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

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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