Number Theory, Analysis and Geometry: In Memory of Serge Lang PDF
Preview Number Theory, Analysis and Geometry: In Memory of Serge Lang
Number Theory, Analysis and Geometry Dorian Goldfeld • Jay Jorgenson (cid:129) Peter Jones Dinakar Ramakrishnan (cid:129) Kenneth A. Ribet John Tate Editors Number Theory, Analysis and Geometry In Memory of Serge Lang 123 Editors DorianGoldfeld JayJorgenson DepartmentofMathematics DepartmentofMathematics ColumbiaUniversity CityUniversityofNewYork NewYork,NY10027 NewYork,NY10031 USA USA [email protected] [email protected] PeterJones DinakarRamakrishnan DepartmentofMathematics DepartmentofMathematics YaleUniversity CaliforniaInstituteofTechnology NewHaven,CT06520 Pasadena,CA91125 USA USA [email protected] [email protected] KennethA.Ribet JohnTate DepartmentofMathematics DepartmentofMathematics UniversityofCaliforniaatBerkeley HarvardUniversity Berkeley,CA94720 Cambridge,MA02138 USA USA [email protected] [email protected] ISBN978-1-4614-1259-5 e-ISBN978-1-4614-1260-1 DOI10.1007/978-1-4614-1260-1 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2011941121 ©SpringerScience+BusinessMedia,LLC2012 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) SergeLang(PhotoprovidedcourtesyofKennethA.Ribet.) Preface SergeLangwasaniconicfigureinmathematics,bothforhisownimportantwork and, perhaps even more crucially, for the indelible impact he left on the field, and on his students and colleagues. It would be difficult to find a mathematician who came of age in the past forty years, who had not been exposed to Serge’s articles, monographs, and textbooks. Serge’s writing shaped the mathematical perspectives of all who came in contact with them. Many were challenged by the glimpses of open problems and conjectures that Serge interweaved with his expositions of established subjects. Serge’s exposition invariably transcended our discipline’spreferenceforbrevityandperfection,whichoftenobscurestheintuition underlyingthesubject.Sergewasneveronetoconform. OneofSerge’supliftingqualitieswashisopennesstonewareasofmathematics and his concurrent willingness, even eagerness, to learn novel concepts and techniques.He was constantlyreinventinghimself, while sharinghis accumulated wisdom with students and young mathematicians. Over the course of his career, he traversed a tremendous amount of mathematical ground. As he moved from subjecttosubject,hefoundanalogiesthatledtoimportantquestionsinsuchareas asnumbertheory,arithmeticgeometry,andthetheoryofnegativelycurvedspaces. Lang’sconjectureswillkeepmanymathematiciansoccupiedfarintothefuture. This memorial volume contains articles in a variety of areas of mathematics, attempting to represent Serge’s breadth of interest and impact. We are happy to publishhere(forthefirsttime)Serge’sfinalpaper,Theheatkernel,thetainversion, and zetas on (cid:2)nG=K, written jointly with one of us (J. Jorgenson). Except for thatonearticle,whichwasleftinthe formitassumedjustbeforeSerge’spassing, everyother entry here was thoroughlyrefereed.We thankall the authorsfor their contributionstothevolumeandfortheirwillingnesstoputupandcomplywithour demandsforrevision.Thanksalsototheanonymousrefereesfortheirexcellentand timelywork. We, the editors, are pleasedto be a partofthis production,especially since we wereallfortunateenoughtoknowSergepersonally.WethankStaceyCroomes,the math administratorat Caltech, for her invaluablehelp in organizingthe receiptof vii viii Preface thearticles,therefereeingprocess,andtherevisions.WearegratefultoAnnKostant andElizabethLoewofSpringerfortheirenthusiasmandhelpfuladviceduringthe manymonthsofeditorialpreparation.Ittookavillagetoproducethisvolume. ColumbiaUniversity DorianGoldfeld YaleUniversity PeterJones TheCityCollegeofNewYork JayJorgenson Caltech DinakarRamakrishnan UniversityofCalifornia,Berkeley KennethA.Ribet HarvardUniversity JohnTate Contents Preface............................................................................. vii PublicationsofSergeLang:from2000andbeyond.......................... xiii Introduction ...................................................................... xv JohnTate Raynaud’sgroup-schemeandreductionofcoverings........................ 1 DanAbramovichwithanAppendixbyJonathanLubin Themodulardegree,congruenceprimes,andmultiplicityone............. 19 AmodAgashe,KennethA.Ribet,andWilliamA.Stein Lethe´ore`medeSiegel–Shidlovskyrevisite´ .................................... 51 DanielBertrand Someaspectsofharmonicanalysisonlocallysymmetricspaces relatedtoreal-formembeddings ............................................... 69 EliotBrennerandAndrewSinton Differentialcharactersoncurves............................................... 111 AlexandruBuium WeylgroupmultipleDirichletseriesoftypeA .............................. 125 2 GautamChintaandPaulE.Gunnells Onthegeometryofthediffeomorphismgroupofthecircle ................ 143 AdrianConstantinandBorisKolev Harmonicrepresentativesforcuspidalcohomologyclasses................. 161 Jo´zefDodziuk,JeffreyMcGowan,andPeterPerry AbouttheABCConjectureandanalternative............................... 169 MachielvanFrankenhuijsen ix
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