Selected Title s i n Thi s Serie s

93 Gor o Shirnura , Eule r product s an d Eisenstei n series , 199 7

92 Fa n R . K . Chung , Spectra l grap h theory , 199 7

91 J . P . Ma y e t a l . , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e

memory o f Rober t J . Piacenza , 199 6

90 Joh n Roe , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J imb o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 Hug h L . Montgomery , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d

harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3

81 S teve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjor n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d th e

study o f functio n spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . T h o m a s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y D y m , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9

70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7

68 J . Wil l ia m Hel ton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics ,

1987

66 G . Andrews , g-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physic s an d compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6

64 Donal d S . Passman , Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6

62 Bel a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r How e an d Al le n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s t o

invariant subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n operato r algebras , 198 4

{Continued in the back of this publication)

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Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Serie s in Mathematic s

Number 9 3

Euler Products an d Eisenstein Serie s

Goro Shimur a

Published fo r th e Conference Boar d of the Mathematica l Science s

by th e American Mathematica l Societ y

^ Providence , Rhod e Islan d with suppor t fro m th e

^^^" Nationa l Scienc e Foundatio n °**D^ X°

http://dx.doi.org/10.1090/cbms/093

CBMS Conferenc e o n Eule r Product s and Eisenstei n Serie s

held a t Texa s Christia n Universit y May 19-24 , 199 6

Research partiall y supporte d b y th e National Scienc e Foundatio n

1991 Mathematics Subject Classification. Primar y l lFxx , l lExx , 20Gxx .

Library o f Congres s Cataloging- in-Publicat io n D a t a

Shimura, Goro , 1930 -Euler product s an d Eisenstei n serie s / Gor o Shimura .

p. cm . — (Conferenc e Boar d o f th e Mathematica l Science s regiona l conferenc e serie s i n mathematics, ISS N 0160-764 2 ; no. 93 )

Includes bibliographica l reference s an d index . ISBN 0-8218-0574- 6 1. Eule r products . 2 . Eisenstei n series . I . Title . II . Series : Regiona l conferenc e serie s i n

mathematics ; no. 93 . QA1.R33 no . 9 3 [QA243] 510 s—dc21 [515/.243] 97-812 9

CIP

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10 9 8 7 6 5 4 3 2 1 0 2 0 1 0 0 9 9 9 8 9 7

TABLE O F CONTENT S

Preface vi i

Notation i x

Frequently use d Symbol s x i

Introduction xii i

Chapter I . Algebrai c an d Loca l Theorie s o f Generalize d Unitary Group s 1 1. Elementar y propertie s o f hermitia n form s an d

unitary group s 1 2. Paraboli c subgroup s an d som e cose t decomposition s 7 3. Th e denominato r idea l o f a matri x 1 7 4. Hermitia n form s ove r a commutativ e semisimpl e algebr a

of rank 2 and quadrati c form s 2 3 5. Quadrati c an d hermitia n form s ove r a nonarchimedea n

local field 3 0 6. Unitar y group s ove r C 3 8 7. Symplecti c groups and spli t unitary groups over

local fields 4 8

Chapter II . Adelizatio n o f Algebrai c Group s an d Auto -morphic Form s 5 7 8. Adelizatio n o f algebrai c group s 5 7 9. Som e cose t decomposition s relativ e t o a paraboli c sub -

group 6 8 10. Automorphi c form s 7 5 11. Heck e operators i n a n arbitrar y grou p an d i n G ^ 8 3 12. Elementar y theor y o f Eisenstein serie s 9 2

Chapter III . Eule r Factor s o n Loca l Group s an d Eisen -stein Serie s 10 1 13. Th e serie s a associate d wit h a hermitia n matri x 10 1 14. Th e serie s a^ wit h nonsingula r £ 10 9 15. Th e explici t for m o f OJ(0 , s) 11 8 16. Th e explici t for m o f a loca l Eule r facto r 12 5 17. Som e loca l grou p indice s 13 5 18. Eisenstei n serie s on G v 14 5 19. Th e pole s an d residue s o f Eisenstein serie s o n G v 15 7

Chapter IV . Mai n Theorem s o n Eule r Products , Eisen -stein Series , an d th e Mas s Formul a 16 7 20. Mai n theorem s o n analyti c continuatio n 16 7

vi CONTENT S

21. Pullbac k o f Eisenstein series : Arithmeti c par t 17 5 22. Pullbac k o f Eisenstein series : Analyti c par t 18 1 23. Differentia l operator s 19 0 24. Th e mas s formul a fo r a unitar y grou p 19 9

Appendix 20 9 Al. Som e elementary fact s o n rea l an d comple x analysi s 20 9 A2. Bounde d domain s an d kerne l function s 21 4 A3. Convergenc e o f Eisenstein serie s an d som e related fact s 22 2 A4. Fourie r expansio n o f automorphic form s 22 7 A5. Paraboli c subgroups and compac t subgroup s of some clas-

sical groups ove r R 23 6 A6. L- functions o f Hecke character s 23 7 A7. Thet a function s o f hermitian form s 24 1 A8. Centra l simpl e algebra s ove r a loca l field 25 1

References 25 7

Index 25 9

PREFACE

A substantia l portio n o f this volum e i s based o n my lecture s a t th e NSF-CBM S Regional Researc h Conferenc e hel d a t th e Texa s Christia n University , Ma y 19-24 , 1996. Whe n I was asked to give lectures at the conference and to write up eventually the content s o f the lecture s i n monograph form , m y idea was rather differen t from , if no t unrelate d to , wha t I a m presentin g now . A t tha t tim e I though t I woul d include th e result s I ha d publishe d i n a serie s o f papers , whic h concerne d Eule r products an d Eisenstei n serie s on symplecti c an d metaplecti c groups , an d I woul d also discuss th e arithmeticit y problem s o f the specia l value s o f the Eule r products .

After thinkin g abou t thi s projec t fo r a fe w weeks , I foun d th e ide a unexciting . Though th e questio n o f arithmeticit y ha d neve r bee n full y explore d i n thos e case s and I stil l inten d t o trea t i t o n a futur e occasion , th e whol e progra m lacke d th e allure o f making m e brave th e burde n o f writing a book o f fai r length . Therefor e I decided t o take up something ne w and mor e challenging which had bee n occupyin g my min d fo r som e time , an d o n whic h I had onl y incomplet e result s bu t fel t tha t I had enoug h technica l idea s t o complet e them . However , i n additio n t o th e obviou s question o f whethe r thos e idea s wer e enough , ther e wa s anothe r problem , namely , whether th e propose d boo k coul d b e accessibl e t o man y readers . Afte r a fe w mor e months o f experimenting , I convince d mysel f tha t I woul d b e abl e t o accomplis h my aim s satisfactorily , an d bega n th e wor k o f whic h th e outcom e i s th e presen t volume.

What ar e the n th e mai n feature s o f the book ? Leavin g th e detail s t o th e Intro -duction, le t u s merel y sa y tha t ther e ar e thre e chie f objectives : (i ) th e determina -tion of local Euler factor s o n classical groups, in an explici t rationa l form ; (ii ) Eule r products an d Eisenstei n serie s o n a unitar y grou p o f a n arbitrar y signature ; (iii ) a class numbe r formul a fo r a totall y definit e hermitia n form .

Though thes e for m th e principa l ne w result s obtaine d i n th e book , w e star t with quit e a genera l setting , an d includ e man y topic s o f expository natur e s o tha t the boo k ca n b e viewe d a s a n introductio n t o th e theor y o f automorphi c form s o f several variables. W e eventually specializ e our exposition t o unitary groups , but w e treat the m as a model case so that th e reader can easily formulate th e correspondin g facts i n other cases . Fo r that purpos e we find unitary groups better tha n symplecti c groups a s will be explaine d i n the Introduction .

Princeton, October, 199 6 Gor o Shimur a

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N O T A T I O N

We denote by Z , Q , R , an d C th e rin g of rational integers , the fields of rationa l numbers, rea l numbers , an d comple x numbers , respectively . W e pu t

T = {*ec| |* | = i }.

If p i s a rationa l prime , Z p an d Q p denot e th e rin g of p-adi c integer s an d th e field of p-adi c numbers , respectively . Fo r a n associativ e rin g R wit h identit y elemen t and a n .R-modul e M w e denote b y R x th e grou p o f al l it s invertibl e element s an d by M™ the i?-modul e o f all m x n-matrices wit h entrie s i n M ; w e put M m = Mf 1

for simplicity . Sometime s a n objec t wit h a superscrip t suc h a s S r o f (2.9.3 ) belo w is used wit h a different meaning , bu t th e distinctio n wil l be clea r fro m th e context . For x £ R™ and a n idea l a o f R w e write x - < a i f al l th e entrie s o f x belon g t o a. (Ther e i s a variatio n o f this ; se e §9.1. ) Th e transpose , determinant , an d trac e of a matri x x ar e denote d b y *sc , det(x), an d tr(rr) . Th e zer o elemen t o f R™ is denoted b y 0™ o r simpl y b y 0 , an d th e identit y elemen t o f R™ b y l n o r simpl y b y 1. Th e siz e o f a zer o matri x bloc k writte n simpl y 0 shoul d b e determine d b y th e size of adjacen t nonzer o matri x blocks . W e put GL n(R) = (i?™)

x, an d

SLn(R) = {ae GL n{R) | det(a) = 1 }

if R i s commutative . If xi, . . . , xr ar e squar e matrices , diag[a?i , . . . , xr] denote s th e matri x wit h

#i , . . . , xr i n th e diagona l block s an d 0 i n al l othe r blocks . W e shal l b e consid -ering matrice s x wit h entrie s i n a rin g wit h a n anti-automorphis m p (comple x conjugation, fo r example) , includin g th e identit y map . W e then pu t x* = f xp', an d x = (x*) - 1 i f x i s squar e an d invertible . Fo r a comple x numbe r o r mor e gener -ally fo r a comple x matri x a w e denote b y Re(a) , Im(o;) , an d a th e rea l part , th e imaginary part , an d th e comple x conjugat e o f a. Fo r comple x hermitia n matrice s x an d y w e write x > y an d y < x i f x — y i s positive definite , an d x > y an d y < x i f x — y i s nonnegative. Fo r r £ R w e denote by [r] the larges t intege r < r .

Given a se t A, th e identit y ma p o f A ont o itsel f i s denote d b y id ^ o r 1^ . T o indicate that a union X = \JieI Yi is disjoint, w e write X = [_\ieI Y{. We understand

that nf= a = * an /? . For a finite se t X w e denot e b y # X o r # ( X ) th e numbe r o f element s i n X. I f H i s a subgrou p o f a grou p G , w e pu t [G : H] = #(G/H). Howeve r w e us e als o th e symbo l [K : F] fo r th e degre e o f a n algebraic extensio n K o f a field F. Th e distinctio n wil l be clea r fro m th e context .

As fo r th e notatio n an d terminolog y concernin g Heck e character s o f a numbe r field and Heck e L-function s associate d wit h them , th e reade r i s referred t o Sectio n A6 of the Appendix . W e note here only that a Hecke character \ o f a number field K mean s a continuou s T-value d characte r o f th e idel e grou p o f K trivia l o n K x, and x * denote s th e idea l characte r associate d wit h \.

IX

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F R E Q U E N T L Y USE D SYMBOL S

a^, a( , s) . . . 13. 4

B(z) . . . 6.1 , 7.9,42. 1

cfc( ) . . . 42. 9

C* ... 6.4 , 11.1 0

C* . . . 12.4 , 20.6

D[ , } ... 9.1 , 18.4

d( ) . . . 1.1 0

S() . . . 6.3 , 7.9 , 10. 3

S{ , ) . . . 6.6 , 7.1 4

0 . . . 4. 6

Dv ... 11.1 0

£>* . . . 12.6 , 20.6

£>" . . . 21. 2

dw, d z . . . 10.9,42. 1

e . . . 20. 2

T,T{,) ... 10. 7

F ... 1.10,4.1 , 8.1 , 10.1

g ... 4.6,8. 1

Gv,G(V,tp) ... 1. 1

C . . . 2.11 , 18.1

r„(s) . . . 18.1 2

if . . . 9.11 , 10. 1

(Hr, T) r) ... 1. 4

£ . . . 6.5,7.9 , 18. 1

rj,r)n ... 1.4 , 1. 7

i . . . 6.4 , 10. 3

t . . . 4.2 , 48. 3

i( , ) . . . 6.10,6.1 4

tt/( , ) . . . 22. 2

il6, il , . . . 9.3 , 18. 4

Inj( , ) . . . 1. 1

ja{z),j(a,z) ... 6.3 , 10. 3

&" •• • 10- 4

K ... 1.1 , 2.13,4.1 , 10. 1

Lc(s, ) . . . 19.1 , 46.1

A0 . . . 12.3 , 12.4

A^ . . . 2.5,2.1 0

Ac, A? , A h,t . . . 19. 2

H(L),K(L) •• • 4. 7

m( , ) . . . 24.1 , 24.6

A*£„ . . . 10. 5

i/( ) . . . 3.1 1

i/[ ] . . . 13. 4

i*( ) . . . 3.7,9. 3

v°{ ) . . . 11.1 1

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I N T R O D U C T I O N

The firs t mai n them e o f thi s boo k i s t o associat e a n Eule r produc t t o a n auto -morphic for m tha t i s a Heck e eigenfor m o n a classica l grou p i n a suitabl e sense . As the nam e indicates , Heck e treated th e cas e o f holomorphic modula r form s wit h respect t o a congruenc e subgrou p r o f 51/ 2 (Z). Eac h Heck e operato r i s given b y a double cose t Far wit h a belongin g t o a semigrou p E o f matrices , containin g r, whose entrie s ar e integer s an d determinant s ar e positive . Takin g a n eigenfor m / in th e sens e tha t f\rar — A (a)f wit h a comple x numbe r A (a) fo r ever y a £ E, one consider s a Dirichle t serie s o f the for m

(1) !(*) = £ A(a)det(a)-* . aer\s/r

With a suitabl e choic e o f E on e ca n sho w tha t X ha s a n Eule r produc t whos e Euler factor s hav e degree 2 ; moreover F(s)1(s) ca n be continued t o a meromorphi c function o n th e whol e s-plan e wit h a t mos t tw o simpl e poles ; F(s)%(s) i s entire i f / i s a cus p form .

Various kind s o f generalizatio n o f thi s theor y o f Heck e hav e bee n attempte d and carrie d out , bu t an y attemp t mus t b e precede d b y a choic e o f formulatio n that i s practicable . Thoug h on e migh t b e abl e t o presen t a framewor k includin g every imaginabl e Eule r product , i t i s o f littl e us e i f on e canno t indee d prov e th e desired results . Therefor e i n this book we choose one type of Euler product whic h is somewhat differen t fro m Hecke' s type. T o be explicit , w e take a n algebrai c numbe r field K wit h an automorphism p o f order 1 or 2; we then put F = {x€K\xp = x} and

(2) G(

X I V INTRODUCTION

sense. Bu t t o avoi d excessiv e details , w e d o no t giv e her e th e precis e definitio n i n the genera l case . Thoug h thi s ma y no t b e th e bes t wa y to produc e a zet a functio n in othe r cases , i t ha s th e advantag e tha t w e ca n connec t th e Heck e eigenvalue s naturally an d directl y t o th e desire d Eule r produc t fo r a larg e clas s o f classica l groups. Also , in certain case s this definitio n allow s us to expres s our Eule r produc t in terms o f the Fourie r coefficient s o f / , thoug h w e do no t touc h o n tha t aspec t i n this book .

For th e purpos e o f illustration , le t n = 2r , G^ = Sp(r, Q) , an d F = Sp(r, Z) ; then w e tak e E — G? an d v{a) t o b e th e produc t o f th e denominator s o f th e elementary divisor s o f a . I f r = 1 , the n th e grou p i s SX2(Q ) an d th e serie s ha s the for m

oo

(4) T(s ) = J2 A(diag[m- \ m])m~ s, 7 7 2 = 1

which produce s a n Eule r produc t o f degree 3 , that i s usually calle d th e symmetri c square zet a functio n associate d t o a serie s o f type (1) , but i t i s also natura l t o cal l it a n Eule r produc t o n 51/ 2 (Q), wit h n o reference t o GL2(Q) .

Now ou r first mai n tas k i s t o sho w tha t thi s typ e o f serie s wit h v define d i n a similar manne r ha s a n expressio n

(5) A(s)%(s) = Y[w p(N(p)-°y\ p

where A i s a produc t o f L-function s o f F , p run s ove r al l th e prim e ideal s i n F , and W p i s a polynomia l wit h constan t ter m 1 whose degre e i s n[K : F] fo r almos t all p , excep t whe n G^ i s a symplectic grou p o r an orthogona l grou p o f odd degree , in whic h cas e th e degre e o f W p i s n + 1 o r n — 1 accordingly . Thi s fac t i s purel y algebraic, or rather local , and ca n be formulated withou t th e notion of automorphi c forms. Therefor e w e obtai n th e resul t fo r a n arbitrar y G^ o f typ e (2 ) (Theore m 16.16).

Now, assumin g tha t K ^ F, tak e a n arbitrar y Heck e characte r x o f X an d denote b y Xi it s restriction t o F ; denot e als o by T(s , x) a n d Z(s, x) th e twist s of T an d th e right-han d sid e o f (5 ) b y x b y viewing the m a s Eule r product s ove r K\ further denot e b y A(s , xi) th e twis t o f A by xi - The n

(6) A(a , xi)£(s , x ) = Z(s, x) -

Our nex t proble m i s to prov e tha t i f / i s a holomorphi c cus p for m i n th e unitar y case, then Z(s, x ) time s suitable gamma factors ca n be continued to a meromorphi c function o n th e whol e plan e wit h finitely man y possibl e simpl e poles . I n principl e our method s ar e applicabl e t o nonholomorphi c forms , bu t th e incomplet e stat e o f knowledge o f suc h form s make s i t difficul t t o find explici t form s o f th e gamm a factors. I n th e holomorphi c case , however , i t i s relativel y eas y t o calculat e th e gamma factors , whic h i s th e mai n reaso n wh y w e restric t ou r expositio n i n late r sections t o holomorphi c forms .

We prove the desired resul t o n Z fo r G^ whe n K i s a totally imaginary quadrati c extension of totally real F, in which case G^ i s a unitary group acting on a hermitia n symmetric spac e 3^ - Ther e ar e severa l reason s wh y w e conside r unitar y group s

INTRODUCTION xv

instead o f symplecti c o r orthogona l groups : (i ) Unitar y group s ca n b e spli t o r nonsplit, an d therefor e the y hav e th e characteristic s o f genera l reductiv e algebrai c groups, whil e symplecti c group s ar e spli t an d specia l i n tha t sense , (ii ) Besides , the symplecti c cas e ha s bee n treate d i n detai l i n a serie s o f recen t paper s b y th e author, an d s o i t seem s desirabl e t o presen t th e "nonspli t aspect " o f th e theory . (iii) Holomorphi c form s ca n be considered als o on orthogonal group s o f a restricte d type. A unifor m treatmen t o f al l thes e classica l group s i s no t impossible , bu t a t some poin t th e tas k wil l becom e cumbersome . Fo r example , i t require s a carefu l analysis o f certain Eisenstei n serie s in the orthogona l case , which would hav e mad e the boo k much longer . Fo r this reason , we discuss only the arithmeti c aspec t o f th e orthogonal case , bu t no t it s analyti c aspect .

Now the function Z i s closely connected with an Eisenstein serie s of the followin g type. Give n G^ a s abov e wit h

(z)/6v{p(z))]'-"/2 .

Here Xo(a) i s the determinan t o f the lowe r righ t m x m-block o f a , p i s a certai n projection ma p o f 3 ^ int o 3^ , 8^ i s th e functio n o n 3 ^ suc h tha t 8^{^w) = \j1{w)\~

28ip{w) fo r ever y 7 G G^ wit h th e standar d scala r facto r o f automorph y j1{w)^ 8^ i s similarl y define d o n 3 ^ , k i s th e weigh t o f / , an d (p||^a)(z ) = j1(z)~

kg(az) fo r a functio n g o n 3^-In orde r t o stud y th e analyti c natur e o f thi s series , w e conside r anothe r grou p

Gu wit h u = diag[^ , —

X V I INTRODUCTION

where d i s the functio n c i n the presen t case . No w we can find a product A ' o f L-functions o f F an d a product Q of gamma factor s suc h tha t h!QS ca n b e continue d to a meromorphic functio n o f s o n the whol e plane with finitely man y pole s whic h are al l simple . Clearl y w e ca n sa y th e sam e fo r K'QH. I n th e settin g o f (8) , thi s A' coincide s wit h A(s , xi) > an d henc e fro m (8 ) w e obtain th e desire d meromorphi c continuation o f d(s)G(s)Z(s, x) (Theore m 20.5) . I n a simila r way , multiplyin g by a facto r o f th e typ e A'Q, w e ca n sho w tha t Z(s, x)E(z, s; / , \) time s suitabl e gamma factor s ca n b e continue d t o a meromorphi c functio n o n th e whol e plan e with finitely man y pole s whic h ar e al l simpl e (Theore m 20.7) .

Strictly speaking , (8 ) i s valid only for the characte r \ whos e archimedean facto r is consisten t wit h th e weigh t o f / i n a certai n sense . T o obtai n X(s , \) f° r a n

arbitrary x , w e have t o replac e £' b y AS' wit h a differentia l operato r A whic h i s not s o simple .

It shoul d b e note d tha t Garret t gav e i n [Ga ] a formul a fo r th e pullbac k o f th e standard Eisenstei n serie s o n Sp(r, Z) , fro m whic h on e coul d deriv e a n equalit y o f type (7 ) i n tha t case . However , h e di d no t carr y ou t th e calculation , whic h wa s later don e b y Bochere r i n [Bo] . I t ma y b e note d als o tha t equalitie s o f typ e (8 ) were employe d i n a fe w earlie r paper s o f the autho r whe n th e grou p i n questio n i s obtained fro m a quaternio n algebra .

The final mai n theore m o f the boo k concern s a generalizatio n o f the clas s num -ber o f a hermitia n form , whic h w e cal l the mass of G^ relativ e t o a specifie d ope n subgroup o f G^ . T o explai n th e concept , denot e b y V th e vecto r spac e o f al l n-dimensional ro w vector s wit h component s i n K o n whic h G ^ act s b y righ t multi -plication, an d b y x th e maxima l orde r o f K. The n w e can find a finitely generate d r-submodule M o f V wit h th e propert y tha t xip • txp € r fo r ever y x e M an d M is maximal amon g such submodules o f V. To make a transparent formulatio n o f th e problem, w e now have to consider the adelization G ^ o f G^, whic h we have avoide d so far . Takin g a n arbitrar y integra l idea l c i n F , w e defin e a n ope n subgrou p D of G ^ containin g th e archimedea n facto r o f G ^ suc h tha t it s f-facto r D v fo r eac h nonarchimedean prim e v o f F i s defined b y

Dv = { a e G% | Mva = M v, M v(a - 1 ) C cvMv } .

Then w e ca n find a finite se t B s o tha t G ^ = LLet f G^aD. Le t T b e th e se t o f elements o f G ^ tha t ac t triviall y o n 3^ ; le t F a = G^ n aDa' 1 fo r eac h a e B. W e then pu t

(9) m(y> , c ) = Y} pa n T • ir^ourw, aeB

where w e understand tha t T — G ̂an d vol(.T a\3^) = 1 if

INTRODUCTION xvn

if (p is anisotropic . Namely , fo r suc h a

XV111 INTRODUCTION

quantity v(a) o f (3) . Mos t noteworth y amon g severa l fact s prove d i n thi s sectio n are Proposition s 3. 9 an d 3.1 0 concernin g v(a) fo r a belongin g t o a paraboli c subgroup o f a genera l linea r grou p an d als o fo r a o f "degenerat e type. " Section s 4 an d 5 concer n quadrati c an d hermitia n form s ove r th e field o f quotient s o f a ring which i s first a Dedekin d domain , late r a principal idea l domain , an d finally a discrete valuatio n ring . W e introduce th e notio n o f maxima l lattices , an d describ e them i n term s o f a refine d for m o f Witt' s decomposition . W e prov e a produc t expression o f the typ e G^ — PjC wit h th e stabilize r C o f a maxima l lattice .

In Sectio n 6 w e defin e th e spac e 3 ^ an d als o basi c factor s o f automorph y i n the unitar y case . Variou s elementar y fact s concernin g th e archimedea n versio n o f Q(f _ p^C an d th e projectio n ma p p : 3^ — • 3^ ar e collecte d i n thi s section . Th e symplectic case , no t include d i n these thre e sections , i s treated i n Sectio n 7 .

The adelizatio n G A o f a n algebrai c grou p G an d som e relate d concept s ar e introduced i n Sectio n 8 . Fo r ou r purpose s i t i s essentia l t o examin e th e cose t decompositions o f G A relativ e t o a n ope n subgrou p an d a paraboli c subgroup . This wil l b e don e i n Sectio n 9 . W e introduc e th e notio n o f automorphi c form s i n Section 10 , prove easy facts o n Hecke operators i n Section 11 , and defin e Eisenstei n series i n Sectio n 12 . W e trea t thes e a s function s an d operator s o n 3^ , an d als o a s objects o n G^ . Ou r expositio n i n thes e section s i s restricte d t o th e unitar y case , though w e add som e comment s i n th e symplecti c case .

Sections 1 3 through 1 5 are devoted to the investigation of a type of local Dirichle t series that appear s a s an Eule r facto r o f a Fourier coefficien t o f an Eisenstei n serie s on a split group . Thi s local series plays also a crucial role in the computation o f the Euler factor s o f our zet a functions . I n Section 1 6 we determine th e explici t rationa l expression fo r W p o f (5) . Th e key fact i n this is Proposition 16.10 , which gives v(a) for a G Pj. Sectio n 1 7 concerns severa l formula s fo r th e grou p indice s whic h ar e necessary fo r th e proof o f (11) . W e investigate i n Sections 1 8 and 1 9 the Eisenstei n series o n G 1 whic h w e denote d b y £ i n th e above . W e first giv e a n explici t for m for eac h Fourie r coefficien t an d determin e a produc t A 7 of //-functions an d anothe r product Q of gamm a factor s suc h tha t K'Q£ ha s onl y finitely man y pole s o n th e whole plane. W e then give an explici t formul a fo r the residue a t a special pole when the weigh t i s 0 .

We state our main theorems onZ(s, \) an d E(z, s, / , \) m Sectio n 20, and prove them i n the nex t thre e sections . On e o f the mai n technica l difficultie s arise s i n th e analysis of the pullback denoted by H(z, w; s) in the above. Thoug h the descriptio n of P u ; \Ga ; / (G^ x G^ ) give n i n Sectio n 2 is not complicated , w e have to describ e i t in connection wit h various open subgroups o f the adelized groups . I t shoul d als o be mentioned tha t i n order t o obtain (7 ) an d (8) , we must choos e £' carefully , becaus e an arbitrar y o r a seemingly natura l choic e of £' ofte n produce s a vanishing integra l or ambiguou s factors . I t i s on e o f th e mai n point s o f ou r treatmen t t o giv e suc h formulas i n nonvanishin g exac t form s wit h al l factor s explicitl y determined .

As w e said earlier , t o establis h th e meromorphi c continuatio n o f Z(s, \) m th e most genera l case , i t i s necessar y t o appl y a certai n differentia l operato r t o £. I n Section 23 we define such an operator an d prove a formula o n its effect o n each ter m of £, whic h eventuall y lead s t o th e proo f o f th e desire d fact . Finall y i n Sectio n 2 4 we prove a formul a fo r m(

INTRODUCTION xix

the smoot h flow of the principa l ideas . Som e of these sections ar e quit e elementar y and contain only the results which are either well known or essentially known. The y are intende d fo r th e reade r wh o i s not familia r wit h suc h standar d facts . However , it seem s that som e results i n less elementary sections , A4 and A 7 for example , hav e never bee n state d i n the form s w e present them .

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[AD] Algebraic groups and discontinuou s groups , Proc . Symposi a i n Pure Math . vol.9, Amer . Math . Soc . 1966 .

[An] A . N . Andrianov , Th e multiplicativ e arithmeti c o f Siege l modula r forms , Usp. Mat . Nauk , 3 4 (1979) , 67-135 ; Englis h transl . Russia n Math . Surv . 3 4 (1979), 75-148 .

[AK] A. N. Andrianov an d V . L. Kalinin, O n the analyti c propertie s o f standar d zeta functions o f Siegel modular forms , Mat . Sb . 10 6 (148) (1978 ) 323-339 ; English transl. Math . USS R Sb.3 5 (1979) , 1-17 .

[Bo] S . Bocherer , Ube r di e Funktionalgleichun g automorphe r L-Funktione n zu r Siegelschen Modulgruppe , J . R . Ang . Math . 36 2 (1985) , 146-168 .

[B63] A . Borel , Som e finitenes s propertie s o f adel e group s ove r numbe r field, Publ. Math . IHES , No.16 , 1963 .

[B69] A . Borel , Introductio n au x groupe s arithmetiques , Hermann , Paris , 1969 . [C] C . Chevalley , Theori e de s groupe s d e Lie , Tom e II : Groupe s algebriques ,

Hermann, Paris , 1951. [E38] M . Eichler , Allgemein e Kongruenzklasseneinteilunge n de r Ideal e einfache r

Algebren iiber algebraischen Zahlkorper n un d ihr e L-Reihen, J . R . Ang . Math . 17 9 (1938), 227-251 .

[E52] M. Eichler, Quadratisch e Formen und orthogonale Gruppen, Springe r 1952 , 2nd ed . 1974 .

[F] P . Feit , Pole s an d residue s o f Eisenstei n serie s fo r symplecti c an d unitar y groups, Memoirs , Amer . Math . Soc . 61 , No.346 (1986) .

[Ga] P . B . Garrett , Pullback s o f Eisenstei n series : Applications , Automorphi c Forms of Several Variables, Taniguchi Symposium , Katata , 1983 , Birkhauser, 1984 , 114-137.

[Go] R . Godement , Fonction s automorphes , Seminair e Carta n 1957-58 , expose s 6 an d 10 .

[Ha] Harish-Chandra , Automorphi c form s o n semisimpl e Li e groups , Lectur e notes i n Math . 62 , Springer , 1968 .

[Hec] E . Hecke , Mathematishe Werke , Gottingen , 1959 . [Hel] S . Helgason , Differentia l Geometry , Li e Groups , an d Symmetri c Spaces ,

Academic Press , 1978 . [Hu] L.K. Hua , Harmonic analysis of functions o f several complex variables in the

classical domains , Translation s o f mathematica l monographs , vol.6 , Amer . Math . Soc. 1963 .

[L] R . P . Langlands , O n th e functiona l equation s satisfie d b y Eisenstei n series , Lecture note s i n Math . 544 , Springer , 1976 .

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[MS] Y . Matsushim a an d G . Shimura , O n th e cohomolog y group s attache d t o certain vecto r value d differentia l form s o n th e produc t o f th e uppe r hal f planes , Ann. o f Math . 7 8 (1963) , 417-449.

[S63] G . Shimura , Arithmeti c o f alternatin g form s an d quaternio n hermitia n forms, J . Math . Soc . o f Japan , 1 5 (1963) , 33-65.

[S64] G . Shimura , Arithmeti c o f unitar y groups , Ann . o f Math . 8 3 (1964) , 369-409.

[S75] G . Shimura , O n th e holomorph y o f certai n Dirichle t sries , Proc . Londo n Math. Soc , 3r d ser . 3 1 (1975) , 79-98.

[S78] G. Shimura, Th e special values of the zeta functions associate d with Hilber t modular forms , Duk e Math . J . 4 5 (1978) , 637-679.

[S82] G . Shimura , Confluen t hypergeometri c function s o n tub e domains , Math . Ann. 26 0 (1982) , 269-302 .

[S83] G . Shimura , O n Eisenstei n series , Duk e Math . J . 5 0 (1983) , 417-476. [S84] G . Shimura , O n differentia l operator s attache d t o certai n representation s

of classica l groups , Inv . math . 7 7 (1984) , 463-488. [S85] G. Shimura , O n Eisenstein serie s of half-integral weight , Duk e Math . J . 5 2

(1985), 281-324 . [S86] G. Shimura , O n a clas s of nearly holomorphi c automorphi c forms , Ann . o f

Math. 12 3 (1986) , 347-406. [S87] G. Shimura , Nearl y holomorphic function s o n hermit ian symmetric spaces ,

Math. Ann . 27 8 (1987) , 1-28 . [S93] G . Shimura , O n th e transformatio n formula s o f thet a series , Amer . J . o f

Math. 11 5 (1993) , 1011-1052 . [S94a] G. Shimura , Eule r product s an d Fourie r coefficient s o f automorphic form s

on symplecti c groups , Inv . math . 11 6 (1994) , 531-576 . [S94b] G . Shimura , Differentia l operators , holomorphi c projection , an d singula r

forms, Duk e Math . J . 7 6 (1994) , 141-173 . [S95a] G . Shimura , Eisenstei n serie s an d zet a function s o n symplecti c groups ,

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groups, Duk e Math . J . 8 2 (1996) , 327-347 . [Si] C . L . Siegel , Gesammelt e Abhandlungen , I-III , 1966 ; IV, 1979 , Springer . [W58] A. Weil, Introduction a l'etude de s varietes Kahleriennes, Hermann , Paris ,

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(1964), 143-211 .

I N D E X

adelization, 58 , 60

algebraic group , 5 8

anisotropic, 1

automorphic form , 7 8

bounded domain , 214ff .

Cayley transformation , 5- 6

central simpl e algebra , 25 1

class o f lattices , 6 2

conductor, 23 8

congruence subgroup , 6 2

cusp form , 7 9

denominator ideal , 1 9

discriminant, 6

Eisenstein series , 94 , 96, 15 0

Fourier expansio n of , 15 1

elementary divisors , 1 7

elliptic modula r type , 7 8

£-hermitian form , 1

Euler facto r

local, 133- 5

Euler product , 168 , 17 4

factor o f automorphy , 40 , 7 8

Fourier expansion , 20 9

of a n automorphi c form , 148 , 227fT.

Gauss sum , 239-4 0

genus o f lattices , 6 2

limited, 20 5

Hecke algebra , 8 5

Hecke character , 23 8

Hecke operator , 8 6

ideal character , 23 8

inner product , 8 1

involution, 1

main, 24 , 252

isotropic, 1 259

kernel function , 220 , 22 4

//-function, 23 8

lattice, 2 6

maximal, 2 6

mass, 9 9

of a genus , 20 5

of a limite d genus , 20 5

maximal order , 25 2

measure

Haar, 6 5

invariant, 42 , 54 , 8 1

modular form , 7 8

normalized Heck e character , 23 8

origin, 41 , 77

parabolic subgroup , 7 , 68

minimal, 7

primitive matrix , 1 7

quaternion algbra , 24 , 252

radical, 6 8

reduced expression , 1 9

reductive group , 6 8

regular matrix , 10 4

semisimple group , 6 8

Siegel set , 23 4

strong approximation , 6 2

subspace, 1

theta function , 241fT .

totally isotropic , 2

unipotent radical , 10 , 68

weak approximation , 6 5

weight, 7 8

Witt decomposition , 4 , 26

weak, 4 , 2 6

Witt's theorem , 2

Selected Title s i n Thi s Serie s (Continued from the front of this publication)

54 Wil l ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wi lhe l m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Ts i t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . EfFros , Dimension s an d C*-algebras , 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finit e Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affin e spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finite groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g operators ,

1977

29 A . Weinstein , Lecture s o n symplecti c manifolds , 197 7

28 T . A . Chapman , Lecture s o n Hilber t cub e manifolds , 197 6

27 H . Blain e Lawson , Jr. , Th e quantitativ e theor y o f foliations , 197 7

26 I . Reiner , Clas s group s an d Picar d group s o f grou p ring s an d orders , 197 6

25 K . W . Gruenberg , Relatio n module s o f finit e groups , 197 6

24 M . Hochster , Topic s i n th e homologica l theor y o f module s ove r commutativ e rings , 197 5

23 M . E . Rudin , Lecture s o n se t theoreti c topology , 197 5

22 O . T . O'Meara , Lecture s o n linea r groups , 197 4

21 W . Stoll , Holomorphi c function s o f finite orde r i n severa l comple x variables , 197 4

20 H . Bass , Introductio n t o som e method s o f algebrai c X-theory , 197 4

19 B . Sz . -Nagy , Unitar y dilation s o f Hilber t spac e operator s an d relate d topics , 197 4

18 A . Friedman , Differentia l games , 197 4

17 L . Nirenberg , Lecture s o n linea r partia l differentia l equations , 197 3

16 J . L . Taylor , Measur e algebras , 197 3

15 R . G . Douglas , Banac h algebr a technique s i n th e theor y o f Toeplit z operators , 197 3

14 S . Helgason , Analysi s o n Li e group s an d homogeneou s spaces , 197 2

13 M . Rabin , Automat a o n infinit e object s an d Church' s problem , 197 2

12 B . Osofsky , Homologica l dimension s o f modules , 197 3

11 I . Glicksberg , Recen t result s o n functio n algebras , 197 2

10 B . Gri inbaum , Arrangement s an d spreads , 197 2

9 I . N . Herstein , Note s fro m a rin g theor y conference , 197 1

8 P . Hil ton , Lecture s i n homologica l algebra , 197 1

(See th e AM S catalo g fo r earlie r titles )

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