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Theta Functions : From the Classical to the Modern
 9780821869970, 0821869973

Table of contents :
Cover
Title page
Contents
Preface
Ramanujan's Theory of Theta-Functions, Bruce C. Berndt
1. Definitions
2. Ramanujan's ₁?₁ summation and the Jacobi triple product identity
3. Basic multiplicative and additiveproperties, including Schröter's formulas
4. Quintuple product identity
5. Applications to Lambert series and sums of squares
6. Elliptic integrals
7. Inversion formulas
8. Catalogue of theta-function evaluations
9. Modular equations
10. Eta-function identities
11. Theta-function identities and work of R. J. Evans
12. Ramanujan's Eisenstein series P, Q, and R
13. Ramanujan's theories of elliptic functions to alternative bases
References
Eisenstein Series and Theta Functions on the Metaplectic Group, Jeff Hoffstein
Introduction
1. Eisenstein series and theta functions on the double cover of GL(2)
2. The n-fold cover of GL(2)
3. Eisenstein series and theta functions over function fields
4. Some GL(3) examples
5. Theta functions on the n-fold cover of GL(r)
6. The Group GSp(2n)
References
Weil Representation, Howe Duality, and the Theta Correspondence, Dipendra Prasad
1. Heisenberg group
2. Metaplectic group and the Weil representation
3. Dual reductive pairs
4. Howe duality
5. Howe conjecture in the Archimedean case
6. The spherical case
7. Seesaw pairs
8. The theta correspondence
9. Questions
References
On Theta-Series Liftings for Unitary Groups, Stephen Gelbart
Introductory remarks
1. Weil's representation and theta-series
2. Howe's correspondence and theta-series liftings
3. Specialization to the unitary group U(3)
4. Trace formula results
5. L-functions for U(3)
6. Characterization of endoscopic representations
Appendix. Explicit formulas for the Weil representation restricted to unipotent elements of U(V) X U(Φ')
References
Back Cover

Citation preview

CRM PROCEEDINGS & LECTURE NOTES

Theta Function s From th e Classica l to th e Moder n

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Volume 1

CRM PROCEEDINGS & LECTURE NOTES Centre d e Recherche s Mathematique s Universite d e Montrea l

Theta Function s From th e Classica l to th e Moder n M. Ra m Murty , Editor B. Berndt , S . Gelbart , J . Hoffstein , M. Ra m Murty , an d D . Prasa d

The Centr e d e Recherche s Mathematique s (CRM ) o f l'Universite d e Montrea l wa s create d i n 1 96 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l an d publica tion programs . CR M i s supporte d b y l'Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , an d th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h l'lnstitu t de s Science s Mathematiques d e Montrea l (ISM) , whos e constituen t members ar e Concordi a University , McGil l University , l'Universite d e Montreal , l'Universit e d u Quebe c a Montreal, an d l'Ecol e Polytechnique .

American Mathematical Societ y Providence, Rhode Island US A ^VDED

T h e p r o d u c t i o n o f thi s volum e wa s s u p p o r t e d i n p a r t b y t h e Fond s p o u r l a F o r m a t i o n de Chercheur s e t l'Aid e a l a Recherch e (Fond s F C A R ) a n d t h e N a t u r a l Science s a n d Engineering Researc h Counci l o f C a n a d a ( N S E R C ) . 1991 Mathematics Subject Classification P r i m a r y 1 1 F27 , 1 1 F37 , 11F57, 1 1 F70 ; Secondar y 22E55 , 33D1 0 , 33E25 .

Library o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Theta functions : fro m th e classica l t o th e moder n / M . Ra m Murty , editor . p. cm . - (CR M proceeding s & lectur e notes , ISS N 1 065-8580 ; v . 1 ) Includes bibliographica l references . ISBN 0-821 8-6997- 3 1. Functions , Theta . I . Murty , Marut i Ram . II . Series . QA345.T471 99 3 515 / .984-dc20 93- 500

8 CIP

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 n articl e fo r use i n teachin g o r research . Permissio n i s granted t o quot e brie f passage s fro m thi s publicatio n in 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 publi cation (includin g abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society. Request s fo r suc h permissio n shoul d b e addresse d t o th e Manage r o f Editorial Services , American Mathematica l Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . The appearanc e o f the cod e on th e firs t pag e o f an articl e i n thi s boo k indicate s th e copyrigh t owner's consen t fo r copyin g beyon d tha t permitte d b y Section s 1 0 7 or 1 0 8 of the U.S . Copyrigh t Law, provide d tha t th e fe e o f $1 .0 0 plu s $.2 5 pe r pag e fo r eac h cop y b e pai d directl y t o th e Copyright Clearanc e Center , Inc. , 2 7 Congres s Street , Salem , Massachusett s 01 970 . Thi s consent doe s no t exten d t o othe r kind s o f copying , suc h a s copyin g fo r genera l distribution , fo r advertising o r promotiona l purposes , fo r creatin g ne w collectiv e works , o r fo r resale . Copyright © 1 9 9 3 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 excep t thos e grante d to th e Unite d State s Government . Printed i n th e Unite d State s o f America . The 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 typese t usin g A\zfi-Tf?}{, the America n Mathematica l Society' s T^ X macr o system , and submitte d t o th e America n Mathematica l Societ y i n camera-read y form b y th e Centr e d e Recherche s Mathematiques . 10 9 8 7 6 5 4 3 2 1 9

8 97 96 95 94 93

Contents Preface vi Ram Murty

i

Ramanujan's1 Theor y o f Theta-Function s Bruce C. Berndt 1. Definition s 2. Ramanujan' s i^ i summatio n an d th e Jacob i tripl e produc t identity 4 3. Basi c multiplicativ e an d additiv e properties , includin g Schro ter's formula s 6 4. Quintupl e produc t identit y 9 5. Application s t o Lamber t serie1 s an d sum s o f square s 2 6. Ellipti c integral s 2 1 7. Inversio n formula s 2 4 8. Catalogu e o f theta-functio n evaluation s 2 7 9. Modula r equation s 2 9 10. Eta-functio n identitie s 3 7 11. Theta-functio n identitie s an d wor k o f R . J . Evan s 4 0 12. Ramanujan' s Eisenstei n serie s P , Q , an d R 4 3 13. Ramanujan' s theorie s o f elliptic function s t o alternativ e base s 5 1 References 6 1 Eisenstein Serie s an d Thet a Function s o n th e Metaplecti c Group 6 Jeff Hoffstein Introduction 6 1. Eisenstei n serie s an d thet a function s o n th e doubl e cove r o f GL(2) 6 2. Th e n-fol d cove r o f GL{2) 7 3. Eisenstei n serie s an d thet a function s ove r functio n field s 7 4. Som e GL(3) example s 8 5. Thet a function s o n th e n-fol d cove r o f GL(r) 9 6. Th e grou p GSp{2n) 9 References 0 Weil Representation , How e Duality , an d th e Thet a Cor respondence 0 Dipendra Prasad 1. Heisenber g grou p 0 2. Metaplecti c grou p an1 d th e Wei l representatio n 0 3. Dua l reductiv e pair s 4. How e dualit y

5 5 7 3 8 5 2 8 3 5 5 7 0 2

CONTENTS

5. How e conjectur e1 i1 n th e Archimedea n cas e 6 6. Th e spherica l cas e 9 7. Seesa w pair s 9 1 8. Th e thet a correspondenc e 2 1 9. Question s 2 5 References 2 6 On Theta-Serie s Lifting s fo r1 Unitar y Group s 2 9 Stephen Gelbart Introductory remark s 2 9 1. Weil' s representatio 1 n an d theta-serie s 3 0 2. Howe' s correspondenc e an 1 d theta-serie s lifting s 3 6 3. Specializatio n1 t o th e unitar y grou p U(3) 4 5 4. Trac e formul a result s 5 6 5. L-function s fo r £7(3 ) 6 1 6. Characterizatio n o f endoscopi 1 c representation s 6 4 Appendix 6 9 References 7 2

PREFACE R a m Murt y

Theta function s pervad e al l o f mathematic s rangin g fro m th e theor y o f partia l differential equations , mathematica l physics , t o algebrai c geometry , numbe r theor y and mor e recentl y t o representatio n theory . I t i s th e them e represente d b y th e last tw o discipline s tha t i s th e concer n o f thi s volume . Th e lecture s represen t th e content o f four course s given at th e Centr e d e Recherches Mathematiques , Montrea l during th e academi c yea r 1 991 -9 2 devote d t o th e stud y o f automorphi c forms . In numbe r theory , th e classica l thet a functio n oo

e(z)= Yl

e

^iv?z

n = —oo

made it s appearanc e i n a t leas t tw o ways . First , i t wa s use d t o determin e exac t formulas fo r th e numbe r o f representation s o f a n intege r a s a su m o f r squares . Second, it was fundamental i n Riemann's derivation o f the functional equatio n of the ^-function. I n al l instances, th e centra l propert y i s the modula r transformatio n la w satisfied b y th e thet a function . Thi s classica l them e a s embodie d b y Ramanujan' s theory o f thet a function s i s develope d i n th e lecture s o f Bruc e Berndt . The theor y o f integra l weigh t modula r form s wa s firs t derive d b y E . Hecke . He wa s successfu l i n associatin g a n L-functio n t o a modula r for m an d usin g th e modular transformatio n propert y t o deriv e a n analyti c continuatio n an d functiona l equation fo r it , i n muc h th e sam e spiri t a s Rieman n ha d don e wit h th e 0 an d £ function. However , i t shoul d b e stresse d tha t Riemann' s derivatio n i s not a specia l case o f th e Heck e theory . Thi s i s because th e ^-functio n i s a modula r for m o f half integral weight . Heck e clearl y kne w o f th e difficultie s i n developin g hi s theor y t o include th e ^-functio n becaus e h e deal t wit h onl y a specia l cas e i n hi s las t pape r written i n 1 944 . Twenty year s later , Wei l suggeste d tha t th e classica l 0-functio n shoul d b e nat urally regarde d a s a n automorphi c form , no t o n th e uppe r half-plan e (a s i n th e case o f integra l weigh t modula r forms) , bu t rathe r o n a 2-fol d cove r o f it . Viewin g the uppe r an d lowe r half-plane s a s a quotien t o f G = GI/2(M) , Wei l considere d a certain non-trivia l extensio n G o f group s 1 - > {±1 } - > G - > G -* 1 , and th e ^-functio n a s a n automorphi c for m o n G . Thi s wa s th e beginnin g o f th e representation theoreti c poin t o f vie w whic h ha s sinc e le d t o deepe r insights . Th e initial difficultie s face d b y Heck e i n 1 94 4 wer e resolve d b y Shimur a i n 1 973 . I n hi s 1991 Mathematics Subject

Classification. Primary : 1 1 F27 ; Secondary : 1 1 F37 .

vii

viii R A

M MURT Y

important Annals paper , h e derive d a systemati c theor y o f modula r form s o f half integral weigh t whic h matche d th e eleganc e an d th e beaut y o f th e classica l Heck e theory. Shimura' s theor y le d t o a remarkabl e correspondenc e betwee n modula r forms o f half-integra l weigh t an d form s o f integra l weight . A specia l cas e o f thi s correspondence wil l serv e t o revea l th e brillianc e o f thi s discovery . Let k b e a positiv e intege r an d g(z) a cus p for m o f weigh t k + 1 / 2 o n ro(4AT) . Suppose g{z) ha s a Fourie r expansio n o f th e typ e 9(z)

= j2 71=1

k

with c(n) = 0 unless ( — l) n = 0 or 1

(mo d 4) . Defin e

a(n) = J2d

1 k

- c(n2/d2).

d\n

Then

oo

/(z) = £>(n)e 2™" n=l

is a cus p for m o f weigh t 2k o n TQ(N). Moreover of Waldspurger : fo r squarefre e Z) ,

n = lx

, ther e i s th e astoundin g formul a

/

where Q is a certai n non-vanishin g transcendenta l facto r an d ( ^ ) i s the Kronecke r symbol. Apar t fro m it s intrinsi c beauty , thi s formul a i s o f arithmeti c interes t fo r at leas t tw o reasons . Firstly , th e famou s Birc h an d Swinnerton-Dye r conjecture s relate th e orde r o f vanishin g o f th e L-functio n o f a n ellipti c curv e a t s = 1 to th e rank o f th e Mordell-Wei l group . I f th e ellipti c curv e come s fro m a modula r for m / (as is conjectured b y Taniyama fo r al l curves over Q), then th e abov e formula state s that th e curv e ha s infinitel y man y rationa l point s i f an d onl y i f c(l ) = 0 , wher e c(D) i s th e D-t h Fourie r coefficien t o f th e Shimur a corresponden t o f / . Secondly , the formul a re-interpret s th e modula r analogu e o f the classica l Lindelo f hypothesi s as th e Ramanuja n conjectur e fo r cus p form s o f half-integra l weight . Though Shimur a derive d hi s theor y alon g classica l lines , i t wa s the representa tion theoreti c poin t o f vie w tha t reveale d highe r generalization s o f th e ^-function . Instead o f considerin g a 2-fol d cove r o f GL2 , on e ca n conside r th e n-fol d cove r o f GL2 an d stud y automorphi c form s o n it . Thi s wa s don e b y Kubot a an d i t le d hi m to th e discover y o f generalize d thet a functions . Suc h a discover y wa s no t withou t its implication s t o classica l problems . Indeed , Patterso n an d Heath-Brow n utilise d the "cubic " thet a functio n (tha t is , the Kubot a thet a functio n whe n n = 3 ) i n con junction wit h method s fro m analyti c numbe r theor y t o sho w tha t th e argument s o f cubic Gaus s sum s ar e uniforml y distributed , thu s disprovin g a n ol d conjectur e o f Kummer. More generally , le t F b e a globa l field an d A it s adel e ring . Suppos e th e n-t h roots o f unity fjL n(F) lie in F an d n i s coprime t o th e characteristi c o f F. Then , th e n-fold cove r o f G = GL r i s define d a s a certai n non-trivia l extensio n o f group s l^/Xn(F)-GA->GA-l.

PREFACE i

x

GA i s calle d th e metaplecti c grou p an d a n automorphi c for m o n G A is calle d a metaplectic form . Thi s concep t wa s firs t derive d b y Matsumot o an d C.C . Moor e (independently). Furthe r development s appea r i n a fundamental pape r b y Kazhda n and Patterson . Th e lecture s o f Jef f Hoffstei n describ e thi s theor y an d ho w th e generalized thet a function s appea r a s residue s o f Eisenstei n series . From thi s poin t o f view , w e ca n as k fo r a n explanatio n o f th e Shimur a cor respondence. Mor e generally , ar e ther e othe r correspondence s tha t aris e i n thi s way. Th e prope r contex t fo r understandin g th e Shimur a correspondenc e seem s t o be Howe' s theor y o f dua l reductiv e pairs . Onc e described , th e theor y i n tur n lead s to mor e conjectura l correspondences . Perhap s som e specia l case s o f thes e corre spondences ar e accessibl e b y existin g methods . Th e lecture s o f Dipendr a Prasa d elucidate thes e ideas . As i s well-known , th e celebrate d "Langland s program " mad e a conceptua l breakthrough b y introducin g automorphi c L-function s an d relatin g the m (conjec turally) t o non-abelia n reciprocit y laws , clas s field theor y an d th e Hasse-Wei l L functions attache d t o algebrai c varieties . However , th e progra m wa s constructe d and develope d i n th e contex t o f algebrai c groups . Th e metaplecti c grou p i s not a n algebraic grou p an d s o th e theor y o f thet a function s an d it s generalization s doe s not naturall y fit int o th e "Langland s program" . Th e lin k betwee n thes e theorie s is the Shimur a correspondence , o r mor e generally , Howe' s theor y o f dua l reductiv e pairs an d th e theor y o f theta-series liftings . Stephe n Gelbart' s lecture s ar e devote d to thi s them e wit h specia l emphasi s o n form s o f t/(3) . There ma y b e a littl e overla p betwee n th e lecture s o f Prasa d an d Gelbart . W e have no t trie d t o remov e thi s overlap , partl y becaus e o f pedagogica l reasons , an d partly becaus e each autho r ha s his special point o f view and manne r o f presentatio n that convey s differen t aspect s o f thi s ric h theory . Besides , i t di d no t see m correc t to tampe r wit h work s o f art . (W e have , o f course , correcte d som e typographica l errors o f the artists. ) It i s becomin g increasingl y clea r tha t thet a function s wil l hav e a significan t role i n th e Langland s program . Perhap s the y wil l b e instrumenta l i n solvin g th e Ramanujan conjectur e fo r Maas s forms . Indeed , i f TT i s a cuspida l automorphi c representation, Langland s outline d i n 1 96 7 ho w knowledg e o f th e analyti c contin uation an d functiona l equatio n o f th e L-function s attache d t o Sym fc(7r) fo r al l k lead t o no t onl y th e Ramanuja n conjectur e bu t als o t o th e celebrate d eigenvalu e conjecture o f Selberg . I t i s well-known (largel y du e t o th e wor k o f Henryk Iwaniec ) that bot h o f thes e conjecture s hav e implication s t o problem s i n analyti c numbe r theory. I n th e cas e k = 2 an d 7 r a classica l modula r form , Shimur a showe d tha t the symmetri c squar e L-functio n i s reall y th e Rankin-Selber g convolutio n o f th e modular for m wit h th e classica l thet a function . Shahid i wa s successful i n obtainin g the meromorphi c continuatio n o f thes e symmetri c powe r L-functions , bu t onl y fo r k < 5 and hi s metho d doe s no t see m t o b e capabl e o f generalization . However, th e method o f Shimura does seem capable of generalization. B y mean s of converse theory, Gelbar t an d Jacque t i n fact showe d tha t Sym 2(7r) i s an automor phic for m o n GL 3 a s predicte d b y th e Langland s program . Meanwhile , Patterso n and Piatetski-Shapir o note d tha t Shimura' s metho d ca n b e generalize d t o GL3 . This means , i n particular , tha t on e ca n conside r Sym 2(Sym2(7r)) an d on e ca n ob tain a n analyti c continuatio n an d functiona l equatio n attache d t o thi s object . Thi s would giv e a meromorphi c continuatio n o f th e symmetri c fourt h powe r L-function .

x RA

M MURT Y

If again , b y convers e theory , thi s ne w objec t ca n b e show n t o b e automorphic , (a s predicted b y th e Langland s program ) the n th e proces s ca n b e repeated , provide d of cours e w e kne w tha t th e symmetri c squar e //-functio n o f a n arbitrar y automor phic for m o n GL n ha s analyti c continuatio n an d functiona l equation . I n fact , i n a recen t pape r i n th e Annals, Bum p an d Ginzbur g derive d th e latte r resul t usin g the theor y o f generalize d thet a functions . Invokin g a n analyti c metho d o f Duk e and Iwaniec , thi s lead s t o significan t progres s toward s th e Ramanuja n conjectur e for Maas s forms . Thi s give s som e ide a o f th e importan t rol e t o b e playe d b y thi s emerging theory . I woul d lik e t o expres s m y gratitud e t o Franci s Clark e an d th e CR M fo r thei r financial suppor t fo r th e specia l yea r an d t o Sylvi e Chenever t an d Jacque s Blai s for co-ordinatin g th e program . I woul d als o lik e t o than k Tar a Ashtakala , Masat o Kuwata, Lie m Ma i an d Rub y Musri e fo r thei r T^Xpertise . I t i s hope d tha t thi s volume wil l b e beneficia l t o graduat e student s an d professiona l mathematician s i n acquainting the m wit h th e richnes s o f th e theor y o f thet a functions . Montreal, 1 99 2 D E P A R T M E N T O F MATHEMATICS , M C G I L L UNIVERSITY , 80 5 S H E R B R O O K E W E S T , M O N T REAL, Q U E B E C H3 A 2K

6

E-mail address: [email protected] a

Centre d e Recherches Math^matique s CRM Proceeding s an d Lecture Note s Volume 1 , 1 99 3

Ramanujan's Theor y o f Theta-Function s Bruce C. Berndt Next i n order yo u certainly ough t On function-theor y besto w you r thought , And penetrat e wit h contemplatio n What resist s you r attempt s a t integration . You'll find n o dearth o f theorems ther e — To vanishing-points giv e prope r car e — Enumerate, reciprocate , Nor forge t t o delineate, Traverse th e plane fro m en d to end, And theta-function s freel y spend . Mephistopheles It i s presumptuous t o title thes e lectur e note s "Ramanujan' s Theor y of ThetaFunctions"; Ramanuja n ha d several theories , an d although w e can often surmis e the argument s tha t Ramanuja n migh t hav e used , i n mos t instances , w e do not know Ramanujan' s methods . Mos t o f the result s offere d i n the sequel ar e found in Ramanujan's notebook s [73] , in particular, i n Chapters 1 6-2 1 and among th e 1 0 0 unorganized page s in his second notebook . Th e proof s tha t w e have chosen t o give below ar e either shor t o r elegant, an d in some case s mor e detail s coul d hav e bee n given. Fo r the proofs tha t ar e not given or presented i n sufficient detail , w e always indicate wher e reader s ma y find complete proofs . 1. Definition s The classical theta-functions # n , 1 < n < 4, are usually defined b y [83, pp. 463464] oo

J (_l)n g (n+l/2) 2 e (2n + lH Z

^{Z,q):=-i V

n= — oo oo

= 2 ^ ( - l ) Vn + 1/ 2 ) 2 s m {(2n + l)z) , n=0 oo o

M*,q)~ E