Automorphic forms and L-Functions for the group GL(n,R)

508 Pages·2006·4.41 MB·English

by  Dorian Goldfeld

CAMBRIDGESTUDIESINADVANCEDMATHEMATICS
EditorialBoard
B.Bollobas,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
AUTOMORPHICFORMSANDL-FUNCTIONSFOR
THEGROUPGL(n,R)
L-functions associated with automorphic forms encode all classical num-
ber theoretic information. They are akin to elementary particles in physics.
This book provides an entirely self-contained introduction to the theory of
L-functionsinastyleaccessibletograduatestudentswithabasicknowledge
of classical analysis, complex variable theory, and algebra. Also within the
volume are many new results not yet found in the literature. The exposition
provides complete detailed proofs of results in an easy-to-read format using
many examples and without the need to know and remember many complex
definitions.ThemainthemesofthebookarefirstworkedoutforGL(2,R)and
GL(3,R),andthenforthegeneralcaseofGL(n,R).Inanappendixtothebook,
asetofMathematica® functionsispresented,designedtoallowthereaderto
explorethetheoryfromacomputationalpointofview.
CAMBRIDGESTUDIESINADVANCEDMATHEMATICS
EditorialBoard:
B.Bollobas,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
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33 C.Souleetal.LecturesonArakelovgeometry
34 A.Ambrosetti&G.ProdiAprimerofnonlinearanalysis
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38 C.WeibelAnintroductiontohomologicalalgebra
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41 G.LaumonCohomologyofDrinfeldmodularvarietiesI
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49 R.StanleyEnumerativecombinatoricsI
50 I.PorteousCliffordalgebrasandtheclassicalgroups
51 M.AudinSpinningtops
52 V.JurdjevicGeometriccontroltheory
53 H.VolkleinGroupsasGaloisgroups
54 J.LePotierLecturesonvectorbundles
55 D.BumpAutomorphicformsandrepresentations
56 G.LaumonCohomologyofDrinfeldmodularvarietiesII
57 D.M.Clark&B.A.DaveyNaturaldualitiesfortheworkingalgebraist
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59 P.TaylorPracticalfoundationsofmathematics
60 M.P.Brodmann&R.Y.SharpLocalcohomology
61 J.D.Dixonetal.Analyticpro-Pgroups
62 R.StanleyEnumerativecombinatoricsII
63 R.M.DudleyUniformcentrallimittheorems
64 J.Jost&X.Li-JostCalculusofvariations
65 A.J.Berrick&M.E.KeatingAnintroductiontoringsandmodules
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Automorphic Forms and L-Functions for
the Group GL (n, R)
DORIAN GOLDFELD
ColumbiaUniversity
WithanAppendixbyKevinA.Broughan
UniversityofWaikato
  
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Contents
Introduction pagexi
1 Discretegroupactions 1
1.1 Actionofagrouponatopologicalspace 3
1.2 Iwasawadecomposition 8
1.3 Siegelsets 15
1.4 Haarmeasure 19
1.5 Invariantmeasureoncosetspaces 23
1.6 VolumeofSL(n,Z) SL(n,R)/SO(n,R) 27
\
2 Invariantdifferentialoperators 38
2.1 Liealgebras 39
2.2 Universalenvelopingalgebraofgl(n,R) 42
2.3 Thecenteroftheuniversalenvelopingalgebraofgl(n,R) 46
2.4 Eigenfunctionsofinvariantdifferentialoperators 50
3 AutomorphicformsandL–functionsforSL(2,Z) 54
3.1 Eisensteinseries 55
3.2 HyperbolicFourierexpansionofEisensteinseries 59
3.3 Maassforms 62
3.4 WhittakerexpansionsandmultiplicityoneforGL(2,R) 63
3.5 Fourier–WhittakerexpansionsonGL(2,R) 67
3.6 Ramanujan–Peterssonconjecture 68
3.7 Selbergeigenvalueconjecture 70
3.8 Finitedimensionalityoftheeigenspaces 71
3.9 EvenandoddMaassforms 73
3.10 Heckeoperators 74
3.11 HermiteandSmithnormalforms 77
3.12 HeckeoperatorsforL2(SL(2,Z)) h2 80
\
vii
viii Contents
3.13 L–functionsassociatedtoMaassforms 84
3.14 L-functionsassociatedtoEisensteinseries 89
3.15 ConversetheoremsforSL(2,Z) 91
3.16 TheSelbergspectraldecomposition 94
4 ExistenceofMaassforms 99
4.1 TheinfinitudeofoddMaassformsforSL(2,Z) 100
4.2 Integraloperators 101
4.3 Theendomorphism 105
♥
4.4 Howtointerpret :anexplicitoperatorwithpurely
♥
cuspidalimage 106
4.5 Thereexistinfinitelymanyevencuspformsfor SL(2,Z) 108
4.6 AweakWeyllaw 110
4.7 Interpretationviawaveequationandtheroleoffinite
propagationspeed 111
4.8 Interpretationviawaveequation:higherrankcase 111
5 MaassformsandWhittakerfunctionsforSL(n,Z) 114
5.1 Maassforms 114
5.2 WhittakerfunctionsassociatedtoMaassforms 116
5.3 FourierexpansionsonSL(n,Z) hn 118
\
5.4 WhittakerfunctionsforSL(n,R) 128
5.5 Jacquet’sWhittakerfunction 129
5.6 Theexteriorpowerofavectorspace 134
5.7 ConstructionoftheI functionusingwedgeproducts 138
ν
5.8 ConvergenceofJacquet’sWhittakerfunction 141
5.9 FunctionalequationsofJacquet’sWhittakerfunction 144
5.10 DegenerateWhittakerfunctions 150
6 AutomorphicformsandL-functionsforSL(3,Z) 153
6.1 WhittakerfunctionsandmultiplicityoneforSL(3,Z) 153
6.2 MaassformsforSL(3,Z) 159
6.3 ThedualandsymmetricMaassforms 161
6.4 HeckeoperatorsforSL(3,Z) 163
6.5 TheGodement–JacquetL-function 172
6.6 Bump’sdoubleDirichletseries 186
7 TheGelbart–Jacquetlift 194
7.1 ConversetheoremforSL(3,Z) 194
7.2 Rankin–SelbergconvolutionforGL(2) 210
7.3 StatementandproofoftheGelbart–Jacquetlift 213
7.4 Rankin–SelbergconvolutionforGL(3) 223
Contents ix
8 BoundsforL-functionsandSiegelzeros 235
8.1 TheSelbergclass 235
8.2 ConvexityboundsfortheSelbergclass 238
8.3 Approximatefunctionalequations 241
8.4 SiegelzerosintheSelbergclass 245
8.5 Siegel’stheorem 249
8.6 TheSiegelzerolemma 251
8.7 Non-existenceofSiegelzerosforGelbart–Jacquetlifts 252
8.8 Non-existenceofSiegelzerosonGL(n) 256
9 TheGodement–JacquetL-function 259
9.1 MaassformsforSL(n,Z) 259
9.2 ThedualandsymmetricMaassforms 261
9.3 HeckeoperatorsforSL(n,Z) 266
9.4 TheGodement–JacquetL-function 277
10 LanglandsEisensteinseries 285
10.1 Parabolicsubgroups 286
10.2 Langlandsdecompositionofparabolicsubgroups 288
10.3 Bruhatdecomposition 292
10.4 Minimal,maximal,andgeneralparabolicEisensteinseries 295
10.5 EisensteinseriestwistedbyMaassforms 301
10.6 FourierexpansionofminimalparabolicEisensteinseries 303
10.7 Meromorphiccontinuationandfunctionalequationof
maximalparabolicEisensteinseries 307
10.8 TheL-functionassociatedtoaminimalparabolic
Eisensteinseries 310
10.9 FouriercoefficientsofEisensteinseriestwistedby
Maassforms 315
10.10Theconstantterm 319
10.11The constant term of SL(3,Z) Eisenstein series twisted by
SL(2,Z)-Maassforms 321
10.12AnapplicationofthetheoryofEisensteinseriestothe
non-vanishingofL-functionsontheline (s) 1 322
ℜ =
10.13LanglandsspectraldecompositionforSL(3,Z) h3 324
\
11 Poincare´ seriesandKloostermansums 337
11.1 Poincare´ seriesforSL(n,Z) 337
11.2 Kloostermansums 339
11.3 Plu¨ckercoordinatesandtheevaluationofKloostermansums 343
11.4 PropertiesofKloostermansums 350