On the theory of Maass wave forms PDF
Preview On the theory of Maass wave forms
Universitext Tobias Mühlenbruch Wissam Raji On the Theory of Maass Wave Forms Universitext Universitext SeriesEditors SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA CarlesCasacuberta UniversitatdeBarcelona,Barcelona,Spain JohnGreenlees UniversityofWarwick,Coventry,UK AngusMacIntyre QueenMaryUniversityofLondon,London,UK KennethRibet UniversityofCalifornia,Berkeley,CA,USA ClaudeSabbah ÉcolePolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau,France EndreSüli UniversityofOxford,Oxford,UK WojborA.Woyczyn´ski CaseWesternReserveUniversity,Cleveland,OH,USA Universitext is a series of textbooks that presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teachingcurricula,intoverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Moreinformationaboutthisseriesathttp://www.springer.com/series/223 ¨ Tobias Muhlenbruch (cid:129) Wissam Raji On the Theory of Maass Wave Forms TobiasMu¨hlenbruch WissamRaji DepartmentofMathematics DepartmentofMathematics andComputerScience AmericanUniversityofBeirut FernUniversita¨tinHagen Beirut,Lebanon Hagen,Germany ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-3-030-40477-2 ISBN978-3-030-40475-8 (eBook) https://doi.org/10.1007/978-3-030-40475-8 MathematicsSubjectClassification(2010):11F11,11F12,11F99 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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 “WhywriteatextbookonMaasswaveforms?”Thisquestionwastheusualreaction to our plan for writing this textbook. There are several textbooks on Maass wave formsalreadyavailable.Whatdistinguishesourtextbookfromothers? WebelievethatmostoftheavailabletextbooksonMaasswaveformsandrelated objects fall into two categories. Textbooks in the first category deal with classical modularforms(whatwemeanby“classical”isexplainedinRemark1.63)andtheir arithmeticproperties.Thatis,thefocusinthesebooksisontheconnectionbetween holomorphicmodularformsandrelatedobjectsinarithmeticgeometry(e.g.,elliptic curvesandGaloisrepresentations).Somegoodexamplesoftextbooksherearethose ofApostol[12],Serre[182],Shimura[184],Lang[116],Schoenberg[176],Koblitz [109],andDiamondandShurman[66].Booksinthiscategory,byandlarge,donot tacklethesubjectofMaasswaveforms,butsomedobrieflydiscussit. ThesecondcategoryoftextbooksconsistsofthosethatdiscussGL (·)orSL (·) 2 2 and their associated Lie algebras. The main focus of books here is usually on the representations of Lie algebras, where holomorphic modular forms often appear. Maass wave forms are natural counterparts to those representations. Examples of textbooks in this category are those of Borel [17], Bump [52], Iwaniec [101], and Kubota[111]. Our textbook takes a different approach. We focus on Maass wave forms as the main objects of interest and strive to give Maass wave forms a firm analytical treatment, only briefly pointing out connections with other fields in mathematics likerepresentationtheory. Maass wave forms are introduced as functions on the complex upper half- planethatareeigenfunctionsofthehyperbolicLaplaceoperator,haveaprescribed behavior under transformations by elements of the full modular group SL (Z), 2 andadmitanexpansionofacertainformthatcontainsexponentialsandWhittaker functions.InsteadoffirstplacingMaasswaveformsinthecontextofrepresentation theory,ideasfromgeometricanalysisandthetheoryofspecialfunctionsareusedto obtainresults. v vi Preface The textbook is structured as follows. We start with an informal exploratory IntroductoryRoadmapthatservesasaguidetothetopicspresented. Chapters1and2giveabriefintroductiontothetheoryofclassicalmodularforms and related objects. The goal of these chapters is to briefly discuss modular forms andtointroducethenecessaryrelatedobjectsandresults(e.g.,L-seriesandperiod polynomials)thatareneededinlaterchapters.Theselectionofthepresentedresults isbiasedtosuitourneeds,andwealsoomitseveralproofsthatcanbeeasilyfound in the references mentioned above. In many cases, we skip proofs in this chapter. However, we prove the analogous results for the Maass wave forms case, which usuallycomewithgreatergenerality. Chapter3introducesMaasswaveformsofrealweight.Afterdefiningmultiplier systems, we discuss three differential operators: the hyperbolic Laplace operator andthetwoMaassoperators.Then,weintroduceMaasswaveformsofrealweight and give some examples. We conclude this chapter by discussing Hilbert spaces of Maass forms, Hecke operators, and the Friedrichs extension of the hyperbolic Laplaceoperator,endingwithafewremarksontheassociatedSelberg’sconjecture. Chapters4and5arethemainchaptersofthistextbook.Weintroducetheconcept offamiliesofMaasswaveformsanddiscusstheassociatedL-seriesinChapter4. Chapter 5 introduces period functions associated with (families of) Maass wave forms.Theaboveobjectsarepresentedinanalogywithclassicalmodularformsand theirassociatedL-seriesandperiodpolynomials. Chapter 6 connects Maass cusp forms (of weight 0) with discrete dynamical systemsassociatedwiththeArtinbilliard. The last chapter, Chapter 7, introduces weak harmonic Maass wave forms and their associated objects like mock modular period functions and regularized L- series. Readers of this textbook are required to have basic knowledge in complex analysis,realanalysis,andabstractalgebra.Althoughelementaryconceptsarebeing introducedineverychapter,somespecialknowledgeofcertaintopicsinanalysisis assumed.ReadersmayrefertoAppendixA,wheresometheoremsandresultsare brieflypresentedwithhelpfulreferencesindicatedincasemoredetailsaredesired. Some exercises are included. Detailed solutions to those exercises are also available online.Thesesolutionscanbefoundatourpersonalhomepages andour ResearchGatepages. Acknowledgements We would like to express our sincere appreciation to all who have participated in improving this textbook. We are particularly grateful to our colleagues who encouragedustopursuethisproject.Wethankourreviewersforgivingusnumerous suggestions on how to improve the manuscript and for the encouraging and insightfulcomments.WealsosayabigthankyoutoKenOnoforhissupport. Preface vii AnotherbigthankyougoestoJeffBreeding-Allison,LloydKilford,LarryRolen, and M. Shane Tutwiler who assisted in proofreading and in the linguistic editing. Wealsothankourfamiliesfortheirunderstandingandsupport. Hagen,Germany TobiasMühlenbruch Beirut,Lebanon WissamRaji May2020 Contents 1 ABriefIntroductiontoModularForms .................................. 1 1.1 NotationsandSomeSimpleResults ................................. 1 1.2 TheFullModularGroupandItsSubgroups......................... 6 1.2.1 TheFullModularGroup ..................................... 6 1.2.2 FundamentalDomainsfortheFullModularGroup......... 8 1.2.3 FundamentalDomainsforSubgroupsof(cid:2)(1).............. 9 1.2.4 CuspidalPoints ............................................... 11 1.3 ModularForms–PropertiesandExamples.......................... 15 1.3.1 Introduction–ModularForms............................... 15 1.3.2 MultiplierSystems............................................ 17 1.3.3 ModularForms................................................ 21 1.3.4 EisensteinSeries.............................................. 26 1.3.5 TheDiscriminantFunction(cid:3)(z)............................. 28 1.3.6 Petersson Scalar Product and the Hilbert Space ofCuspForms ................................................ 30 1.4 HolomorphicPoincaréSeries......................................... 32 1.4.1 FurtherRemarksonModularForms......................... 37 1.5 ThetaSeries ........................................................... 38 1.5.1 JacobiThetaFunction ........................................ 38 1.5.2 ThetaSeriesofIndexm ...................................... 52 1.5.3 ThetaSeriesandSumsofSquares........................... 53 1.6 HeckeOperators ...................................................... 67 1.6.1 TheSlashOperator ........................................... 67 1.6.2 HeckeOperatorsofIndexn∈N............................. 69 1.6.3 Eigenforms.................................................... 72 1.7 L-FunctionsAssociatedwithCuspForms........................... 74 1.7.1 TheMellinTransform........................................ 74 1.7.2 L-SeriesAssociatedwithCuspFormson(cid:2)(1)............. 78 1.7.3 L-FunctionsofHeckeEigenforms........................... 86 1.8 ConcludingRemarks.................................................. 92 ix x Contents 2 PeriodPolynomials .......................................................... 93 2.1 VariousWaystoPeriodPolynomials................................. 93 2.1.1 PeriodPolynomialsviaIntegralTransformations........... 94 2.1.2 EichlerIntegralsandPeriodPolynomials ................... 98 2.1.3 The(k−1)-FoldAntiderivative............................. 102 2.1.4 AnIntegralRepresentationoftheEichlerIntegral forCuspForms ............................................... 104 2.1.5 AnIntegralRepresentationoftheEichlerIntegral forEntireForms .............................................. 107 2.1.6 TheNieburIntegralRepresentationforEntireForms ...... 110 2.2 TheEichlerCohomologyGroup ..................................... 112 2.3 TheEichlerCohomologyTheorem .................................. 118 2.3.1 InjectivityoftheMapμ...................................... 120 2.3.2 SurjectivityoftheMapμ..................................... 124 2.4 HeckeOperatorsandPeriodPolynomials ........................... 133 2.5 ConcludingRemarks.................................................. 147 3 MaassWaveFormsofRealWeight........................................ 149 3.1 MultiplierSystemsandUnitaryAutomorphicFactors.............. 149 3.2 TheDifferentialOperators(cid:3) andE ............................... 152 k k 3.2.1 WhittakerFunctions.......................................... 159 3.2.2 Whittaker-FourierExpansionson(cid:2)(1)...................... 162 3.2.3 Whittaker-FourierExpansionsfor(cid:2) ⊂(cid:2)(1)................ 164 ± 3.2.4 MaassOperatorsE andtheFourier-Whittaker k Expansion..................................................... 169 3.3 MaassWaveFormsofRealWeight.................................. 172 3.4 ExamplesofMaassWaveForms..................................... 180 3.4.1 TheEmbeddingofClassicalModularForms ............... 180 3.4.2 Non-holomorphic Eisenstein Series for the Full ModularGroup................................................ 186 3.4.3 MaassCuspFormsDerivedfromtheDedekind η-Function..................................................... 189 3.5 ScalarProductandtheHilbertSpaceofCuspForms............... 191 3.5.1 DefinitionofthePeterssonScalarProduct .................. 191 3.5.2 PropertiesofthePeterssonScalarProduct .................. 192 3.5.3 ThePeterssonScalarProductandMaassCuspForms...... 196 3.6 HeckeOperatorsonMaassCuspForms............................. 199 3.7 TheLaplaceOperatorandItsSelf-AdjointFriedrichs Extension.............................................................. 203 3.7.1 TheFriedrichsExtensionof(cid:3) fortheFullModular 0 Group.......................................................... 204 3.7.2 TheFriedrichsExtensionof(cid:3) .............................. 207 k 3.8 Selberg’sConjecture.................................................. 212 3.9 ConcludingRemarks.................................................. 213
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