Lectures on Automorphic L-functions

Table of contents :
Cover......Page 1
Title page......Page 4
Contents......Page 6
Preface......Page 12
Lectures on ��-functions, converse theorems, and functoriality for ����_{��}, by James W. Cogdell......Page 14
Preface......Page 16
Lecture 1. Modular forms and their ��-functions......Page 18
Lecture 2. Automorphic forms......Page 26
Lecture 3. Automorphic representations......Page 34
Lecture 4. Fourier expansions and multiplicity one theorems......Page 42
Lecture 5. Eulerian integral representations......Page 50
Lecture 6. Local ��-functions: The non-Archimedean case......Page 58
Lecture 7. The unramified calculation......Page 64
Lecture 8. Local ��-functions: The Archimedean case......Page 72
Lecture 9. Global ��-functions......Page 78
Lecture 10. Converse theorems......Page 86
Lecture 11. Functoriality......Page 94
Lecture 12. Functoriality for the classical groups......Page 100
Lecture 13. Functoriality for the classical groups, II......Page 104
Automorphic ��-functions, by Henry H. Kim......Page 110
Introduction......Page 112
Chevalley groups and their properties......Page 114
Cuspidal representations......Page 126
��-groups and automorphic ��-functions......Page 128
Induced representations......Page 132
Eisenstein series and constant terms......Page 142
��-functions in the constant terms......Page 150
Meromorphic continuation of ��-functions......Page 158
Generic representations and their Whittaker models......Page 160
Local coefficients and non-constant terms......Page 166
Local Langlands correspondence......Page 174
Local ��-functions and functional equations......Page 178
Normalization of intertwining operators......Page 184
Holomorphy and bounded in vertical strips......Page 190
Langlands functoriality conjecture......Page 194
Converse theorem of Cogdell and Piatetski-Shapiro......Page 196
Functoriality of the symmetric cube......Page 200
Functoriality of the symmetric fourth......Page 206
Bibliography......Page 212
Applications of symmetric power ��-functions, by M. Ram Murty......Page 216
Preface......Page 218
Lecture 1. The Sato-Tate conjecture......Page 220
Lecture 2. Maass wave forms......Page 226
Lecture 3. The Rankin-Selberg method......Page 232
Lecture 4. Oscillations of Fourier coefficients of cusp forms......Page 240
Lecture 5. Poincaré series......Page 250
Lecture 6. Kloosterman sums and Selberg’s conjecture......Page 256
Lecture 7. Refined estimates for Fourier coefficients of cusp forms......Page 260
Lecture 8. Twisting and averaging of ��-series......Page 266
Lecture 9. The Kim-Sarnak theorem......Page 270
Lecture 10. Introduction to Artin ��-functions......Page 278
Lecture 11. Zeros and poles of Artin ��-functions......Page 284
Lecture 12. The Langlands-Tunnell theorem......Page 288
Bibliography......Page 294
Back Cover......Page 298

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Lectures o n Automorphic L-functions

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http://dx.doi.org/10.1090/fim/020

FIELDS INSTITUTE MONOGRAPHS T H E FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S

Lectures o n Automorphic Jv-functions J a m e s W . Cogdel l Henry H . Ki m M. Ra m Murt y

American Mathematica l Societ y Providence, Rhod e Islan d

The Field s Institut e for Researc h i n Mathematica l Science s T h e Field s I n s t i t u t e i s n a m e d i n h o n o u r o f t h e C a n a d i a n m a t h e m a t i c i a n J o h n Charle s Fields (1 863-1 932) . Field s wa s a visionar y wh o receive d m a n y h o n o u r s fo r hi s scientifi c work, includin g electio n t o t h e Roya l Societ y o f C a n a d a i n 1 90 9 a n d t o t h e Roya l Societ y o f London i n 1 91 3 . A m o n g o t h e r accomplishment s i n t h e servic e o f t h e i n t e r n a t i o n a l m a t h ematics community , Field s wa s responsibl e fo r establishin g t h e world' s m o s t prestigiou s prize fo r m a t h e m a t i c s r e s e a r c h — t h e Field s Medal . T h e Field s I n s t i t u t e fo r Researc h i n M a t h e m a t i c a l Science s i s s u p p o r t e d b y g r a n t s fro m t h e O n t a r i o Ministr y o f Training , College s a n d Universities , a n d t h e N a t u r a l Science s a n d Engineerin g Researc h Counci l o f C a n a d a . T h e I n s t i t u t e i s sponsore d b y Carleto n University, M c M a s t e r University , t h e Universit y o f O t t a w a , t h e Universit y o f Toronto , t h e Universit y o f Waterloo , t h e Universit y o f W e s t e r n O n t a r i o , a n d Yor k University . I n addition t h e r e ar e severa l affiliate d universitie s a n d c o r p o r a t e sponsors , fro m C a n a d a an d t h e Unite d S t a t e s .

2000 Mathematics Subject

Classification.

Primar

y 1 1 F70 , 22E55 .

For additiona l informatio n a n d u p d a t e s o n thi s book , visi t www.ams.org/bookpages/fim-20

Library o f Congres s Cataloging-in-Publicatio n Dat a Cogdell, Jame s M. , 1 953 Lectures o n automorphi c L-function s / Jame s M . Cogdell , Henr y H . Kim , M . Ra m Murty . p. cm . (Field s Institut e monographs , ISS N 1 069-5273 ; 20 ) Includes bibliographica l references . ISBN 0-821 8-351 6- 5 (alk . paper ) 1. L-functions . 2 . Automorphi c functions . I . Kim , Henr y Hyeongsin , 1 964 - II . Murty , Maruti Ram . III . Titie . IV . Series . QA246.C64 200 4 512.7 / 3-dc22 2004046 6 6 CIP AMS softcove r ISB N 978-0-821 8-4800- 5 C o p y i n g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given . Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n thi s publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 200 4 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepare d b y Th e Field s Institute . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 21 1

4 1 3 1 2 1 1 1 0 09

Contents Preface x

i

Lectures o n L-functions , Convers e Theorems , an d Functoriality fo r GL(n) James W . Cogdel l Preface 3 Lecture 1 . Modula r Form s an d Thei r L- functions 5 1. Example s 6 2. Growt h estimate s o n cus p form s 7 3. Th e L-functio n o f a cus p for m 8 4. Th e Eule r produc t 5. Reference s 1 Lecture 2 . Automorphi c Form s 1. Automorphi c form s o n GL2 1 1 2. Automorphi c form s o n GL n 1 3. Smoot h automorphi c form s 4. L 2 -automorphic form s 5. Cus p form s 6. Reference s

0 2 3 3 6 7 8 8 9

Lecture 3 . Automorphi c Representation s 2 1 1. (K-finite ) automorphi c representation s 2 1 2. Smoot h automorphi c representation s 2 4 3. L 2 -automorphic representation s 2 5 4. Cuspida l representation s 2 5 5. Connection s wit h classica l form s 2 6 6. Reference s 2 7 Lecture 4 . Fourie r Expansion s an d Multiplicit y On e Theorem s 2 9 1. Th e Fourie r expansio n o f a cus p for m 2 9 2. Whittake r model s 3 1 3. Multiplicit y on e fo r GL n 3 3 4. Stron g multiplicit y one s fo r GL n 3 4 5. Reference s 3 5

vi Content

s

Lecture 5 . Euleria

n Integra l Representation s 3

7

1 1. GL 2 x GL 3 7 2. GL n x GLm wit h m < n 3 8 3. GL n x GL n 4 1 4. Summar y 4 3 5. Reference s 4 3 Lecture 6 . Loca

l L-functions : Th e Non-Archimedea n Cas e 4

1. Whittake r function s 4 2. Th e loca l L-functio n ( m < n) 4 3. Th e loca l functiona l equatio n 4 4. Th e conducto r o f IT 4 5. Multiplicativit y an d stabilit y o f 7-factor s 4 6. Reference s 5 Lecture 7 . Th

5 5 6 8 9 9 0

e Unramifie d Calculatio n 5 1

1. Unramifie d representation s 5 2. Unramifie d Whittake r function s 5 3. Calculatin g th e integra l 5 4. Reference s 5 Lecture 8 . Loca

l L-functions : Th e Archimedea n Cas e 5

2 3 5 7 9

1. Th e arithmeti c Langland s classificatio n 5 9 2. Th e L-function s 5 9 3. Th e integral s ( m < n) 6 1 4. I s th e L-facto r correct ? 6 2 5. Reference s 6 4 Lecture 9 . Globa

l L-function s 6

1. Convergenc e 6 2. Meromorphi c continuatio n 6 3. Pole s o f L-function s 6 4. Th e globa l functiona l equatio n 6 5. Boundednes s i n vertica l strip s 6 6. Summar y 6 7. Stron g Multiplicit y On e revisite d 6 8. Generalize d Stron g Multiplicit y On e 7 9. Reference s 7 Lecture 1 0 . Convers

e Theorem s 7

1. Convers e Theorem s fo r GL n 7 2. Invertin g th e integra l representatio n 7 3. Proo f o f Theore m 1 0. 1 (i ) 7 4. Proo f o f Theore m 1 0. 1 (ii ) 7 5. Theore m 1 0. 2 an d beyon d 7 6. A usefu l varian t 7 7. Conjecture s 7

5 5 6 7 7 8 9 9 0 0 3 3 4 7 7 8 9 9

Contents vi

i

8. Reference s 8

0

Lecture 1 1 . Functorialit y 8 1 1. Th e Weil-Delign e grou p 8 1 2. Th e dua l grou p 8 2 3. Th e loca l Langland s conjectur e 8 2 4. Loca l functorialit y 8 3 5. Globa l functorialit y 8 3 6. Functorialit y an d th e Convers e Theore m 8 4 7. Reference s 8 5 Lecture 1 2 . Functorialit y fo r th e Classica l Group s 8 1. Th e result s 8 2. Constructio n o f a candidat e lif t 8 3. Analyti c propertie s o f L-function s 9 4. Appl y th e Convers e Theore m 9 5. Reference s 9

7 7 8 0 0 0

Lecture 1 3 . Functorialit y fo r th e Classica l Groups , I I 9 1 1. Functorialit y 9 1 2. Descen t 9 2 3. Bound s toward s Ramanuja n 9 4 4. Th e loca l convers e theore m 9 4 5. Furthe r application s 9 5 6. Reference s 9 6 Automorphic L-function s Henry H . Ki m Introduction 9

9

Chapter 1 . Chevalle y Group 1 s an d thei r Propertie s 0 1 1. Algebrai c group s 0 1 2. Root s an d coroot s 0 3 1 3. Classificatio n o f roo t system s 0 4 4. Constructio n o f Chevalle y groups : simpl y connecte 1 d typ e 0 7 1 5. Structur e o f paraboli c subgroup s 0 8 Chapter 1 1 2 . Cuspida l Representation s

3

Chapter 3 . L-group

5

s an d Automorphi 1 1 c L-function s

1 Chapter 1 4 . Induce d Representation s 9 1 1 1. Harish-Chandr a homomorphism s 9 1 2. Induce d representations : F loca l 2 1 1 3. Intertwinin g operator s fo r I(s,ir) 2 2 4. Digressio n 1 o n admissibl e representation s 2 3 5.1 Induce d representations : F globa l 2 6

viii C o n t e n t

s

6. Induce d representation s a s holomorphi 1 c fiber bundle s 2 Chapter 5 . Eisenstei n Serie 1 s an d Constan t Term s 2 1 1. Definitio n o f Eisenstei n serie s 2 2. Constan t term s 3 3. Psuedo-Eisenstei n serie s 3

6 9 9 0 2

Chapter 6 . L-function1 s i n th e Constan t Term s 3 List o f L-function s vi a Langlands-Shahid 1 i metho d 4

7 3

Chapter 7 . Meromorphi

5

c Continuatio 1 n o f L-function s 4

Chapter 8 . Generi c Representation s an d thei r Whittake1 r Model s 4 1. Genera l cas e 4 2. Whittake r model s fo r 1 induce d representation s 4

7 7 9

Chapter 9 . Loca l Coefficient s an d Non-constan 1 t Term s 5 1. Non-constan 1 t term s o f Eisenstei n serie s 5 2. Loca l coefficient s an d crud1 e functiona l equatio n 5

3 3 8

Chapter 1 0 . Loca 1 l Langland s Correspondenc e 6 1 Chapter 1 1 . Loca l L-function s an d Functiona 1 l Equation s 6 1 1. Definitio n o f loca l L-function s 6 2. Propertie s o f loca l L-functions ; supercuspida l representation 1 s 7

5 9 0

Chapter 1 2 . Normalizatio n o f Intertwinin 1 g Operator s 7 1 1. 7 r is supercuspidal 7 1 2. 7 r is tempered, generi c 7 1 1 3. 7 r is non-tempered , generi c 7 2 4. Applicatio 1 n t o reducibilit y criterio n 7 5 Chapter 1 3 . Holomorph y an d Bounde d1 i n Vertica l Strip s 7 1 1. Holomorph y o f L-function s 7 2. Boundednes s i n vertica 1 l strip s o f L-function s 7

7 7 7

Chapter 1 4 . Langland 1 s Functorialit y Conjectur e 8 1 Chapter 1 5 . Convers

e Theore m o f Cogdel l an d Piatetski-Shapir 1 o 8

Chapter 1 6 . Functorialit y1 o f th e Symmetri c Cub e 8 1 1. Wea k Ramanuja n propert y 8 2. Functorialit 1 y o f th e symmetri c squar e 8 3. Functorialit y o f th e tenso r produc 1 t o f GL2 x GL3 8 4. Functorialit 1 y o f th e symmetri c cub e 9 Chapter 1 7 . Functorialit y o 1 f th e Symmetri c Fourt h 9 1. Functorialit 1 y o f th e exterio r squar e 9

3 7 7 7 8 0 3 3

Contents i

x

2. Functorialit 1 y o f th e symmetri c fourt h 9 Bibliography 9

4 9

Applications o f Symmetri c Powe r L-function s M. Ra m Murt y Preface 20

5

Lecture 1 . Th e Sato-Tat e Conjectur e 20 1. Introductio n 20 2. Unifor m distributio n 20 3. Wiener-Ikehar a Tauberia n theore m 20 4. Weyl' 1 s theore m fo r compac t group s 2

7 7 8 9 0

1 Lecture 2 . Maas s Wav e Form s 2 1 1. Maas s form s o f weigh t zer o 2 1 2. Maas s form s wit h weigh t 2 3. Eisenstei n serie s 2 4. Uppe r boun d fo r Fourie r coefficient s an d eigenvalu e estimator 1 s2

3 3 4 4 6

Lecture 1 3 . Th e Rankin-Selber g Metho d 2 9 1. Eisenstei n serie s an d non-vanishin g o f £(s ) 1 o n 9t(s ) = 1 2 9 2. Explici t constructio n o f Maas s cus p form s 22 1 3. Th e Rankin-Selber g L-functio n 22 2 4. Rankin-Selber g L-function s fo r GL n 22 5 Lecture 4 . Oscillation

s o f Fourie r Coefficient s o f Cus p Form s 22

1. Preliminarie s 22 2. Rankin' s theore m 22 3. A revie w o f symmetri c powe r L-function s 23 4. Proo f o f Theore m 4. 1 23 Lecture 5 . Poincar

e Serie s 23

1. Poincar e serie s fo r SLz^L) 23 2. Fourie r coefficient s an d Kloosterma n sum s 23 3. Th e Kloosterman-Selber g zet a functio n 24

7 7 8 0 2 7 7 9 2

Lecture 6 . Kloosterma n Sum s an d Selberg' s Conjectur e 24 1. Petersson' s formul a 24 2. Selberg' s theore m 24 3. Th e Selberg-Linni k conjectur e 24

3 3 4 5

Lecture 7 . Refine d Estimate s fo r Fourie r Coefficient s o f Cus p Form s 24 1. Siev e theor y an d Kloosterma n sum s 24 2. Gaus s sum s an d hyper-Kloosterma n su m 24 3. Th e Duke-Iwanie c metho d 24

7 7 8 8

x Content

s

Lecture 8 . Twistin g an d Averagin g o f L-serie s 25 1. Selber g conjecture s fo r GL n 25 2. Ramanuja n conjectur e fo r Gl n 25 3. Th e metho d o f averagin g L-function s 25

3 3 4 5

Lecture 9 . Th e Kim-Sarna k Theore m 25 1. Preliminarie s 25 2. Rankin-Selber g theor y 25 3. A n applicatio n o f th e Duke-Iwanie c metho d 25

7 7 8 9

Lecture 1 0 . Introductio n t o Arti n L-function s 26 1. Heck e L-function s 26 2. Arti n L-function s 26 3. Automorphi c inductio n an d Artin' s conjectur e 26

5 5 6 8

Lecture 1 1 . Zero s an d Pole s o f Arti n L-function s 27 1 1. Th e Heilbron n characte r 27 1 2. Th e fundamenta l inequalit y 27 2 3. Rankin-Selber g propert y fo r Galoi s representation s 27 3 Lecture 1 2 . Th e Langlands-Tunnel l Theore m 27 1. Revie w o f som e grou p theor y 27 2. Som e representatio n theor y 27 3. A n applicatio n o f th e Deligne-Serr e theor y 27 4. Th e genera l cas e 27 5. Sarnak' s theore m 27 Bibliography

5 5 6 7 7 8 281

Preface "Mathematics goes to great pains to create expressions for relationships which pass empirical comprehension." Carl Gustav Jung in "Memories, Dreams, Reflections" This monograp h i s base d o n graduat e course s whic h th e author s gav e a t th e Thematic Progra m o n Automorphi c Form s a t th e Field s Institut e i n th e sprin g o f 2003. Th e progra m wa s organize d b y J . Arthur , T . Haines , H . Kim , R . Murty , G. Pappas , an d F . Shahidi . Thes e course s wer e intende d fo r pos t docs an d ad vanced graduat e student s i n order t o introduc e the m t o the Langland s functorialit y conjecture an d it s consequence s i n numbe r theor y an d representatio n theory . I n particular, w e wante d t o sho w the m ho w automorphi c //-function s pla y a crucia l role i n th e theory . Ther e hav e bee n som e ne w development s i n th e theory , mos t notably, functorialit y o f the symmetri c cub e an d symmetri c fourt h o f cuspida l rep resentations o f GL(2) an d functorialit y o f classica l groups . Thes e development s make us e o f automorphi c L- functions, namely , th e combinatio n o f convers e theo rems o f Cogdel l an d Piatetski-Shapir o an d th e Langlands-Shahid i method . Th e aim o f th e themati c progra m wa s t o revie w thes e development s an d encourag e th e discovery o f a s ye t unknow n implication s o f functorialit y t o numbe r theory , an d vice versa . Besides th e courses , ther e wa s a weekl y semina r o n automorphi c form s give n by members . Ther e wer e tw o workshops : on e o n Shimur a varietie s an d relate d topics, organize d b y T . Haine s an d G . Pappas , an d th e othe r o n Automorphi c Lfunctions, organize d b y H . Ki m an d R . Murty . I n addition , S . Kudl a gav e th e Coxeter Lecture s o n Arithmeti c thet a serie s an d P . Sarna k gav e th e Distinguishe d Lectures o n Automorphi c L-function s an d equidistribution . This monograp h i s no t a t al l a comprehensiv e accoun t o f automorphi c forms . The mos t seriou s omission s ar e trac e formula s an d Shimur a varieties , namely , th e geometric poin t o f vie w o f automorphi c forms . However , i n orde r t o compensat e for this , ther e wa s a summer schoo l o n Harmoni c Analysis , th e Trac e Formul a an d Shimura Varieties , sponsore d b y th e Cla y Mathematic s Institut e i n th e summe r o f 2003. I t wa s organize d b y J . Arthur , D . Ellwoo d an d R . Kottwitz . Thei r lectur e notes will be published soo n an d w e believe that i t wil l complement ou r monograp h very nicely . Let u s describ e ou r monograp h i n detail . Th e Langland s functorialit y conjec ture ca n b e roughl y formulate d as : i f H an d G ar e tw o reductiv e group s ove r a number field F , the n t o eac h homomorphis m o f L-group s (ft : LH — • L G, ther e i s associated a lif t o f automorphi c representation s o f H t o automorphi c representa tions o f G. On e exampl e woul d be : tak e H t o b e th e grou p consistin g o f a singl e element an d G t o b e GL(2) . The n L H i s a Galoi s grou p an d th e proble m i s tha t of associatin g a n automorphi c for m t o a two-dimensiona l Galoi s representation . xi

Xll

Preface

A partia l solutio n ha s bee n use d b y Andre w Wile s i n hi s proo f o f Fermat' s Las t Theorem. I t ha d bee n though t tha t onl y th e trac e formul a develope d b y Arthu r and other s wa s promisin g fo r th e functorialit y conjecture . Indee d th e trac e for mula metho d ha s bee n successfu l i n som e cases , mos t notably , cycli c bas e chang e of GL(n) du e t o Arthu r an d Clozel . At thi s moment , th e convers e theorem s o f Cogdel l an d Piatetski-Shapiro , i f combined wit h th e Langlands-Shahid i method , provid e severa l instance s o f functo riality. Thes e ar e L-function techniques . Th e convers e theorems determin e whethe r a certai n globa l representatio n o f GL(n) whic h i s just a produc t o f al l loca l ones , is a n automorphi c representation . I n orde r t o us e th e convers e theorem , w e nee d to stud y certai n automorphi c L-functions , namely , w e need t o prov e that thes e au tomorphic L-function s ar e entire , satisf y functiona l equations , an d ar e bounde d i n vertical strips . Ther e ar e tw o way s o f studyin g automorphi c L-functions . The first on e i s called th e metho d o f integra l representations . I t expresse s cer tain automorphi c L-function s a s integral s o f automorphi c forms , ofte n integrate d against Eisenstei n series . Sometimes , i t i s called th e Rankin-Selber g method . Thi s method ha s bee n investigate d b y a larg e numbe r o f mathematicians , goin g bac k to Hecke , Ranki n an d Selberg . However , on e run s int o seriou s difficultie s upo n studying th e metho d a t archimedea n places . Bu t th e theor y i s complete fo r GL(n). This i s th e subjec t o f th e firs t cours e b y J . Cogdell . I n Cours e On e w e presen t a comprehensiv e accoun t o f th e theor y o f L-function s fo r GL(n) vi a integra l rep resentations. W e begi n wit h th e classica l theor y o f Hecke . The n w e tur n t o th e modern theor y o f automorphi c representation s an d thei r L-functions , bot h loca l and global , a s develope d b y Jacquet , Piatetski-Shapir o an d Shalika . W e conclud e with a n expositio n o f the convers e theorem s fo r GL(n) an d thei r applicatio n t o th e question o f globa l functoriality . The secon d metho d o f studyin g automorphi c L-function s i s calle d th e Langlands-Shahidi method . Thi s i s th e subjec t o f th e secon d cours e b y H . Kim . Many automorphi c L-function s appea r a s normalizing factor s o f intertwinin g oper ators i n th e constan t term s o f Eisenstei n series . W e ca n stud y the m wit h th e hel p of many propertie s o f Eisenstein series . Eve n thoug h th e goa l o f this cours e wa s t o explain recen t strikin g result s suc h a s functoriality o f the symmetri c cub e an d sym metric fourt h o f cuspida l representation s o f GL(2) , w e try t o giv e a comprehensiv e account o f th e Langlands-Shahid i method , withou t assumin g an y previou s knowl edge of automorphic representations . Becaus e o f time constraints , man y proof s ha d to b e omitted . I n thos e cases , w e alway s provide d th e reference s fo r th e proofs . In th e thir d cours e b y R . Murty , w e loo k a t th e application s o f th e Langland s functoriality conjectur e t o analyti c numbe r theory , especiall y t o th e Sato-Tat e con jecture, th e Ramanuja n conjecture , th e Selber g eigenvalu e conjecture , Artin' s holo morphy conjectur e an d th e Langland s reciprocit y conjecture . W e emphasiz e ho w the Langland s progra m propose s t o solv e eac h o f thes e conjectures . W e the n ap ply th e recen t wor k o f Ki m an d Shahid i o n symmetri c powe r L-function s t o thes e conjectures a s wel l a s relate d question s i n analyti c numbe r theory . James W . Cogdel l Henry H . Ki m M. Ra m Murt y January 200 4

Lectures o n L-functions , Convers e Theorems , and Functorialit y fo r GL n James W . Cogdel l Department o f Mathematic s Oklahoma Stat e Universit y Stillwater, O K 7407 8 U.S.A .

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http://dx.doi.org/10.1090/fim/020/01

Preface These ar e th e lectur e note s tha t accompanie d my lectur e serie s a t th e Field s Institute i n th e Sprin g o f 200 3 a s par t o f th e Themati c Progra m o n Automorphi c Forms. Th e poste d descriptio n o f th e cours e wa s th e following . "The theor y o f L-function s o f automorphi c form s (o r modula r forms ) vi a in tegral representation s ha s it s origi n i n th e pape r o f Rieman n o n th e zeta-function . However th e theor y wa s reall y develope d i n th e classica l contex t o f L-function s of modula r form s fo r congruenc e subgroup s o f 5L(2 , Z) b y Heck e an d hi s school . Much o f ou r curren t theor y i s a direc t outgrowt h o f Hecke's . L-function s o f auto morphic representation s wer e first develope d b y Jacque t an d Langland s fo r GL(2). Their approac h followe d Heck e combined wit h th e local-globa l technique s o f Tate' s thesis. Th e theor y fo r GL(n) wa s the n develope d alon g th e sam e line s i n a lon g series of papers b y various combination s o f Jacquet, Piatetski-Shapiro , an d Shalika . In additio n t o associatin g a n L-functio n t o a n automorphi c form , Heck e als o gave a criterion fo r a Dirichle t serie s t o com e from a modular form , th e so-calle d Convers e Theorem o f Hecke . I n th e contex t o f automorphi c representations , th e Convers e Theorem fo r GL(2 ) wa s develope d b y Jacque t an d Langlands , extende d an d sig nificantly strengthene d t o GL(3 ) b y Jacquet , Piatetski-Shapiro , an d Shalika , an d then extende d t o GL(n). n "In thes e lecture s w e hop e t o presen t a synopsi s o f thi s wor k an d i n doin g s o present th e paradig m fo r th e analysi s o f genera l automorphi c L-function s vi a inte gral representations . W e wil l begi n wit h th e classica l theor y o f Heck e an d the n a description o f its translation int o automorphi c representation s o f GL(2) b y Jacque t and Langlands . W e wil l the n tur n t o th e theor y o f automorphi c representation s of GL(n), particularl y cuspida l representations . W e wil l first develo p th e Fourie r expansion o f a cus p for m an d presen t result s o n Whittake r model s sinc e thes e ar e essential fo r definin g Euleria n integrals . W e wil l the n develo p integra l represen tations fo r L-function s fo r GL(n) x GL(m) whic h hav e nic e analyti c propertie s (meromorphic continuation , boundednes s i n vertica l strips , functiona l equations ) and hav e Euleria n factorizatio n int o product s o f loca l integrals. " "We nex t tur n t o th e loca l theor y o f L-function s fo r GL(n), i n bot h th e archimedean an d non-archimedea n loca l contexts , whic h come s ou t o f th e Eule r factors o f the globa l integrals. W e finally combine the global Eulerian integral s wit h the definitio n an d analysi s o f th e loca l L-function s t o defin e th e globa l L-functio n of a n automorphi c representatio n an d deriv e thei r majo r analyti c properties. " "We will then tur n t o the variou s Convers e Theorem s fo r GL(n). W e will begi n with th e simpl e inversio n o f th e integra l representation . The n w e wil l sho w ho w to procee d fro m thi s t o th e proo f o f th e basi c Convers e Theorems , thos e requirin g twists b y cuspida l representation s o f GL(m) wit h m a t mos t n — 1 . W e wil l the n 3

4

Preface

discuss how one can reduc e the twistin g to m a t mos t n — 2. Finall y w e will conside r what i s conjecturally tru e abou t th e amoun t o f twistin g necesssar y fo r a Convers e Theorem." "We will end wit h a description o f the application s o f these Convers e Theorem s to ne w case s o f Langland s Functoriality . W e wil l discus s bot h th e basi c paradig m for usin g th e Convers e Theore m t o establis h lifting s t o GL(n) an d th e specific s o f the lift s fro m th e spli t classica l group s SO(2n + 1 ) , SO(2n), an d Sp(2n) t o th e appropriate GL(N)" I have chosen to keep the informal forma t o f the actual lectures; what follow s ar e the texe d version s of the note s that I lectured from . Othe r tha n makin g correction s they remain a s they were when posted weekly on the web to accompany the recorde d lectures. I n particular , I hav e lef t eac h lectur e wit h it s individua l references , bu t there ar e n o citation s withi n th e bod y o f th e notes . Fo r ful l detail s o f th e proofs , many o f which ar e onl y sketche d i n th e note s an d man y other s omitted , th e reade r should consul t th e reference s fo r tha t section . Of course , ther e wil l b e som e overlap wit h othe r survey s I hav e writte n o n thi s subject, particularl y m y PCM I Lectur e note s L-functions and Converse Theorems for GL n. Howeve r ther e ar e severa l lectures , particularl y amon g the earl y one s an d later ones , that appea r i n surve y form, a t leas t b y me, for th e firs t time . I hope thi s more informa l presentatio n o f th e material , i n conjunctio n wit h th e accompanyin g Lectures o f Henr y Ki m an d Ra m Murty , ad d valu e t o thi s contribution . I woul d lik e t o than k th e staf f o f th e Field s Institute , an d particularl y th e program manager s fo r ou r specia l progra m - Aliso n Conwa y an d Soni a Houl e - fo r taking suc h goo d car e of u s durin g th e Themati c Progra m o n Automorphi c Forms . J. W . Cogdell , Stillwater , Oklahoma .

http://dx.doi.org/10.1090/fim/020/02

LECTURE 1

Modular Form s an d Thei r L-function s I want t o begi n b y describing the classical theory of holomorphic modula r form s and thei r L-function s mor e or les s in the term s i n which i t wa s developed b y Hecke. Let f ) = {z — x + iy | y > 0} denote th e uppe r hal f plane . Th e grou p PSX 2 W or PGZ/2~(M ) a c t s o n # b y linea r fractiona l transformation s a b\ c dj

az cz- z =

+b + d'

We wil l b e intereste d i n certai n discret e group s o f motion s V whic h hav e finite volume quotient s T\S). W e wil l conside r tw o mai n examples . 1. The full modular group 5^ 2 (Z). Thi s grou p i s generate d b y th e tw o trans nations T — I( J formations

and 5 = 1 )

. I t ha s th e usua l (closed ) fundamenta l

domain give n b y T={z =

> 1}.

x + iy\=± C satisfyin g (i) [modularity ] fo r eac h 7 = ( ,

E T we hav e th e modula r transformatio n

\;wf(1z) = (cz + d) kf(z); (ii) [regularity ] / i s holomorphic o n S)\ (iii) [growt h condition ] / extend s holomorphicall y t o ever y cus p o f V. Let u s explai n th e conditio n (iii ) fo r th e cus p a t infinity . Th e elemen t T = ) G T an d T generate s th e stabilize r T^ o f th e poin t 0 0 i n T . O n modula r

6

1. Modula r Form s an d Thei r L-function s

forms T ac t a s

f(Tz) = f(z + l) = f{z) so an y modula r for m i s periodi c i n z — i » z + 1 . f(z) the n define s a holomorphi c function o n r ^ f ) whic h ca n b e viewe d a s eithe r a cylinde r o r a puncture d dis k "centered a t oo" . W e ca n tak e a s a loca l paramete r o n thi s dis k D th e paramete r q = q^ =: e 2'Klz. The n z \-> q maps T\9) — > D x = D — {0} . Sinc e / i s holomorphi c o n D x w e can writ e i t i n a Lauren t expansio n i n th e variabl e q: oo

f(z)= Y.

a

^n-

n= — oo

For / t o b e holomorphi c a t th e cus p o o mean s tha t a n = 0 for al l n < 0 , i.e. , oo

oo

f(z) = Y,anq n = n=0

2 inz ne " .

Y,a

n=

0

This expansio n i s called th e Fourier expansion (o r q-expansion) o f f(z) a t th e cus p oo. Ther e i s a simila r expansio n a t an y cusp . A modula r for m i s calle d a cusp form i f i n fac t f(z) vanishe s a t eac h o f th e cusps o f T . I n th e Fourie r expansio n o f f(z) a t th e cus p o o this take s th e for m oo

71=1

Traditionally on e lets M/ C(T) denot e the space of all holomorphic modula r form s of weigh t k fo r T an d £jt(r ) th e subspac e o f cus p forms . I t i s a fundamenta l fac t that th e impose d condition s o n modula r form s ar e stron g enoug h t o giv e a basi c finiteness result . T h e o r e m 1 . 1 dim cMjfe(r) < oo . The proof i n thi s contex t i s essentially a n applicatio n o f Riemann-Roc h t o th e powers o f th e canonica l bundl e o n th e compac t Rieman n surfac e T\fi*. 1 Example s Here ar e som e well known example s o f classical modula r forms . Not e the arith metic natur e o f th e Fourie r coefficient s i n eac h case . 1. Eisenstein series. Le t k > 2 b e a n eve n integer . The n Gk(z)= ]

T (mz

+ n)-

k

(m,n)#(0,0)

is a modula r for m o f weigh t k fo r 6X 2 (Z). I t ha s a Fourie r expansio n / „ >.£

Gk(z) = 2((k) +

,0

0

2 ^- £ a

_ f c1

(n)e 2 "" z

where a r(n) = ^2 d\ndr. Th e normalize d Eisenstei n serie s Ek(z) i s define d t o hav e constant Fourie r coefficien t equa l t o 1 so tha t Gk(z) = 2((k)E

k(z).

7

2. Growt h estimate s o n cus p form s

2. The Discriminant function. oo 1

A(z) = e 2 ™ [ ] ( 1 - e 2 — ) 2 4 = Y728 {E4{Z)3 ~

E(i{z)2)

is the uniqu e cus p o f weigh t 1 2 for Shrift) • It ha s th e Fourie r expansio n 2 A( 2 ) = ^ r ( inyto)C ) e Jl-Kinz

n=l

where r(n ) i s the Ramanuja n r-function . S.Theta series. Le t Q b e a positiv e definit e integra l quadrati c fro m i n 2k variables. The n ®Q(Z) =

J2 e 2k

rh£Z

27riQ{rh)z

= l + ^r Q(n)e27rznz

n=

l

is a modula r for m o f weigh t k fo r a n appropriat e congruenc e grou p T. Her e th e Fourier coefficient s ar e th e representatio n number s fo r Q 2k

rQ(n) = \{meZ

\Q(m)=n}\.

2 Growt h estimate s o n cus p form s As preliminarie s t o th e definitio n o f th e L-functio n w e loo k a t tw o estimate s on cus p forms . S o le t f(z) G Sfc(r). 1. Fro m th e Fourie r expansio n oo

f(z) = Y,ane

2 lnz

*

71=1

we obtai n \f(x + iy)\ 0

/

s

s

d xy

0

f(iy)y'd*y + ik J f{iy)y k A(k-s,f).

k

-'dxy

Note tha t fro m th e rapidl y decreas e o f cusp forms , th e integral s fro m 1 to oo ar e all absolutel y convergen t fo r al l s and bounde d i n vertica l strips . Theorem 1 . 2 The completed L-function A(s,f) is nice i.e., it converges absolutely in a half-plane and (i) extends to an entire function of s, (ii) is bounded in vertical strips, (iii) satisfies the functional equation A(s , /) = i kA(k — 5 , /) Moreover, Heck e wa s abl e to invert th e integra l representatio n (vi a th e Melli n inversion formula ) an d prov e a Converse t o thi s Theorem. Theorem 1 . 3 Suppose D(s) — Y^ ~ 7 i> s absolutely convergent for Re(s) ^ > 0 n=l

and, setting A(s) = (2w)-

s

T(s)D(S),

3. T h e L-functio n o f a cus p for m 9

that A(s) is nice , i.e., satisfies (i)-(iii) in Theorem 1 .2. Then oo

/(z) = ^ a

n

e

2

—

71=1

is a cusp form of weight k for 51 / 2 (Z) . Proof: Th e convergenc e o f the Dirichle t serie s gives a n estimat e o n th e coefficient s of the for m \a n\ < C nc whic h i n tur n give s th e convergenc e an d holomorph y o f f(z) as a functio n o n f) . Recal l tha t SL^i^) i s generated b y th e tw o transformation s

o 1 ) an

d 5=

( r "o 1

By construction w e have f(Tz) = f{z-{- 1 ) = f(z) s o we need t o prov e the transfor mation la w fo r f(z) unde r S. Sinc e we already kno w f(z) i s holomorphic i t suffice s to sho w f(S - iy) — f(i/y) = {iy) k f{iy)- Bu t b y usin g th e Melli n inversio n formul a and th e functiona l equatio n fo r A(s ) w e hav e oo

r

= ^rt„4Wd ^G)*it.,=| A(s) G)^ s " \y) \y Note the n tha t f(z) i s cuspidal fro m it s Fourie r expansion . • For T = To(iV ) th e situatio n i s mor e complicated . Th e functiona l equatio n fo r A(s, / ) no w come s fro m th e actio n o f SN

=

(N

o

which onl y normalize s TQ(N). Howeve r i f f(z) E Sk(To(N)) the n on e ca n sho w that th e functio n g(z) obtaine d fro m th e actio n o f S^ o n f(z), namel y

M-N-w.-'lfe) is als o i n 5/ c (r 0 (A r )) an d th e Melli n transfor m no w lead s t o a functiona l equatio n of th e for m

A(s,f)=ikN^-sA(k-s,g) and that thi s function extend s to a n entire function o f s which i s bounded i n vertica l strips, i.e. , i s nice. The convers e t o thi s resul t i s du e t o Weil . On e varian t o f Weil' s statemen t i s the following .

1. Modi ular Form s> a id r Thei r L-function s

10

oo ,

oo

£

71 s

and D

6e absolutely conver-

2(S) =

n=l

71=1

gent in some right half-plane Re(s) ^ > 0 . For any primitive Dirichlet character \ set

^i(«» x) = 2^ „

anc

(

a

n6

^ ( s , x) = 2 ^

n—1 n— 1

and se £

A i (s,x) = (27r)-«r(s)A(*,x) . Suppose that there exists an N such that for all primitive characters x of conductor q prime to N we have (i) the Ai(s,x) extend to entire functions of s, (ii) the Ai(s,x) are bounded in vertical strips, (iii) we have the functional equation Ai(s,x) = i ke(X)Nl~sA2(k withe(X)

=

Tj

s,x),

^x(N).

Then both oo

oo

f(z) = Y ianeMnz and

g(z)

= £ b

2 inz ne *

n=l n=l

are cusp forms of weight k for TQ(N) and are related by 9{z)

= N-Wz->f ( ^

).

4 Th e Eule r produc t One o f Hecke' s crownin g achievement s wa s t o giv e condition s o n a modula r form f(z) tha t woul d guarante e tha t it s L- function woul d hav e a n Eule r produc t factorization. H e di d thi s vi a wha t ar e no w know n a s th e Hecke operators T n fo r n G N. I n essenc e T n act s o n a modula r for m b y averagin g i t ove r intege r matrice s of determinan t n. To mak e thi s precise , introduc e a weigh t k actio n o f GL ^ (M) o n holomorphi c functions o n 9) by

flk9{z)

= WTWkf{gz) fo

r 5=

€GL

( " d)

iw-

(This i s the actio n o f Sjy tha t w e spoke of without definin g i n the previou s section. ) Then th e conditio n o f modularit y o f weigh t k fo r f(z) wit h respec t t o T become s simply /|fc 7 = / fo r al l 7 e I \ If w e le t £ n

l ( c d)

e M 2

^ \ ad-bc =

n\

and ^ n = i ( n1 )

£ £n\

a

d = n an d 0

then

Cn = I I SL 2(Z)5.

-C

The functio n p(g) o n G(A) whic h w e construc t i n thi s wa y wil l b e a smooth function i n th e followin g sense . I f w e writ e g G G(A) = G ^ • Gf a s g — (#oo>#f ) then ip(g) = (f(g^ , g /) wil l b e G° ° i n th e archimedea n #0 0 variabl e an d locall y constant i n th e non-archimedea n gf variables . Moreove r i t wil l satisfy : (i) [automorphy ] y^g) = ip(g) for al l 7 G G(Q); (ii) [if-finite ] ip(gkokf) = e irne(p(g) fo r fc# G if+> an d A; / G Kf, or , mor e gener ally, th e spac e (ip(gk) | fc G X) i s finite dimensional ; (iii) [Z-finite ] ther e exists an ideal J C Z o f finite co-dimension suc h that J-p> — 0, o r equivalently , th e spac e (X