Emil Artin and Beyond: Class Field Theory and L-Functions

245 Pages·2015·62.062 MB·English

by  Della Dumbaugh, Joachim Schwermer

Della Dumbaugh
Joachim Schwermer
with contributions by
James Cogdell and Robert Langlands
Emil Artin and Beyond –
Class Field Theory
and L-Functions
Authors:
Della Dumbaugh Joachim Schwermer
Department of Mathematics Faculty of Mathematics
University of Richmond University of Vienna
Richmond, VA, USA Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
E-mail: [email protected]
E-mail: [email protected]
Authors of Chapters V and VI:
James W. Cogdell Robert P. Langlands
Department of Mathematics School of Mathematics
Ohio State University Institute for Advanced Study
231 W. 18th Ave. Einstein Drive
Columbus, OH, USA Princeton, NJ, USA
2010 Mathematics Subject Classification: 01A60, 01A70, 11R37, 11R39, 11S37, 11S39, 11Fxx, 11Mxx
Key words: Number theory, class field theory, L-functions, automorphic L-functions; history of
mathematics, Emil Artin
ISBN 978-3-03719-146-0
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Preface
Intheearlydecadesofthe20thcentury,theobjectiveofclassfieldtheorywastorelate
thearithmeticofagivenfiniteextensionfieldLofanalgebraicnumberfieldK with
abelian Galois group to the properties of the base field K. Two fundamental lines
ofresearchregardingthisproblemwereintimatelyinterwovenwithoneanother,one
beingthequestforalawofdecompositioninLfortheprimeidealsinK,theother
beingthewishtoobtainageneralreciprocitylaw,similartothelawofCarlFriedrich
Gauss in the case of a quadratic extension L=K. In 1927, EmilArtin’s reciprocity
lawprovidedthedecisivelinkbetweenthesetwotheories. Itis,initsabstractform,
theheartofthetheoryofabelianextensionsofalgebraicnumberfields. However,his
workreliedtoalargeextentonthestudyofanalyticinvariantssuchas(cid:2)-functionsor
L-functionsattachedtoalgebraicnumberfields.
Atthattime,Artin,borninViennaonMarch3,1898,wasalreadyOrdinariusat
therecentlyfoundedUniversityofHamburg. HehadcompletedhisPh.D.inLeipzig
with Gustav Herglotz in 1921 and his Habilitation in Hamburg in 1923, and was
finally appointed as Ordentlicher Professor in 1926. In the following years Artin
was,togetherwithhiscolleaguesErichHeckeandWilhelmBlaschke,amostactive
member of the Mathematisches Seminar der Universität Hamburg. His works in
variousareasinnumbertheory,algebraandtopologyduringthattimeearnedhima
reputation as a distinguished scholar who was held in high esteem in the scientific
communityasbothteacherandresearcher.
Unfortunately, however, the political events in Germany would disrupt Artin’s
progress and plans. The German government prevented EmilArtin from attending
theInternationalCongressofMathematiciansinOslo,Norway,in1936andrefused
to grant Artin permission to deliver a series of lectures at Stanford University in
theU.S.A.in1937. Theseeventswereconsequencesofthedramaticchangeinthe
politicalsituationinGermanyinJanuary1933whenAdolfHitlerandtheNaziparty
hadassumedcontrolofGermany.
“It was only a question of time”, Richard Brauer would later describe it, “until
[Emil]Artin,withhisfeelingforindividualfreedom,hissenseofjustice,hisabhor-
rence of physical violence would leave Germany”.1 By the time Hitler issued the
edict on January 26, 1937, that removed any employee married to a Jew from their
positionasofJuly1,1937,2ArtinhadalreadybeguntomakeplanstoleaveGermany.
Artin had married his former student, Natalia Jasny, in 1929, and, since her father
practicedtheJewishfaith,theNazisclassifiedherasJewish.3 OnOctober1,1937,
ArtinandhisfamilyarrivedinAmerica.
1Brauer,Richard,EmilArtin,BulletinoftheAmericanMathematicalSociety73(1967),p.28.
2Art.59inthe“NeuesDeutschesBeamtengesetz”(NewGermanCivilServiceLaw),lateronsupplemented
bytheso-called“Flaggenerlass”ofApril19,1937.
3AlreadyonSeptember27,1934,Artinhadtodeclareinanofficialstatementthathiswifewasof“non-Aryan
descent”, see Reich, Karin, EmilArtin – Mathematiker vonWeltruf, in Das Hauptgebäude der Universität
HamburgalsGedächtnisort,editedbyRainerNicolaysen,HamburgUniversityPress,Hamburg2011,p.57.
vi Preface
These moments in EmilArtin’s life naturally raise some compelling questions.
Howdidclassfieldtheorydevelopinthe1930s? HowdidArtin’scontributionsin-
fluenceothermathematiciansatthetimeandinsubsequentyears? Giventhedifficult
political climate and his forced emigration as it were, how didArtin create a life in
Americawithintheexistinginstitutionalframework? DidArtincontinuehiseduca-
tionofandcloseconnectionwithgraduatestudents? Finally, isitpossibletobegin
to come to terms with the influence ofArtin on present day work on a non-abelian
classfieldtheory?
Ourattempttoinvestigatethesequestionsledtoindividualessaysbytheauthors
andtwocontributors,JamesCogdellandRobertLanglands,thatnowformthechap-
tersofthisvolume.4 LikeCarlSchorske’scompilationofwritingsforhiscelebrated
study of fin-de-siècle Vienna, each chapter “issued from a separate foray into the
terrain, varying in scale and focus according to the nature of the problem”.5 Taken
together,thesechaptersofferaviewofboththelifeofArtininthe1930sand1940s
andthedevelopmentofclassfieldtheoryatthattime. Theyalsoprovideinsightinto
thetransmissionofmathematicalideas,thecarefulstepsrequiredtopreservealifein
mathematicsatadifficultmomentinhistory,andtheinterplaybetweenmathematics
and politics (in more ways than one). Some of the technical points in this volume
require a sophisticated understanding of algebra and number theory. The broader
topics,however,willappealtoawideraudiencethatextendsbeyondmathematicians
andhistoriansofmathematicstoincludehistoricallymindedindividuals,particularly
thosewithaninterestinthetimeperiod.
We take Claude Chevalley’s presence inArtin’s 1931 Hamburg lectures on the
developmentofclassfieldtheoryasourstartingpoint. WeconsiderChevalley’searlier
workinclassfieldtheory,histhesis,andArtin’searlyinfluenceonthesemathematical
developments. WegiveespecialattentiontoChevalley’s1935lettertoHelmutHasse
wherehepresentstheconceptofélémentsidéaux. Wethenturnourattentiontothe
intersectionofArtin’spersonalandprofessionallivesashemadehiswaytotheU.S.
in 1937 through the time of his appointment at Princeton in 1946. Specifically, we
explorethebehind-the-scenesinitiativestosecureArtinhisfirsttemporaryposition
at the University of Notre Dame and his move to a permanent position at Indiana
University in 1938 where he worked with George Whaples on the foundations of
algebraic number theory. We introduce the reader to Margaret Matchett, Artin’s
Ph.D.studentatIndiana,andherthesis“Onthezetafunctionforideles”. Thebook
alsoconsiderstheinfluenceofArtinoncontemporaryworkonnon-abelianclassfield
theory as found in the seminal letter of Robert Langlands to André Weil. In this
epistolaryconversation, LanglandsdescribeshisgeneralconceptofEulerproducts,
incorporating the known L-series of Hecke andArtin in one notion. In his chapter
on L-functions, James Cogdell provides the bridge betweenArtin’s L-functions as
4InpreparingChaptersIIandIII,wehavedrawnfrompastcollaborativeandindividualsources[64],[60],
and[59].
5Schorske,CarlE.,Fin-de-SiècleVienna,AlfredA.Knopf,NewYork,1980,p.xxviii.
Preface vii
theyappearinhisgeneralreciprocitylawandtheautomorphicL-functionsencoded
inLanglands’letter. ThebookconcludeswithLanglands’personalcommentaryon
hislettertoWeil,revealinghisearlymathematicalthoughtsandchallenges.
Thefirstchapter,“Classfieldtheory: FromArtin’scourseinHamburgtoCheval-
ley’s‘Élémentsidéaux’”,presents,forthefirsttime,theletterfromClaudeChevalley
toHelmutHasseoutlininghisnotionofélémentsidéaux. Italsoprovidesthehistori-
calcontextfortheseideasand,insodoing,exposesthejourneynotonlyofanideabut
also of a mathematician. Chevalley attendedArtin’s course on class field theory in
1931inHamburg. Hesubmittedhisthesis“Surlathéorieducorpsdeclassesdansles
corpsfinisetlecorpslocaux”in1932. By1935,ChevalleyhadbeeninvitedbyHasse
andErichHecke,intheirnewrolesaseditorsofVolumeI“AlgebraundZahlenthe-
orie” of the Encyklopädie der MathematischenWissenschaften mit Einschluss ihrer
Anwendungen, to contribute an article on class field theory. In his seminal publi-
cations of 1936 and 1940 on this topic, Chevalley replaced classical ideal-theoretic
approaches to number theory with his “les éléments idéaux”, or as it was called in
1940, “notion de idèle”. In less than a decade, Chevalley progressed from student
to specialist on class field theory. In terms of geography, Chevalley traveled from
France to Germany to the United States, interacting with Japanese mathematicians
alongtheway,callingattentiontoarelativelynewinternationalphenomenoninthe
lifeofamathematician.
Chapter II, “Creating a life: EmilArtin inAmerica”, acquaints the reader with
Artinatthemomentwhenpoliticalturbulenceaffectedhislifeinarealandtangible
way. The news events that splashed across the top of newspapers were more than
headlines for Artin; they were vivid historical events that manifested themselves
inthemomentsofhislife, withhisfamilyandwithhismathematics. Inhisroleas
PresidentoftheAmericanMathematicalSociety,SolomonLefschetzwasparticularly
“wellplacedtoknowwhatwasgoingon”relativetoworldwidepoliticaleventsand
the subsequent needs of mathematicians, in this case, the needs of EmilArtin and
his family. On January 12, 1937, Lefschetz wrote to Father O’Hara to urge him to
considerapositionforEmilArtinonhisfacultyatUniversityofNotreDame. After
oneyearatNotreDame,ArtinmovedtoapermanentpositionatIndianaUniversity
where, it seems, the teaching grounded him. The steady, methodical rhythm of the
daily life of imparting mathematics to fresh faces established a routine for him, a
patternnotsounlikewhathesearchedforinmathematics.
At the same time, the teaching helped restore a sense of balance to Artin. It
extendedhim,intheformofGeorgeWhaplesandMargaretMatchettwhogavehis
larger ideas – those beyond trigonometry and calculus, for example – a chance to
growanddevelopintoresearchlevelmathematics.
ChapterIIIfocusesonArtin’sworkwithWhaples,whichwasverymuchinspired
by Chevalley’s notion of the idele group. Whaples came to Indiana by way of the
University of Wisconsin, where he earned his Ph.D., under the direction of Mark
H.Ingraham,withathesis“Onthestructureofmoduleswithacommutativealgebra
viii Preface
as operator domain”. Since Ingraham studied with E.H. Moore at the University
of Chicago, Whaples enjoyed a distinctly American heritage in his mathematical
training. ArtinandWhaplesdefinedvaluationvectorsastheadditivecounterpartof
the group of ideles attached to an algebraic number field. This association enabled
themtoderivethefundamentalresultsofnumbertheoryfromsimpleaxioms. Their
main result is the use of the product formula for valuations to derive an axiomatic
characterizationofbothalgebraicnumberfieldsandfunctionfieldswithafinitefield
ofconstants. Theseareexactlythetwofamiliesoffieldsforwhichclassfieldtheory
isknowntohold.
Whaples’workandhisassociationwithArtinopenedthedoorforhimtovisitthe
InstituteforAdvancedStudyinPrinceton. Hisapplication,anastonishinglyconcise,
handwrittendocumentrevealsWhaples’summaryandanalysisofthisworkaswell
ashisfutureresearchplans.
Chapter IV introduces the reader to Margaret Matchett, Artin’s second Ph.D.
studentatIndiana,andherthesis“Onthezetafunctionforideles”. WhileMatchett
pursuedherPh.D.work,ArtinandWhaplesusedthetheoryofvaluationstoinvestigate
classicalquestionsinalgebraicnumbertheory. Itwasonlynatural,then,forArtinto
suggesttoMatchettthatsheconsiderthepossibilityofinterpretingthetheoryofzeta
functions and L-series through the lens of ideles and valuation vectors. Her Ph.D.
in 1946 was the only Ph.D. in mathematics awarded to a woman at Indiana in the
1940s. Exploringherlifemoregenerallyprovidesfurtherinsightsintobroadertrends
inemploymentforwomenandpersonalramificationsofpoliticalideologies.
InChapterV,inhiscontribution“L-functionsandnon-abelianclassfieldtheory,
from Artin to Langlands”, James Cogdell rather poetically begins his study with
Weil’sassertionthatArtinhada“loveaffairwiththezetafunction”whilehewasat
Hamburg. Infact,itseems,Artinhadaloveaffairwiththezetafunctionthroughout
most of his mathematical life. Cogdell follows Chevalley’s suggestion that Artin
madeuseofzetafunctionstodiscoverprecisealgebraicfactsratherthanestimatesor
approximations. In particular, Cogdell argues thatArtin used L-functions as a tool
to study non-abelian class field theory. His chapter in this volume begins with an
overviewofL-functionsbeforeArtin,thenfollowsthecourseofArtin’sL-functions
andtheparalleldevelopmentoftheL-functionsofHeckeandtheir“reconciliation”
intheLanglandsprogram.
That brings us to the final chapter of the volume, “Automorphic L-functions”,
contributedbyRobertP.Langlands. Inthischapter,wepresenttheletterfromLang-
lands toAndréWeil outlining his ideas concerning the definition of a new class of
Eulerproducts. Rightatthebeginningoftheletter,thenewconceptofwhatisnow
known as the L-group appears in a quite sophisticated form. The theory of auto-
morphicforms,inparticular,thetheoryofEisensteinseriesinthecontextofgeneral
reductivegroups,servesasasourceofinspirationforthiscircleofideas. Inaddition
tothisletter,inthetreatise“FunktorialitätinderTheoriederautomorphenFormen:
Ihre Entdeckung und ihre Ziele”, Langlands describes how his letter to Weil came
Preface ix
into existence, comments on its mathematical content, and provides personal and
professional insights about how his ideas unfolded, in particular, within the theory
ofautomorphicforms. Inhismorepersonalremarks,Langlandsdescribeshismath-
ematical youth when he read Courant and Dickson; his desire to learn too fast; his
frustration with a year that was not particularly fruitful in terms of mathematical
results;histhoughtsaboutgivingupmathematics;andhissuccessduringtheChrist-
masbreakof1966–67,inhisofficeinFineHallatPrinceton,whenhefoundtheidea
toprovetheanalyticcontinuationofautomorphicL-functions. Theebbandflowof
Langlands’professionallifenotonlyoffersguidepostsforacareerinmathematicsbut
alsoinspiration. Quiteunexpectedly,then,thisvolume,thatwasoriginallyintended
toshedlightonthedevelopmentofclassfieldtheoryasitpassedthroughthehands
ofChevalley,ArtinandLanglands,grewintoarichexpositiononthearcofalifein
mathematics.
RichmondandVienna,October2014 DellaDumbaugh
JoachimSchwermer
Acknowledgements
This book has benefitted from the generosity and wisdom of many others. We are
gratefultoJamesW.CogdellandRobertP.Langlandsfortheircontributionstothis
volumeandtheirpatienceinthelongevolutionofitspublication. Weareespecially
indebted to James Cogdell, who read every chapter of this volume, some of them
multiple times, and shared many hours of meaningful conversation about how to
improve this work. Andrew Matchett of the University of Wisconsin, La Crosse,
offered invaluable insights about his mother, Margaret Matchett, and gave permis-
sion to publish her thesis in this volume. Jean-Pierre Labesse, Marseille, checked
ourtranslationsfromtheFrench. Manyarchivistsworkedwithustolocatematerial
and reproduce it in the highest possible quality. Among them: Erica Mosner at the
Shelby White and Leon LevyArchives Center at the Institute forAdvanced Study,
Princeton; Kevin Cawley at the Archives at the University of Notre Dame; Carrie
Schwier, Anthony Barger, and Sherri Michaels at the Indiana UniversityArchives,
Bloomington;andDr. HelmutRohlfingoftheNiedersächsischeStaats-undUniver-
sitätsbibliothekGöttingen. JoachimHiltmannoftheMuseumfürKunstundGewerbe
HamburgprovidedphotographstakenbyNataliaArtininthe1930s. Thiscollection
of photographs appears under her later name NataschaArtin-Brunswick. Her chil-
dren Karin, Michael and TomArtin graciously granted permission to publish these
photographs. PeterSykesandJenniferWrightSharpattheAmericanMathematical
SocietyhelpedcreateabettercopyoftheSolomonLefschetzletter. TheUniversity
ofRichmond,theUniversityofVienna,andtheErwinSchrödingerInternationalIn-
stituteforMathematicalPhysicsinViennaprovidedfinancialsupportforthisproject.
AnespecialthankstoDr.ManfredKarbeoftheEuropeanMathematicalSociety
Publishing House who believed in this project from the start and provided helpful
insight during the final stages, particularly with regard to the preface. Dr. Irene
Zimmermanngaveextraordinarycaretotheactualproductionofthisvolume. That
youholdthisbookinyourhandsisadirectresultoftheircombinedattentiontoour
work.
Finally,thisvolumetestifiestotheongoingtourdeforceoftheViennacoffeehouse
culture. Thatvibrantatmosphere–combinedwithmélange,EarlGreyteaandtruffle
torte–createdjusttherightvenueforunexpecteddetailstounfoldandideastotake
shape.