Automorphic Forms on Adele Groups. (AM-83), Volume 83 9781400881611

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Annals of Mathematics Studies Number 83

AUTOMORPHIC FORMS ON A D ELE GROUPS

BY

STEPHEN S. GELBART

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

C o p y rig h t © 1 9 7 5 , by P rin ce to n U n iv ersity P re ss A LL RIG H TS R E SE R V E D

P rinted in the U n ited S tates o f A m e ric a

L IB R A R Y O F CO N G RESS CATALOGING IN PUBLICATIO N DATA

G e lb a rt, S. A u to m o rp h ic fo rm s on A d e le group s. (A n n a ls o f m a th e m a tics stu d ies; no. 8 3 ) “ E xp an d ed from notes m im eog rap h ed at C o rn ell in M a y o f 1 9 7 2 and entitled A u to m o rp h ic form s and re p resen tatio n s o f A d e le g ro u p s.” B ib lio g ra p h y : p. In clu d es in d ex. 1.

R e p re se n ta tio n s o f grou p s.

2 . A u to m o ip h ic form s.

3 . L in e a r a lg e b ra ic grou p s. I. Q A 1 7 1 .G 3 9

T itle . 1975

II. S e rie s. 512\22

7 4 -2 3 3 8 8

IS B N 0 - 6 9 1 - 0 8 1 5 6 - 5 3 5 7 9

10

8 6 4 2

T o my fath er, Abe G elbart

PREFACE

S e ctio n s 1 through 7 of th e se N otes are b ased on le c tu re s I gav e at C o rn ell U n iv ersity in the Spring of 1 9 7 2 .

They are expanded from N otes

mimeographed a t C orn ell in May of 1 9 7 2 and en titled A utom orphic Forms a n d R e p re s e n ta tio n s of A d e le G roups.

1 am gratefu l to E . M. Stein for s u g ­

g estin g th at I expand th o se N otes for p ub lication by the P rin ce to n U ni­ v e rsity P r e s s and th at I in co rp orate into them the m aterial of S e ctio n s 8 through 10.

T h e s e la s t th ree s e c tio n s are b ased on le c tu re s I gav e at the

In stitu te for A dvanced Study, P rin ce to n , in the Spring of 1 9 7 3 .

I am in­

debted to the In stitu te for its h o sp ita lity a s w ell a s for the atm osphere it c re a te d for s e rio u s work. Th e s u b je c t m atter of th e se N otes is the in terp lay betw een the theory of autom orphic forms and group re p re se n ta tio n s.

One go al is to in terp ret

som e re ce n t developm ents in th is a re a , m ost sig n ifica n tly the theory of Ja c q u e t-L a n g la n d s , working out, w henever p o s sib le , e x p lic it co n se q u e n c e s and c o n n e ctio n s with the c l a s s i c a l theory.

Another go al is to c o lle c t a s

much inform ation a s p o ssib le co n cern in g the d ecom p osition of L 2 (G L (2 ,Q )\ G L (2,A (Q )).

Although e a c h p articu lar s e c tio n h as its own introduction

d escrib in g the m aterial co v ered I would like to add the follow ing orienting rem arks to th is P r e f a c e . S e ctio n s 1 through 5 are prelim inary in nature and th eir purpose is to sp e ll out the e x p lic it re la tio n s betw een c l a s s i c a l cu sp forms and c e rta in irredu cib le c o n stitu e n ts of L 2 (G L (2 ,Q )\ G L (2 ,A (Q )).

H ere I c o ll e c t only

th o se fa c ts from rep resen tatio n theory and the c l a s s i c a l theory of forms which are c ru c ia l to the se q u e l.

P a r ts of th e se s e c tio n s are eith er new or

part of the s u b je c t ’s “ fo lk lo re .”

R e fe re n ce s to the e x is tin g literatu re are

to be found in the “ N otes and R e f e r e n c e s ” at the end of e a c h s e c tio n and individual ackn ow led gem ents are made w henever p o ssib le . vii

viii

PR EFA C E

S ectio n s 6 through 10 d eal with Ja c q u e t-L a n g la n d ’s theory and som e important q uestion s related to it.

S ectio n 8 d e sc rib e s the continuous

spectru m of L 2 (G L (2 ,Q )\ G L (2 ,A )) and is perhaps the le a s t se lf-co n ta in e d . T h e rem aining s e c t io n s , including S ectio n 9 on the tra c e form ula, co n cern the d is c re te spectru m .

I have included a com p lete proof of the tra c e

formula for G L (2 ) prim arily b e c a u s e the id eas involved here are s t il l not w ell known.

I a ls o w anted there to be no doubt in the re a d e r’s mind th at

the proof of Ja c q u e t-L a n g la n d s ’ Theorem 1 0 .5 is now com p lete.

In w riting •

S ectio n 9 I h ave followed J . G. A rthur’s as yet unpublished m anuscript on the tra c e formula for rank one groups and I w ish to thank him for allow ing me to do s o . S cattered throughout th e se N otes are som e new re su lts and proofs w hich I have not d escrib ed elsew h ere.

I am indebted to my c o lle a g u e s at

C o rn ell, in p articu lar K. S. Brow n, W. H. J . F u c h s , A. W. Knapp, S. L ich te n baum, 0 . S. R o th au s, R. Stanton, H. C . Wang, and W. C . W aterhouse, for help and en couragem en t, and to J . G. Arthur, P . C a rtier, W. C a s se lm a n , R. Howe, R. H otta, M. K arel, R . P . L a n g la n d s, R . P a rth a sa ra th y , P . J . S ally , J r ., and T . Shintani, for helpful c o n v e rs a tio n s and co rresp o n d en ce related to the re su lts d escrib ed here.

I e s p e c ia lly w ish to thank R. P . L an g lan d s for

much valu ab le information and in sp iration . T he first typing of th e se N otes w as done a t C orn ell by E sth e r Monroe, D olores P en d ell and Ruth H ym es. w as greatly ap p reciated .

ITHACA DECEMBER 1973

T h eir unusual e fficie n cy and e x p e rtis e

CO N TEN TS

P R E F A C E ....................................................................................................................................

v

§1.

...............................................................................

3

E lem en tary N otions .......................................................................................... E xam p le s ......................................... H e c k e ’s T h e o r y .................................................................................................... Com plem ents to H e c k e ’s T h eory ............................................................. N otes and R e fe re n c e s ......................................................................................

3 9 12 17 20

T H E C L A S S IC A L T H E O R Y A. B. C. D.

§2.

AUTOMORPHIC FORMS AND T H E DECOMPOSITION O F L 2( r \ S L ( 2 ,R ) A. B. C.

§3.

AUTOM ORPHIC FORMS AS FU N CTIO N S ON T H E A D E L E G ROUP O F G L (2 ) ..................................................................................................... A. B. C.

§4.

22

40

B a s ic N otions ....................................................................................................... H eck e O p erators .................................................................................................. A rbitrary B a s e F ie ld s ..................................................................................... N otes and R e fe re n c e s ......................................................................................

40 47 50 52

T H E R E P R E S E N T A T IO N S O F G L (2 ) O V ER L O C A L AND G L O B A L F IE L D S ........................................................................................................

54

A. B.

C. §5.

........................................................................................................

Autom orphic Fo rm s a s F u n ctio n s onS L ( 2 ,R ) .................................. 22 Automorphic F o rm s and the D ecom p osition of L 2 (P \ G ) .......... 30 Some M iscellan eo u s R e s u lts C on cern in g the D ecom p osition of L 2 ( F \ S L ( 2 ,R ) ... ........................................................... 37 N otes and R e fe re n ce s ....................................................................................... 39

T h e A rchim edean P l a c e s ............................................................................. T h e p -ad ic T h eory ............................................................................................. 1. A d m issib ility ................................................................................................ 2. C la s s if ic a tio n of A dm issible R e p re s e n ta tio n s ......................... 3 . Some P ro p e rtie s of Irred u cib le A dm issible R e p re s e n ta tio n s .............................................................................................. G lobal T h eory ....................................................................................................... N otes and R e fe re n c e s ......................................................................................

C U S P FORMS AND R E P R E S E N T A T IO N S O F T H E A D E L E G RO U P O F G L (2 ) ......................................................................... A. B. C.

P relim in ary R e s u lts on the D ecom p osition of R ^ (g ) ................... C u sp Form s and H eck e O perators R e v isite d ..................................... Some E x p lic it F e a tu r e s of the C orresp on d en ce B etw een C usp Fo rm s and R e p r e s e n ta tio n s .............................................................. N otes and R e fe re n ce s ...................................................................................... ix

54 60 60 65 71 75 77 79 80 36 92 96

CONTENTS

X

§6.

H E C K E T H E O R Y FO R A. B. C. D.

§7.

C. D.

T h e Weil R ep resen tatio n .............................................................................. T he C o n stru ctio n of C ertain S p e cia l R ep resen tatio n s of G L (2 , A ) ............................................................................................................ An E x p lic it E xam p le ...................................................................................... C o nn ection s with C la s s F ie ld Theory ............................................... N otes and R e fe re n ce s .....................................................................................

134 143 151 154 1 59

EIS EN S T E IN S E R IE S AND T H E CONTINUOUS S P E C T R U M ......... 161 A. B. C. D. E.

§9.

98 99 105 108 121 130

T H E CO NSTRUCTION O F A S P E C IA L CLA SS OF AUTOMORPHIC FORMS ........................................................................................ 133 A. B.

§8.

G L (2 ) ..........................................................................

H ecke T heory for G L ( 1 ) .............................................................................. Fu rth er M otivation ........................................................................................... Ja c q u e t-L a n g la n d s ’ T h e o ry ......................................................................... C o n n ectio n s with the C la s s i c a l Theory ............................................ N otes and R e fe re n ce s .....................................................................................

Some P r e lim in a rie s ........................................................................................... A n aly sis of C ertain Induced R ep resen tatio n s ............................... E is e n s te in S eries .............................................................................................. D escrip tion of the Continuous Spectrum ............................................ Summing Up ........................................................................................................... N otes and R e fe re n ce s .....................................................................................

162 165 168 173 178 179

T H E T R A C E FO RM ULA FO R G L (2 ) ........................................................... A. M otivation ............................................................................................................. 1. T h e R e a l Situation .................................................................................. 2. The C a s e of Com pact Q u o tien t......................................................... 3. The Situation for G L (2 ) ......................................................................

181 181 181 183 186

B. C.

T h e T ra c e of R j( f ) ...........................................................................................1 88 A Second Form of the T ra c e Form ula ................................................ 195 1. C onju gacy C la s s e s in G q ................................................................ 196 2. T ru n catin g K i ( x ,x ) and r((x ,x ) .................................................... 197 3. P lan of A t t a c k ............................................................................................. 2 0 0 4. T h e E llip tic and Singular Term s ................................................... 201 5. T h e F i r s t P a ra b o lic Term ..................................................................... 2 03 6. T h e Second P a ra b o lic Term ................................................................ 2 1 0 7. T h e Third P a ra b o lic Term ................................................................ 21 4 8. F in a l Form of the T ra c e Form ula ................................................. 2 1 8 N otes and R e fe re n ce s .......................................................................................2 2 4

§ 1 0 . AUTOMORPHIC FORMS ON A QUATERNION A L G E B R A .............. 2 2 7 A. P re lim in a rie s ........................................................................................................ 2 2 9 B. Statem ent and P roof of the Fu n dam ental R e su lt ......................... 2 3 4 C . C o n stru ctio n of Some S p e cia l Automorphic Form s in the C a s e of C om pact Q u o tie n t............................................................................. 2 46 D. T h eta S eries A ttach ed to Q uaternary Q uadratic F o r m s 251 1. Weil R e p re se n ta tio n s and T h eta S eries .................................... 2 5 2 2. D ecom p osition of the Weil R e p r e s e n ta tio n ............................... 25 3 3. A pp lication to th e B a s is Problem ................................................... 2 5 6 N otes and R e fe re n ce s ..................................................................................... 2 5 9 BIB L IO G R A P H Y .................................................................................................................... 2 6 0 IND EX ............................................................................................................................................2 6 4

Automorphic Forms on Adele Groups

§1.

T H E C L A S S IC A L T H E O R Y

T h is s e c tio n d e s c rib e s variou s a s p e c ts of H e c k e ’s th eory of D irich let s e r ie s atta ch e d to cu sp forms and som e re ce n t refinem ents of it due to Weil and A tk in -L eh n er.

T h e se re s u lts from the c l a s s i c a l theory of a u to ­

morphic forms play a c ru c ia l role in the modern theory.

S in ce we in clu de

them primarily to provide a co n v en ien t c l a s s i c a l referen ce for our d is c u s ­ sion of J a c q u e t-L a n g la n d s ’ theory no attem pt at c o m p leten ess is made. A.

E lem en tary N otions Throughout th is s e c tio n we s h a ll be d ealin g with non co -co m p a ct a rith ­

m etic subgroups of S L (2 ,R ).

(T h e c a s e of com pa ct fundam ental domain

w ill be con sid ered in S ectio n 1 0 .)

In f a c t,

F

gruen ce subgroup, i .e ., a subgroup of S L (2 ,Z )

w ill u su ally d enote a c o n ­ which co n tain s the h o m o g e ­

n eo u s p rin cip a l c o n g r u e n c e su b g ro u p

for som e p o sitiv e in teger

N.

Im portant exam p les are

S L ( 2 ,Z )

(the 4‘ full

co n gru en ce su b g ro u p ,’ ’ or ‘ ‘co n g ru en ce subgroup of le v e l 1 ” ) and “ H e c k e ’s sub grou p ”

By

G L + ( 2 ,R ) we s h a ll d enote the group of real

p o sitiv e determ inant. ilm (z) > 01,

and

If g =

2x2

m atrices with

b elon gs to G L +(2 ,R ),

k is a p o sitiv e in teg er, we s e t

(l.D

3

z

to

4

AUTOMORPHIC FORMS ON ADELE GROUPS

(1 .2 )

j(g ,z ) = (c z + d )(d e t g)

1/2 ,

and

(1-3)

fkg]|/z) = f(ez)Kg,z)_k -

T h is la s t formula d efin es an operator on the s p a c e of a ll com p lex-valu ed functions

z e llm (z) > Oi.

f(z ),

z l f z2

Tw o points

w ill be c a lle d eq u iv a len t un der T

for som e y t l \

if y z j = z 2 dom ain for V

if F

A s u b se t

are 1 '-eq u iv alen t and e a c h point of

is T -eq u iv alen t to som e point of the c lo s u re of F .

A point s elem en t of F c u sp s of r

of flm (z) > Oi is a fundam ental

is a co n n ected open su b se t of ilm (z) > Ol with the

property th at no two points of F flm (z) > Oi

F

(or r-e q t/iv a /e /if )

in R U i°oj fixing s .

then F

is a c u s p of F

If H

if there e x is ts a pa ra bo lic

d en o tes the union of ilm (z )> Oi and the

a ls o a c ts on H ; the resu ltin g quotient s p a c e

p o s s e s s e s a natural (H ausdorff) topology and a com plex s tru ctu re su ch . ^ that r \ H is a co m p a ct R iem ann s u rfa c e . The cu s p s we s h a ll co n sid er may be taken as various ratio n al points on the real a x is and . Most authors denote the cu sp at 00 by em p h asize that as

z = x + iy ap p roach es the cu sp in F ,

x

i°° to

is bounded,

and y > 00. In g en eral, if T

is an arb itrary d is c re te subgroup of S L (2 ,R ),

ca lle d a F u c h s ia n gro up of the firs t kin d if r \ H

is co m p act.

V

is

A ll

F u c h s ia n groups, and F Q(N) in p articu lar, have (at m ost) a fin ite number of T -in eq u iv alen t c u s p s . The follow ing definition is valid for F

an arb itrary F u c h s ia n group

of the first kind. DEFINITION 1 .1 .

A com p lex-valu ed function

f(z)

is c a lle d a F -autom or­

p h ic form of w eigh t k ( or an autom orphic form of w eigh t is defined in tlm (z) > 0!

k for F )

and s a tis f i e s the follow ing con d itio n s:

if it

§1. THE CLASSICAL THEORY

(i)

f|[y]k = f,

5

i.e .

'(ffra) ■(cz ,d)k[fe) for all

y = jj!

jjj e 1';

th is is the “ automorphy co n d itio n ”

for f; (ii)

f is holom orphic in jlm (z) > Oi;

(iii)

f is holom orphic at every c u sp of F .

T he s p a c e of su ch functions w ill be denoted

and

Mj^ F ) .

F o r con gru en ce sub grou ps, elem en ts of M ^(F) are often c a lle d modular form s (or m odular form s of le v e l

N if F = TfN )).

If s

Im(o-(z)) = Im(z)|j(cr,z)|- 2 ).

T h erefo re, if f f Sk( r ) , g(w ) -» 0 a s * (with re s p e ct to the topology of H ). T h is means th at g is a c o n ­

tinuous function on the com pa ct s p a c e v e rse ly , if

|g(z)| < M, fg ( 0

F\H

and h ence

|g(z)| < M.

must be holomorphic at C = 0,

C on­

and in fa c t it

must van ish there. □ oo

C o ro lla ry

1 .6 .

If f(z) =

1

a

e 27/inz f Sk( D ,

th en

n= 1

( 1 .1 0 )

Pro of.

a n = 0 (n k / 2 ) .

S in ce

f^OD =

^

an £ n