformsconverginginMyk(�;r)arealsoconvergingintheq-expansiontopology.Itisnottoodiculttoconstructsectionsof!kwhichdon'tvanishonXrig[r],andhenceMyk(�;r)'My0(�;r)asBanachspacesforeveryintegerk.Ofcourse,theseisomorphismsdon'tcommutewithHeckeoperators.3.5.Compactoperatorsandspectralexpansions.LetUbealinearoperatoronaniterankvectorspaceV(youcan,ifyouwish,chooseabasisforVandthinkofUasamatrix).HerewesupposethatthecoecientslieinR,orC,orQp,orCp,oranycompletenormedeldF.TheoperatorUhasngeneralizedeigenvaluesinsomeniteextensionofF.Foranyv2V,wemaywritev=Xiviforaneigenbasisvi.Letussupposethatj1j&#x-298;j2j:::jnj:33 3.5.2.Lemma(AsymptoticExpansions).SupposethatUactscompactlyonasep-arableBanachspaceBwithanultrametricnorm.Then,forv2B,thereexistsconstantsiandgeneralizedeigenvectorsviofUwithnon-zeroeigenvaluePivianda\spectralexpansion"vPiviwiththefollowingproperty.Let�0beaxedrealnumber.Then,asngoestoinnity, Unv�XiUnvi =o(n);wherethesumrangesoverthenitelymanygeneralizedeigenvectorsviwhosecor-respondingeigenvalueis.Inparticular,anasymptoticexpansionallowsonetounderstandUnvmoduloanyxedpowerof,withthenecessaryprovisothattheimpliederrorconstantsdependon.3.5.3.Remark.Notethatforaxedeigenvalue6=0,thegeneralizedeigenspaceofUisnitedimensional,butthatnotallgeneralizedeigenfunctionsmaybeactualeigenfunctions.Thishappensalreadyinthenitedimensionalcase.3.5.4.Exercise.LetC(r)denotethecomplexanalyticfunctionsonjzjr.Provethatthecomposition:C(1)!C(2)!C(1)denedbyUf(z)=f(z=2)iscompact.DeterminealltheeigenvectorsofU,andprovethateveryelementinC(1)admitsanabsolutelyconvergentspectralexpansion.ThepointofthisexerciseisthatthemapUpisexactlyoftheform,andhence:3.5.5.Theorem.Supposethatrp p+1:ThenthemapU:H0(Xrig[r];!k)!H0(Xrig[r];!k)iscompact.Theproofisthatitiscomposedofacontinuousmapwhichextendsconvergencewiththerestrictionmapwhichiscompact(thisusesTheorem3.3.2.)Thebigquestionthenis,whattypeofcompactoperatoristhis?3.6.ClassicalForms.ThefollowinglemmaistheanalogofLemma2.1.63.6.1.Lemma.Thereisaninclusion:Mk(�0(pn))Myk(�;r)foranykandsmallenoughr.IfE=Risnottoosupersingular,thenEhasacanonicalsubgroupC.AslongastheHasseinvariantofE=Rissucientlylarge,wededucethatE=Cisalsonottoosupersingular,andthus(byinduction)aslongasrissucientlysmall,forsuitableE=RwemayndacanonicalsubgroupCoforderpn,fromwhencethelemmafollows.35 3.6.2.Someimportantbutnotentirelyrelevantfacts.Supposethatfisaclassicaleigenformofweightkandlevel�.SupposethatTphaseigenvalueap.Considerthepolynomialx2�apx+pk�1;whichistheminimalpolynomialofcrystallineFrobenius(afactwhichisbothhighlyrelevantandcanbeignoredcompletely).Associatedtofisatwodimensionalspaceofold-formsoflevel�0(p),givenexplicitlybyf=f(q)andUpf=apf�pk�1f(qp).Ifandaretherootsofthecharacteristicpolynomial(theyareconjecturallydistinctifk�1)thenf(q)�f(qp)andf(q)�f(qp)havelevel�0(p)andareeigenvaluesofUp.Theyareoverconvergent!Notethatv();v()p�1.Thereis(almost)aconversetothis,namely,iffisanoverconvergenteigenformforUpwithUpf=fandv()k�1,thenfisclassical.ThisisatheoremofColeman[Col96].Whenv()=k�1,itcan(anddoes)goeitherway,althoughtherearemorerenedconjecturespredictingwhatshouldhappeninthiscase.3.7.Thecharacteristicpowerseries.AssociatedtothecompactoperatorUpistheFredholmpowerseriesdet(1�TUp)2Zp[[T]].GeneralizingHida'stheo-rem,Colemanshowsthatastheweightvaries,thecoecientsofthisseriesvarycontinuouslyintheweight.Moreover,theymaybeidentiedwithelementsintheIwasawaalgebra=Zp[[Zp]].Usingthefactthatformsofsmallweightareclassi-cal,theusualtraceformulaallowsonetogiveanexactformulaforthecoecientsofdet(1�TUp)asnitesumsinvolvingclassnumbers.Inparticular,thecoecientsareverycomputable,andthus,viaNewton'sLemma,thevaluationsofthespectraleigenvaluesj1jj2j:::arealsoverycomputable.3.7.1.Exercise.ShowthatanyniteslopeeigenvalueofUpliesinMyk(�;r)foranyrp p+1.3.7.2.Exercise.ProvethatthetraceofU2onMy0(1;r)is7�p �7 28=1+23+24+27+210+212+213+:::3.8.TheSpectralconjecture.Wehaveseenthat,ingeneral,theasymptoticex-pansionwithrespecttoacompactoperatorneednotbeabsolutelyconvergent.OnemayaskwhetherthissequencedoesconvergeinthespecialcaseofoverconvergentmodularformswithrespecttotheUp-operator.Oneobstructiontoconvergenceisasfollows.3.8.1.Lemma.IftheasymptoticexpansionofanoperatorUonaBanachspaceBisconvergenttotheidentityoperator,thenker(U)=0.Proof.Thisisobvious.Ontheotherhand,wehavethefollowing.3.8.2.Lemma.VpdenesamapVp:Myk(�;r)!Myk(�;r=p):TheproofisvirtuallythesameastheproofthatVppreservesp-adicmodularforms.Moreprecisely,VpfevaluatedonEdependsonlyonfevaluatedatE=C,36 whereCisthecanonicalsubgroup.YetthisincreasesthevaluationoftheHasseinvariant.Onq-expansions,wehaveVpXanqn=Xanpqn:Inparticular,thecompositionUpVpistheidentity.(Thisfollowsfromtheq-expansionprinciple.)LetWpbetheoperator1�VpUp.ThenUpWp=Up�UpVpUp=Up�Up=0.Inparticular,iffliesinker(Up)thenWpf=f,andmoreover,theimageofWpliesinthekernelofUp.Onq-expansions,wehaveWpXanqn=X(n;p)=1anqn:3.8.3.Lemma.Supposethatr1 p+1,thenWpdenesamap:Myk(�;r)!Myk(�;r)whichisaprojectionontoker(Up).Thereasontheboundonrisneededisthatforlargerr,itisnotnecessarilythecasethatUpincreasestheradiusofconvergencebyafactorofp,andthusthecompositeVpUpmaydecreasetheradiusofconvergence.Weimmediatelydeducefromthisthefollowing:3.8.4.Lemma.Supposethatr1 p+1:ThenthekernelofUponMyk(�;r)isinnitedimensional.Inparticular,thespectralexpansionofUpforsuchrisnotingeneralconvergent.Ontheotherhand,thereseemstobeatransitionthattakesplaceatr=1=(p+1),asindicatedbythefollowinglemma.3.8.5.Lemma.Ifr�1 p+1,thenthekernelofUponMyk(�;r)istrivial.Proof.Thisis[BC06]Lemma6.13(andRemark6.14).Thekernelofacompactoperatorisnottheonlyobstructiontoconvergence.RecallthatthedampedshiftoperatorUxn�1=nxnconsideredabove(wherelimn=0isasequenceofnon-zeroelements)hastrivialspectralexpansionseventhoughUitselfhasnokernel.Therestill,however,appearstobereasontobelievethefollowing.3.8.6.Conjecture.Supposethatr2(1=(p+1);p=(p+1)).ThenanyF2Myk(�;r)hasaconvergentspectralexpansionwhichconvergestoF.Explicitly,wemaywriteanyF2Myk(�;r)asF=Xi(F)vi;wheretheviareaxedchoiceof(generalized)eigenvectorswitheigenvaluesi.Notethatby\convergence"abovewemeanconvergenceintheBanachspacenormonMyk(�;r)(thatis,thesupremumnorm).Thisisamuchmorerestrictivecondi-tionthanconvergenceintheq-expansiontopology(whichisthesupremumnormontheordinarylocusXrig[0].)37 3.10.2.Theorem.Let123:::denotetheeigenvalueswhichoccurforN=1andp=2ofvaluation3;7;13;:::.Letvidenotethecorrespondingeigenform,normalizedsothattheleadingcoecientis1.IfF2My0(1;1=2),thenthereisanequalityF=Xii;wherei:=hF;ii hi;ii:Moreover,kiik1=2kFk1=2foralli.3.10.3.Lemma.Ifiisanormalizedoverconvergenteigenform,thenkikr1:Proof.Wenotedpreviouslythatkikrkik0.Thelatterisgivenbytheq-expansionnorm,andthus(sinceiisnormalized)itfollowsthatkik0=1.ByLemma3.10.3,wededucethatjijkFk1=2 kik1=2kFk1=2:Inpractice,oneexpectskik1=2toincreaserelativelyquickly.However,thisesti-mateatleastallowsforanexplicitcomputationofi.Wereturntothenumerologyofeigenformsinsectionx5.3.11.Someheuristics.Letusnowreformulatethespectralconjectureinaslightlydierentwayinweight0.First,supposeweareworkingwithclassicalcuspformsinSk(�;C).Then,foracuspformF,onehasanidentity:F=Xhi;Fi hi;iii;wheretherighthandsideisanitesumovercuspidaleigenformsi,andh;iisthePeterssoninnerproduct,givenbyh; i=Z 
ykdxdy y2;whichsatisesh;i=L(1;ad0)foreigenforms.Ontheotherhand,weexpectthatforF2Sy0(�;r)(andrsucientlylarge),onehasanidentity:F=Xhi;Fi hi;iii;wheretherighthandsideisnowaninnitesumoverniteslopeeigenformsi,andh;iistheinvariantpairingdescribedabove.3.11.1.Exercise(?).Showthat,suitablynormalized,theinvariantpairingh;iforaniteslopeeigenformcoincideswiththep-adicL-functionLp(1;ad0)at=0,wheredenotestheColemanfamilyofeigenformsofweightpassingthrough.40 Herearesomethoughtsonthisexercise.Notethatthep-adicadjointL-functionisrelatedtotheramicationoftheColemanfamilyoverweightspace.Inparticular,Lp(1;ad0)shouldhavezerosexactlyattheramicationpoints(resultsofthis avourwereprovedbyKiminhisthesis[Kim06]).Ontheotherhand,assumingtheexistenceofspectralexpansionsoneexpectsthath;i=0whereU=ifandonlyifthereexistsageneralizedeigenform suchthat(U�) =(seethecalculationofx5.0.7foronedirection,andusetheq-expansionprincipleandthefactthath;iisnon-vanishingfortheotherdirection).Yetthenon-semi-simplicityofUisequivalenttotheeigencurvebeingramiedatthispoint.4.ExamplesInthissection,wegivesomeexplicitexamplesinordertoillustratethegeneraltheory.Writej=1 q+744+196884q+:::=Xc(n)qn:Werstshowhowtounderstandcongruencesforc(n)modulopowersoftwousingaclassicalmethod,andweshallreturnlateranduseamodernapproach,whichgivesmoreinformation.4.1.Anexample:N=1andp=2;theWatsonapproach.RecallthatX0(2)isuniformizedbythefunction:f=q1Yn=1(1+qn)24=q+24q2+:::andthatthereisanidentity(1+28f)3 f=j:WerstapplyU2toj,andwendthat:U2j=744+Xc(2n)qn:Formally,U2takesfunctionsonX0(1)toX0(2).ThusU2jisameromorphicfunc-tiononX0(2).Moreover,sinceU2j(E)isasumofj(E=C)forvariousC,thefunctionU2jwillbeholomorphiconX0(2)awayfromthecusps.SinceU2jisholo-morphicat1,itcanonlyhavepolesattheothercuspofX0(2),namelyatf=1,andhenceU2jisapolynomialinf.Indeed:U2j�744=140737488355328f4+3298534883328f3+19730006016f2+21493760f=25�262144g4+393216g3+150528g2+10495g
;whereg=26
f.Ontheotherhand,ifhisameromorphicfunctiononX0(2)thensoisU2h,andifhonlyhasapoleat0thensodoesU2h;thatis,U2takespolynomialsinftopolynomialsinf.Wesee:U2f=24f+2048f2;U2f2=f+1152f2+196608f3+8388608f4;andsoon.Moregenerally,U2fn=1 2f 2+f+1 2:41 HenceU2fnsatisesarecurrencerelationxn�a1xn�1+a2xn�2=0,whereX2�a1X+a2=X�f 2X�f+1 2=X2�(48f+4096f2)X�f:Theclassicalideaisnowtoexplicitlycomputethe\matrix"ofUonsomenicebasis.If,forexample,oneshowsthatthismatrixisdivisibleby8(inthiscase),theniteratingUwillestablishthenecessarilycongruences.4.1.1.Lemma.Leth=8f,andconsidertheringR=Z2[[h]]ofpowerseriesinhwithintegralcoecients.ThentheoperatorF:=U2=8actscontinuouslyonh
R.Proof.ContinuityisequivalenttoaskingthatF(hn)2R,andthedegreeofU2hngoestoinnitywithn.Bothclaimsfollowbyinduction.Fromthecomputationsabove,weseethat:F(h)=3h+32h2;F(h2)=h+144h2+3072h3+16384h4;andthenF(hn)=16(3h+32h2)F(hn�1)+8hF(hn�2).SinceU2j�744228h
R,itfollowsthat1Xn=1c(2mn)qn=Um2j�744=(8F)m�1(U2j�744)(8F)m�1(28h
R)23m+5F(h
R)23m+5h
R23m+8Z2[[q]]:ThisprovesLehmer'scongruenceintheintroduction.4.2.Anexample:N=1andp=2;theColemanapproach.ThefunctionjdenesameromorphicfunctiononXrig[r]withapoleonlyat1,andhenceUpjextendstoanelementofMy0(�;r)foranyrp p+1:TheoperatorUponthisspaceiscompact.Nowletp=2.WemaymanuallycomputetherstfewslopesofthespectrumofU2tobe0,3,7,and13.Ofcourse,1isaneigenvalueforU2withslopezero.Inparticular,foranyoverconvergentformginMy0(�;r)withnoconstanttermwehave,fromtheasymptoticexpansionLemma3.5.2),thatg11+22+33+:::andthus:Um2(U2j�744)=1m11+2m22+o(213m):Howmayonecomparethesearguments?TheWatsonstyleargumentessentiallyprovesbyhandthatU2iscompact,andindeedthatthenormofF=U2=8onthecuspidaloverconvergentformsis1.Thisjustiestheclaimintheintroduction|ColemangivesyouthecompactnessofUbygeometry,whereasWatsongivesittoyoubyexplicitcomputation,butbyacomputationwhichneedstoberedoneeverysingletimetogetthebestbounds.Moreover,suchcomputationsbecomeessentiallyinfeasibleassoonasX0(p)hasgenus�0.Ontheotherhand,thefactthatFhasoperatorbound1isisstrongerthanthefactthatthersteigenvaluehasslope3,evenifitdoesn'tsayanythingaboutthehigherordereigenvalues.Howmaywereconcilethesetwoapproaches?42 acontradiction.Inparticular,theeigenfunctionofweightzerowithlargestslopewhichisnotordinaryhasslopeatleast1.Theresultthenfollowsimmediatelyfromtheasymptoticexpansion.4.3.2.Remark.Notethatexactlythesameargument|andconclusion|appliestoanyoverconvergentp-adicmodularformofweight0andlevel1.4.3.3.Exercise.Letgbeanyp-adicoverconvergentmodularfunctionwhichiscongruentto1modp,forexample,g=1+p
(p)
().Provethatgs:=exp(slog(g))=1+s(g�1)+s2(g�1)2+s3(g�1)3+:::isalsooverconvergentforsucientlysmalls2Cp.Computewhatsucientlysmallmeansexplicitlyinthiscase.Anaturalquestionthatpresentsitselfisasfollows:Canoneeectivelycomputetheconstantc?Supposeoneassumedtheexistenceofaconvergentspectralexpan-sion,togetherwiththeestimatekFkr=supjij
kikrforr=1=2.Aswithp=2,wewouldthenhave:Upj=Upej+11+22+:::anditwouldsucetoobtaineectiveanduniformboundsfori.Yetthereareobviousboundskikr1andkUpjkrpforallr,andthusjijp.Asweshallseelater,itismostlikelythecasethatthenormskikrgrowextremelyrapidly(exponentiallyini)andthustheidecreasetozeroinaconcomitantfashion.4.3.4.Exercise(?).WhatistheoptimalupperboundforkUpjkrforgeneralp?WhatabouttheoptimalupperboundfortheoperatornormkUpkr?4.4.Anexample:convergenceslowerthanO(pn).Theredoexistformsofslopestrictlybetween0and1,whichmayeecttherateofconvergence.Togiveaneasyexample,letf=Xa(n)qn=qYn=1(1�qn)2(1�q11n)22S2(�0(11);Z);whichcorrespondstothemodularcurveX0(11).Thisellipticcurvehassuper-singularreductionat2,andtheminimalpolynomialofcrystallineFrobeniusisx2+2x�2.Itfollowsthatthecorrespondingoldformsf,fwith;=�1p 3oflevel�0(22)eachhaveslope1=2.Inparticular,itisnottoohardtoshowthat:Xa(2mn)qn0mod2dm 2e;butthatthereisnosuchcongruencemoduloanyhigherpowerof2.Ofcourse,thesamethinghappens(withthesameform)foranyoftheinnitelymanyprimespsuchthatX0(11)issupersingular.4.4.1.Exercise.ShowthatifE4
E58=1+240X3(n)qnq1Yn=1(1�qn)24 1�1416 2913228046513104891794716413587449X58(n)qn=:Xd(n)qn;thenXd(59mn)qnconvergestozeronofasterthanO(59m=2).44 4.8.Anexample:congruencesforthepartitionfunctionmodulopowersof5,followingWatson.SupposeinsteadofusingColeman'stheory,onewantedtoprovethecongruenceabovedirectly,evenjusttheconsiderablyweakerclassicalcongruences.ThenonehastoexplicitlydetermineenoughabouttheoperatorU=U25toshowthat(forexample)itisdivisibleby25.Asinx4.1,oneneedstoworkexplicitlywithmodularequations.Forexample,letf()=254s
(5)
()=25(5)6 ()6=25q1Yn=1(1�q5n)6 (1�qn)6;4.8.1.Lemma.Thefollowingidentityholds:4Ym=0X�f+m 5=X5�a1X4�a2X3�a3X2�a4X�a5;wherea1=52f(63+260f+315f2+150f3+25f4)a2=54f(52+63f+30f2+5f3)a3=55f(63+30f+5f2)a4=57f(6+f)a5=58fProof.Theproofisroutine.4.8.2.Lemma.Foranon-negativeintegern,letAn=U5fn�1() �1(5)andBn=U5fn�1(5) �1().ThenAnandBnarepolynomialsinfwhichsatisfytherecurrencerelationXn=a1Xn�1+a2Xn�2+:::+a5Xn�5:Moreover,forsmallvaluesofn,wehavethefollowingequalities:A0=f 5A1=5f(28+245f+525f2+455f3+175f4+25f5)A2=52f(104+9100f+113880f2+528125f3+1232725f4+1660750f5+1376375f6+715000f7+227500f8+40625f9+3125f10)A3=53f(19+13889f+672885f2+9791080f3+66083900f4+252487675f5+608947625f6+988926250f7+1124158750f8+913721875f9+534909375f10+224081250f11+65609375f12+12765625f13+1484375f14+78125f15)A4=56f(1+8375f+1375975f2+52547625f3+831122125f4+7023871875f5+36454450625f6+126528231250f7+310499593750f8+559393046875f9+759056634375f10+788952734375f11+634365468750f12+396053515625f13+191527734375f14+71064453125f15+19855468750f16+4042968750f17+566406250f18+48828125f19+1953125f20)48 B0=1B1=5f(63+260f+315f2+150f3+25f4)B2=53f(104+4095f+32820f2+107300f3+182700f4+180375f5+107500f6+38250f7+7500f8+625f9)B3=54f(189+49230f+1512585f2+15998850f3+83171925f4+251923750f5+488490750f6+640687500f7+586327500f8+379518750f9+173362500f10+54750000f11+11390625f12+1406250f13+78125f14)B4=56f(24+42920f+4266360f2+118018875f3+1455608800f4+9969720300f5+42885018000f6+125026746500f7+259678080000f8+397294462500f9+457754050000f10+402607546875f11+272038500000f12+141147812500f13+55788750000f14+16505156250f15+3540625000f16+520312500f17+46875000f18+1953125f191Xn=0AnTn=q=5+5(28f+182f2+265f3+140f4+25f5)T�52(�104f�20f2+10f3)T2�54(�19f+6f2+5f3)T3+56fT4 1�a1T�a2T2�a3T3�a4T4�a5T51Xn=0BnTn=1�20f(63+260f+315f2+150f3+25f4)T�53f(156+189f+90f2+15f3)T2�54f(126+60f+10f2)T356f(6+f)T4 1�a1T�a2T2�a3T3�a4T4�a5T5Proof.Thesefollowfromthestandardmethods.Notethatsomeoftheseidentitiesarequiteclassical,forexample,A1=f=5isjusttheidentityXp(5n+4)qn=5(1�q5)5(1�q10)5(1�q15)5


(1�q)6(1�q2)6(1�q2)6


Manyoftheseequationsare(inslightlydisguisedform)in[Wat38].Thesere-currencesgiveenoughinformationtoprove(asinx4.1)that,withrespecttosomesuitablebasis,thatU25=52FforsomesuitablecontinuousoperatorF,whichallowsonetoprovethedesiredcongruences,whichiswhatWatsondoes.However,itdoesnotseemobvioushowonecanusethisapproachtounderstandthesecondeigenvalue(andeigenvector)ofU25.WhatoneneedstoshowisthattheactiononU25onsomenaturalspaceisdivisibleby57|yetthisisonlypossibleifonecansomehowprojectawayfromtheeigenform1ofslope2.UnliketheEisensteinseries|whichinweight0isjusttheconstant1|thereisnoobviouswaytoaccountforthein uenceof1whentryingtoestimatetheerrorterm.5.p-adicarithmeticquantumchaos(See[Sar95].)FixamodularcurveX=X(�).DonBlasiussuggestedtomethepossibilitythattherecouldbeanusefulanalogybetweenthediscretespectrumofthehyperbolicLaplacian
=�y2@2 @x2+@2 @y2andtheoperatorUpinweightzero.Inthissection,wediscussvariousconjecturesrelatingtothisquestion.Onenaturalquestionishowtocountsucheigenvalues.Theybothforminnitecountablesets|intheArchimedeancasebyanon-trivialresultofSelberg,andinthep-adiccasebyColeman|itsucestonotethatthecoecientsoftheFredholmdeterminantareallnon-zero,whichfollowsbyatheoremofKoike[Koi75,Koi76].49 Thenaturalwaytocounteigenformsisthusbyboundingthesizeoftheeigenvalues.Intheclassicalcase,theclassicalresultisthefollowing:5.0.3.Theorem(Weyl'sLaw).LetN(T):=#fjv()Tgdenotethecountingfunctionfordiscreteeigenvaluesof
.ThenN(T)Vol(X) 4
T:ThemodularcurvesinheritfromHanaturalmetricofconstantcurvature-1.Recallthat,withrespecttothismetric,Vol(X)=[�0(1):�]
 3:Somewhatbettererrorboundsareknown,butthey,too,arenon-trivial.NotethatvolumeistakenwithrespecttothenaturalmeasureonXwhichmakesitamanifoldofconstantcurvature�1.Whenitcomestocountingniteslopeeigen-forms,thenaturalmeasureofsizeofeigenvaluesisbytheirvaluation.SincetheoperatorUpdependsonthechoiceofasubgroupoforderp,itismorenaturaltoworkwithX0(p)ratherthanX(attheleveloftheappropriaterigidanalyticspacesXrig[r]thereisnodierenceduetotheexistenceofthecanonicalsubgroup).5.0.4.Conjecture(p-adicWeyl'sLaw).LetN(T):=#fjv()TgdenotethecountingfunctionforeigenvaluesofUp.ThenN(T)Vol(X0(p)) 4
T:Howdoesonecountsucheigenvalues?Theslopesoftheeigenvaluesarede-termined(viaNewton'sLemma)tothevaluationsoftheFredholmpowerseriesof1�TU.Wehavethefollowingpartialresult,whichprovesonedirectionofConjecture5.0.4:5.0.5.Theorem.Thereisaninequality:N(T)Vol(X0(p)) 4
T+o(T):ThisfollowsfromtheestimatesofWan[Wan98](inparticular,itfollowsviaaneasycomputationfromLemma3.1ofibid.)Onealsoobtainsfromthisabound:5.0.6.Theorem.Thereisaninequality:N(T)Vol(X0(p)) 4
T;wheretheimplicitconstantdependsonX.Anothernaturalproblemtobeconcernedwithisthebehavioroftheeigenfunc-tionsU=asfunctionsas!1.Forexample:(1)HowfastdoestheL1-normgrowwith?Obviouslythisdependsonsomenormalizationoftheeigenfunctions.SincetheeigenfunctionsareL2,anaturalnormalizationistheL2-norm,i.e.,insistingthatkk22=jh;ij=1:50 Wexaradiusofconvergencer,andletk
kdenotethesupremumnorm.Inthiscase,wehave:(1)Weviewkk=kkrasasubstitutefortheL1-norm(itisasupremumnorm).(a)Thereisnoknownupperboundforkk.(b)Thereisatriviallowerboundkk1,ifwenormalizebyusingq-expansions,butnoknownnon-trivialbounds.(c)Ifr1=2,thereisatriviallowerboundkk1,ifwenormalizebysettingjh;ij=1:(2)Regardingthefunctionforlarge,therearetwonaturalquestionsonecouldaskdependingonthenormalization.(a)Ontheordinarylocus,thefunctionscanbethoughtofaselementsoftheuniversaldeformationringofanitenumberofresidualrepre-sentations.Nothingisknownaboutthedistributionofthesepoints.(b)Onthesupersingularlocus,aresultofBuzzard[Buz03]impliesthateigenformsextendtosectionsofXrig[r]forallr1,andthattheycannotbeextendedbeyondthis(sokkr!1asr!1),butnothingisknownconcerningwhatthesefunctionslooklike.5.0.10.Lemma(Hadamardthree-circletheorem).SupposethatkFka=pAandkFkb=pBforrational0abp=(p+1).Thenthereisaninequality:logpkFkrA+(B�A)(r�A) (b�a)forallarbinQ.IfFhasnozeroesonthecorrespondingannulus,thenequalityholds.Proof.BothAandBarerational.SincenormofanintegralpowerofFisthecorrespondingpowerofthenorm,afterreplacingFbyapowerofitselfandmul-tiplyingtheresultbyapowerofp,wemayassumethatA=maandB=mbforsomem2Z.Nowconsiderthefunction:G=F
t�m:Byconstruction,thenormofGontheannulusjtj=jpajis1,andthenormontheannulusjtj=jpbjisalso1.IfFhasnozeroes,thesameargumentappliestoF�1.OneconsequenceisthattheminimumvalueofkFkskFk1�sfors2(1�r;r)andF2My0(�;r)withr�1=2occursfors=1=2.5.1.Anexplicitexample:N=1andp=2.WhenN=1andp=2,somemileage7maybeobtainedfromthefactthatX0(2)hasgenuszero,aswellasthefactthatU2hassuchanexplicitformonMy0(1;r),namelybyidentifyingthelatterwiththeTatealgebraMy0(1;r)'C2h2r
fi;f=q1Yn=1(1+qn)24: 7orkilometrage,ifyouprefer.52 Withrespecttothenaturalbasisintermsofpowersof2r
f,onehasU2=[sij],wheresij=3i(i+j�1)!22i+2j�1 (2i�j)!(2j�i)!
2(6�12r)(i�j):Whenr=1=2thisisparticularlysymmetric.Wewill(mostly)beconcernedwiththisvalueofr,althoughnotexclusively.Notethatonehasthefollowingrelationshipbetweenthevaluationoffandtheannulijtj=j2rj,whichcanbededucedinasimilarmannertothecomputationsinx3.2.2:5.1.1.Lemma.Supposethat0r1.Then,ontheannulijtj=j2rj,onehasjfj=kfkr=212r.Forconvenience,however,wemakethefollowingdenitions.5.1.2.Denition.Letk
kdenotethenormk
k1=2,andletg=26f.Notethatkgk=1.Letusdenotetheeigenvectorsbynforpositiveintegersn.Akeyresultof[Buz03]impliesthateigenvectorscanbeanalyticallycontinuedtobesectionsofXrig[r]foranyr1.Onehasanexactformulafortheslopesoftheeigenvalues[BC05],andoneknowsthespectralconjecture[Loe07].Forexample,theslopeofthentheigenvaluenoftheeigenfunctionnisv(n)=1+2v(3n)! (n)!:5.1.3.Exercise.Usingtheexplicitformulaefortheslopes,provethe2-adicWeyl'slawforN=1,namely,thatN(T)Vol(X0(2)) 4
T+O(log(T)):RecallthematricesA=[aij]andB=[bij]andthediagonalmatrix[Dii]from[BC05],where:aij=2(j�i)(12�6r)6ij(2j)! 2jj!22ii! (2i)!2(2i�1)! (i+j)!(2j+i�1)! (3j)!ji�j;bij=j iaji;dii=24i+1(3i)!2i!2 3(2i)!4:OnehasafactorizationU=ADB.Whenr=1=2,thematricesAandBlieinZ2andarecongruenttotheidentitymodulo2.Asnotedin[Loe07],thesameholdsforallrintherange56r7.Inparticular,ifwelet=2(6r�12),thentheproofofthemaintheoremofLoeer[Loe07]impliesthefollowing:5.1.4.Lemma.Then-theigenfunctionn,consideredasanelementoftheTatealgebraC2hgi,admits,afternormalization,anexpansion:n=Xai(g)iwherev(ai)&#x]TJ/;ø 9;&#x.962; Tf;&#x 20.;ł ;� Td;&#x [00;0foralli6=n,andv(an)=0.Wederivesomeimmediatecorollariesfromthis.53 (4)Ifk
k0denotestheq-expansionnorm,thenkk2 jh;ij
jj=O(logv()):(5)As!0,arethefunctionsaredistributedinsomenaturalway?Asaspecialcase,isittruethatthe(normalized)sequencekk2onMy0(�;1=2)convergestofunctionswhichareconstantonsupersingularannulijtj=1=2,atleastifonerestrictstosubsequencesforwhichtheresidualrepresentation isconstant?(Or,perhaps,toaconnectedcom-ponentoftheeigencurve?)DothesumofdeltameasuresonXrig[1=2]supportedonthezerosofconvergetoanyexplicitmeasureonthecorre-spondingBerkovichspace(alreadyaninterestingcomputationforN=1andp=2)?Tomakeanevenwilderguess,letSdenotethezerosetofintheregionXrig[1=2].Weexpect(andknowforN=1andp=2)thatjSjgrowslinearlywithrespecttothenaturalorderingoftheeigenvalues.ItmayalsobethecasethatSiscompletelycontainedwithinXrig[r],wherer=1=(p+1).Considerthemeasures:1 jSjXx2Sx:IsitthecasethatthesemeasuresoncomplexvaluedcontinuousfunctionsonXrig[r]havealimitingmeasureontheBerkovichspaceassociatedtotheanoidXrig[r](forr=1=(p+1))?Ifso,doesitconvergetotheGausspointcorrespondingtothesupremumnormontheentirespace?Forexample,isitthecasethatwhenN=1andp=2,andforapolynomialF2C2[[g]],onehaslim!01 jSjXx2SjF(x)j=kF(x)kr;wherer=1=(p+1).Forexample,ifF=g,thisisequivalenttosayingthatalmostallofthezerosof(inXrig[1=2]),theHasseinvarianthasvaluationatleastr�foranyxed�0andr=1=(p+1).(6)As!0,theGaloisrepresentationsaredistributedonthecorrespond-ingglobaldeformationringsSpec(R )withrespecttoanaturalmeasure.NotethatwhenN=1andp=2alltheeigenfunctionshavecoecientsinZ2;Ingeneral,Buzzardraisesthequestion[Buz05]ofwhetherforanyNandpallniteslopeeigenformsinanyparticularweightaredenedoveraxedextensionK=Qp.Hence,bymeasure,weareconsideringsubsetsofthecompactp-adicmanifoldHom(R ;OK)ratherthansomemeasureontheBerkovichspaceassociatedtherigidanalyticspacecorrespondingtoR .AssumingaverystrongversionoftheGouv^ea{Mazurconjecture,onecanrephrasepart6ofthisguessasfollows(andequallyvaguely):5.2.2.Guess.Considertheclassicalmodulareigenformsofweightpk�1(p�1)over Qp.Then,ask!1,theGaloisrepresentationsmodulopkaredistributedonthecorrespondingglobaldeformationringswithrespecttoanaturalmeasure.56 Ontheotherhand,theclaimsconcerningthedistributionofontheannuliareclosetomeaninglesswithoutsomepossiblecandidatedistribution.6.StudentprojectsTherearevariousprojects,dependingontheinclinationofthestudent|somearemoretheoreticalandsomearemorecomputational.(Ofcourse,thecomputa-tionsshouldhelpwiththetheoreticalmusings.)6.1.TurnGuess5.2.1intoaconjecture.oratleastaquestion.Thisrequires:6.1.1.Moredata.SupposethatN=1andp=2.Here'sapracticalwayofcom-putingeigenformsofhighslope.Chooseanarbitrarycuto,sayn=100;letM=[sij]fori;j100denotethecorrespondingmatrix.Computethecharacter-isticpolynomialX100+:::ofM.TherootsofthispolynomialalllieinZ2,sotheyareeasytocomputetohigh2-adicaccuracy.Let0denotearootofthispolyno-mialtohigh2-adicaccuracy.Letdenotethecorrespondinggenuineeigenvalue.Chooseaarandomvectoru2Z1002,andletv=(M�0)�mu:forsomelargishintegerm.Thenvshouldbeagoodapproximationtothegenuineeigenvectorassociatedto.Forexample,theeighteigenvalue8hasvaluation31,and,usingHensel'sLemma,wecomputethat8=1802090304606119228112737467361460811598903762601218215405738446438703552331427086814610754371584+O(2321)Let0denotethisnumber,andletMbethe5050matrix[sij]withi;j=1;:::;50.Letv=(M�0)�100(1;0;0;0;:::;0):(Alargerexponentwouldprobablygiveamoreaccurateapproximation,butIdidn'tdothisinaverycleverwaysoeventhiscomputationwasalittleslow.)Letwdenotethescalarmultipleofvnormalizedsothattherstentryis2�6.Anapproximationto8shouldthenbegivenby850Xi=1vigi:Wecomputethevaluationsofthecoecientsvitobeasfollows:[�6;�9;�9;�14;�13;�16;�16;�23;�16;�16;�13;�14;�9;�9;�6;�7;4;6;11;12;19:::]Totestthisasanapproximationto8,notethatthesquareofthenormk8k2appearstobeequalto246=29
82
223,aspredictedbyGuess5.1.10.Wealsocomputetherstfewtermsoftheq-expansion(omitted,becausetheyareratiosof500000digitnumbers,althoughitisridiculoustocomputetheminthisman-ner,sinceoneshouldworkmodulosomepowerof2|hopefullysomeofyoucanprogrammebetterthanIcan):8'q+a(2)q2+a(3)q3+a(4)q4+a(5)q5+a(6)q6+:::wendthatv(a(2))=31=v(8),andwealsocheckthata(3)a(5)a(15)mod2115;57 [Ram08] ,Thehalf-integralweighteigencurve,AlgebraNumberTheory2(2008),no.7,755{808,WithanappendixbyBrianConrad.MR2460694(2010a:11077)[Ram09] ,TheoverconvergentShimuralifting,Int.Math.Res.Not.IMRN(2009),no.2,193{220.MR2482114(2010i:11060)[Sar95]PeterSarnak,Arithmeticquantumchaos,TheSchurlectures(1992)(TelAviv),IsraelMath.Conf.Proc.,vol.8,Bar-IlanUniv.,RamatGan,1995,pp.183{236.MR1321639(96d:11059)[Ser62]Jean-PierreSerre,EndomorphismescompletementcontinusdesespacesdeBanachp-adiques,Inst.HautesEtudesSci.Publ.Math.(1962),no.12,69{85.MR0144186(26#1733)[Ser73a] ,Congruencesetformesmodulaires[d'apresH.P.F.Swinnerton-Dyer],SeminaireBourbaki,24eannee(1971/1972),Exp.No.416,Springer,Berlin,1973,pp.319{338.LectureNotesinMath.,Vol.317.MR0466020(57#5904a)[Ser73b] ,Formesmodulairesetfonctionsz^etap-adiques,Modularfunctionsofonevariable,III(Proc.Internat.SummerSchool,Univ.Antwerp,1972),Springer,Berlin,1973,pp.191{268.LectureNotesinMath.,Vol.350.MR0404145(53#7949a)[Sil86]JosephH.Silverman,Thearithmeticofellipticcurves,GraduateTextsinMathemat-ics,vol.106,Springer-Verlag,NewYork,1986.MR817210(87g:11070)[Sil94] ,Advancedtopicsinthearithmeticofellipticcurves,GraduateTextsinMath-ematics,vol.151,Springer-Verlag,NewYork,1994.MR1312368(96b:11074)[SS89]A.SeegerandC.D.Sogge,Boundsforeigenfunctionsofdierentialoperators,IndianaUniv.Math.J.38(1989),no.3,669{682.MR1017329(91f:58097)[TO70]JohnTateandFransOort,Groupschemesofprimeorder,Ann.Sci.EcoleNorm.Sup.(4)3(1970),1{21.MR0265368(42#278)[Wan98]DaqingWan,Dimensionvariationofclassicalandp-adicmodularforms,Invent.Math.133(1998),no.2,449{463.MR1632794(99d:11039)[Wat38]G.N.Watson,Ramanujansvermutunguberzerfallungsanzahlen,J.ReineAngew.Math.179(1938),97{128.62