Number Theory, Analysis and Geometry: In Memory of Serge Lang

725 Pages·2012·5.33 MB·English

by  Dan Abramovich

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
[e-mail] [e-mail]
PeterJones DinakarRamakrishnan
DepartmentofMathematics DepartmentofMathematics
YaleUniversity CaliforniaInstituteofTechnology
NewHaven,CT06520 Pasadena,CA91125
USA USA
[e-mail] [e-mail]
KennethA.Ribet JohnTate
DepartmentofMathematics DepartmentofMathematics
UniversityofCaliforniaatBerkeley HarvardUniversity
Berkeley,CA94720 Cambridge,MA02138
USA USA
[e-mail] [e-mail]
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
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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