Selberg's Zeta-, L-, and Eisensteinseries

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1030 Ulrich Christian

Selberg's Zeta-, L-, and Eisensteinseries

Springer-Verlag Berlin Heidelberg New York Tokyo 1983

Author Ulrich Christian Mathematisches Institut, Georg-August-Universit~t Bunsenstr. 3 - 5 , 3 4 0 0 G6ttingen, Federal Republic of Germany

CR Subject Classifications (1982): 3, 10 A M S Subject Classifications (1980): 10 D05, 10 D20, 10 D 2 4 ISBN 3-54042701-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12701-1 Springer-Vertag New York Heidelberg Berlin Tokyo

This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting, re-use of illustrations,broadcasting, reproductionby photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg 1983 Printed in Germany Printing and binding: Bettz Offsetdruck, Hemsbach/Bergstr. 2t 46/3140-543210

PREFACE

This

course

G~ttingen

I thank

of

lectures

in the

Mrs.

was

given

summer-semester

Christiane

of the

troublesome

Ulrich

Christian

Gieseking

manuscript.

at the U n i v e r s i t y

of

1983.

for her

careful

typing

CONTENTS

Introduction Chapter I. Epstein's zetafunction of a binary quadratic form § 1. Preliminaries § 2. Epstein's zetafunctions and L-series § 3. Elementary Eisenstein series

24-

Chapter II. Preparational material § 4. Systems of primitive characters § 5. Matrices § 6. The Riemannian space of positive matrices § 7. Theta functions

28 29 36 53 66

Chapter III. Selberg's zeta- and L-series § 8. Descending chains § 9. Characters § 10. Selberg's zeta- and L-series § 11. Analytic continuation § 12. Functional equations § 13. Residues of Selberg's zetafunctions

89 89

Chapter IV. Selberg's Eisensteinseries § 14. Siegel's upper half-plane § 15. Selberg's Eisensteinseries § 16. Representation with Siegel's Eisensteinseries § 17. Representation with Selberg's zetafunction § 18. Analytic continuation

149

VII 1 1 11

98 102 116 137 139

14-9

155 160 171 177

Chapter V. Siegel's Eisensteinseries § 19. Siegel's Eisensteinseries § 20. Poles and Hecke's summation

181 181

Literature List of symbols Index

187

183

191 195

INTRODUCTION In these lecture notes we prove analytic continuation and functional equations for Selberg's Eisensteinseries, Selberg's zetafunctions, Selberg's L-series, and Siegel's Eisensteinseries. We start with Epstein's

zetafunction for a binary quadratic form and

Epstein's L-functions which are connected with Epstein's zetafunction like Dirichlet's L-series are connected with Riemann's zetafunction.

Then we consider Eisensteinseries

for the elliptic modu-

lar group which are also closely related to Epstein's

zetafunction.

In the next chapters we come to Selberg's zetafunction

(see Maa~

[53], § 17, Selberg [41], and Terras [45], [46]). Furthermore we consider Selberg's L-series which are connected with Selberg's zetafunctions

like Dirichlet's L-series are connected with Rie-

mann's zetafunction. These functions may be described as follows. symmetric,

positive

nxn matrix and

Y

Let

Y

be a real

positive matrices

(~=l,...,n)

which are connected by n ~ ~+1 Y = Y; Y = G ' Y Gv

(a)

with integral matrix.

(~

=

(v+l)x~ matrices G . Here

1 ....

,n-l)

' denotes the transposed

The above mentioned authors then consider the zetafunction

n-1 v -z -~- (Det Y) v) , n 1 "#=1 n-1 1 where the s~Immation is taken over all possible Y,... ,Y for which (b)

~(Y;Zl,...,Zn_1)

=

~

(a) holds. Let

D(n) = GL(n, 2Z)

be the group of unimodular nxn matrices and

&(n) the subgroup of upper triangular matrices.

The above mentioned

authors then show, that the function (b) is closely connected to the function (c)

C

(Y;Zl,''',Zn-1)

=

n-1 -z ( -~-(Det (U' YU)~ ) v), U E n(n)/A(n) v=1

VIII here

A

means generally the left upper

A. The summation By computing

is over all cosets

residues

of zetafunctions

lecture notes we generalize

as follows. Let q Dirichletcharacters G

D(n)/A(n).

Maa~ [33~, pages 279-299 furthermore

taines analytic continuation neral than (b) and (c). In the present

~x~ submatrix of a matrix

ob-

which are more ge-

the functions

(b),

(c)

be a natural number and XI,... ,Xn_ I even mod q. In (a) we assume that the elements of

below the main-diagonal

are divisible

by q. Furthermore,

we

put inside the sum (b) the Dirichletcharacters XI ,...,Xn_ 1. It is difficult to describe in the introduction how this has to be done.

It is simpler for the function

O(n) by the subgroup

U

(c) for which we replace

w(n)

consisting

0

un

=

of all unimodular

matrices

mod q .

Then instead of (c) we consider the function (d)

~ (X1,...,Xn_l;Y;Zl,...,Zn_l) n-1 - ~ - ( X ~ ( u )(Det(U'YU) ~ (n)/A (n) ~=I v

Under the assumption n-l)

-z ) ~)

that all the products

are primitive

characters

nuation and functional For these functions

=

equations

X u ''' X~ (1 ~ u ~ ~

mod q we derive analytic

conti-

for our functions.

we prove results that may be described

as

follows. Choose even Dirichletcharacters ~1,...,~ n with -I ¢ ~ + 1 ~ = X~ (~ = I,...,n-I); introduce new variables Sl,...,s n by z~ = sv+ I - s v + ~ (~ = 1,...,n-I) and put ¢ = ( ¢ 1 ' ' ' ' ' ¢ n ) ; S = (S I .... ,Sn). Let L(X,S) be Dirichlet's L-series and put S

g(X,S)

= (~-~F(~)L(x,S).

If then all characters function

¢~1¢U~ (I ~ U < ~ ~ n)

are primitive,

the

IX

k($,Y,s)

=

( ~T

~(¢~l~H,2(s -su)+1))_

x

~ (Det Y) sn- i(s1+...+Sn)+ n

~* (x,Y, z)

can be holomorphically continued to all s E Cn. Furthermore k($,Y,s) satisfies certain functional equations which we shall now describe. Let = (-Sn'''''-Sl); ~ = (en- I ,. ""$I-1 );

~

_S 1 '-Sn); = ('Sn-l' ....

¢-I $" = ( n_1,...,~11,enl); ~}s = (Sn,S 1 ..... Sn_1) ;

~$

= (¢n,$1,...,$n_1); s = (Sn_1,Sl,...,Sn_2,Sn);

A

~ = ($n_1,~1,...,¢n_2,~n) • Form the Gaussian sum I

q-2

G(X) =

~ x ( d ) e x p ( ' ~ qid) d mod q

and put n-1 ~(~) = TF a(x u)

•

U=I

Form the nxn matrices W(n) -- I~ ."10~ •

9

Q(n) = IW(0n-l)

01

, P(n)

=

q

W

W(n)

Q-I

q and the r

=

qn-2

matrices p K where

=

(nl)

0 1q

0 0

E (n-2) L

L = (12,...,in_l) runs over all

r

v )-I Y = W(n)y-1W(n),Y = (Y[Q(n)]

(~=1 .... ,r) residue classesmod q. Set

Then the following functional equations hold: k(¢,Y,s) = X(¢,Y,s) n

,

[

k($,Y,s) = ~(~)q

V

~=I

V

V.

X($,Y,s)

,

n 1 2(sI- ~ ~ s ) ~($,Y,s) = ~($)q n

X(~Y,s)

~j=1

~(}~,P(n)'-IyP(n) -I, ~s),

4n _ ~ s - 2Sn_ I- ZSn+1 - ~n = ~1-I($)q ~=1

r X(;,K ! YK

9

s).

:~=I The transformations

} and

~

generate the symmetric group

~n"

As in the case of Riemann's zetafunction and Dirichlet's L-series however the L-series have less poles than the zetafunctions. Therefore in the case q > I poles and residues of type Maa~ [33], pages 279-299 do not exist. So it is impossible to get any results about more general series by computing residues. For this reason we start already with more general series and derive the analytic continuation and functional equations for them. Let ~(n) = {Z = Z' = X + iY, Y > 0 1 Siegel's upper halfplane of degree n and F(n) = Sp(n, ~ ) Siegel's modular group of degree n. Set

(e)

M = (A B) E F(n)

with nxn matrices A, B, C, D and (f)

M = (AZ + B)(CZ + 0) -I = X M + iY M .

Set (g)

M{zl

= cz

. o

.

Let s = (Sl,...,s n) a complex variablerow and Selberg's Eisensteinseries

Z E ~(n). We consider

XI

(h)

(n,r,Z,s) =

(Det

M{2l)2r(Det YM)sn+ ~ + r

)su-sv + 1 - 21.

v=1

M 5 FB\T(n)

is the Borel subgroup of

Here FB(n) matrices (i)

-n-1 ~(Det(gM)

F(n) consisting of all 2nx2n

M =

U-1 with integral nxn matrices U, S. Here S = S' and U is an upper triangular matrix. For the functions (h) we prove again analytic continuation and functional equations by applying a method of Diehl [11]. Since the functional equations are very similar to those of Selberg's zetafunctions I do not write them down here. Finally consider Siegel's Eisensteinseries (j)

E(n,r,Z,~) =

~ (Det M{Z})-2r(Det YM )~-r M E Vn(n)\F(n)

Here ~ is a complex variable and Cn(n) the group of matrices (i) where now U is arbitrarily unimodular. We show that Siegel's Eisensteinseries may be obtained by computing residues of Selberg's Eisensteinseries. Hence the analytic continuation and the functional equations of Selberg's Eisensteinseries give us analytic continuation and a functional equation for Siegel's Eisensteinseries. Especially we get the following results A)

It is E(n,r,Z,~) holomorphic at ~ = r for

(k)

r

= 1,2, [ Tn-1 ]'

[

] ,

so for these values of r the Eisensteinseries E(n,r,Z,~) has Hecke summation. B)

It is

C)

Let

E(n,1,Z,1) = 0 (n >_ 3).

Xll

(I)

S(r) --

If 3 < r < F n ~ a pole of order --

m

r-2

n+2 (3 _~ r < -~-)

[~]-r

(~

_~ r ~_ [ ~ ] )

the Eisensteinseries S(r) at the most. 9

E(n,r,Z,~) has at

~ = r

All functions considered in this lecture play an important r~le in the theory of Siegel's modular functions but it seems to me that they are also interesting for themselves. They are eigenfunctions of invarian~ differential operators (see Selberg [39] till [42~ and Maa~ F33]) and they may be used to describe the continuous spectrum of those differential operators. Furthermore, as we have seen they may be used to get analytic continuation of Siegells Eisensteinseries. It is an important open question if E(n,r,2,~) is holomorphic at

~ = r also for

3 ~ r ~ [~].

CHAPTER I. EPSTEIN'S ZETAFUNCTIONS OF A BINARY QUADRATIC FORM In the first chapter we consider Epstein's zetafunction for binary quadratic forms. To this zetafunction

we associate L-series in the

same way as Dirichlet's L-series are associated to Riemann's zetafunction. Furthermore we consider Eisensteinseries for the elliptic modular group which are closely related to Epstein's zetafunction. For all these functions we prove analytic continuation and functional equations with the aid of thetafunctions.

§ I. PRELIMINARIES § 1 contains some preliminary definitions and results on thetafunctions for binary quadratic forms.

A matrix

K = (k)

pxc matrix; Rk K

of

p

rows and a columns is called a is the rank, K'the transposed and ~ the con-

jugate complex matrix. Occasionally we write = [kl,...,~min( elements let

Tr K

,a)~

= ~

=

for the diagonalmatrix formed of the diagonal(~ = 1,...,min(p,q))

be the trace, Det K

absolute value of

Dg K

Det K. Let

matrix of K. With a ~x~ matrix

from K. In case

the determinante and K

be the upper left L

define

abs K

p = a the

vx~ sub-

KILl = L'KL. Let

O,E

be zero- and identity-matrix. The number of rows and columns will either be seen from the connection or it will be written as upper

indices in brackets. A real symmetric ~xp matrix Y is called positive (Y > 0) respectively semipositive (Y ~ 0), if the quadratic form Y[x] is positive respectively nonnegative for all real columns x # 0 with p elements. Let ~ n ) denote the space of all positive symmetric nxn matrices Y. Then ~(n) is real and has (I)

d(n) =

dimensions. Y1 > Y2 is defined by YI - Y2 > 0 and YI ~ Y2 by Y1 - Y2 ~ O. A matrix is called "integral", "rational", "real" or "complex" if all elements are in2Z,~, ~ or ~. Brackets of type < > denote the greatest common de visor of integers. "exp" is the exponential function.

Let

Y E %n).

column

w

By an "isotropic vector" of Y we

mean a complex

satisfiying

(2)

Y[w]

= 0

.

If w is an isotropc vector of vector of y-1.

THEOREM 1:

Let

Y, obviously

Y E ~(n); g E2Z , g > O; u

arbitrary n-rowed complex columns, w

Yw

and

is an isotropc

v

be two

an isotropic vector of Y.

Then

• ((m+v)'Yw)gexp(-~Y[m+v]

(3) m

EZZ n

I exp(-2wiu'v)(Det Y f ~ ig

Here

m

+ 2~im'u) =

LC(m-u ) 'w)gexp(-~Y-1[m-u]+ 2~im'v). m E2z n

runs over all integral columns with

n

elements.

PROOF:

Use Siegel

Let q E ~.

[44], Page 65, formula

The concept

(57).

of a character mod q, a primitive

mod q and an even character mod q is defined like in Hasse or Landau [22], Kapitel form the Gaussian

22. With an even character M mod q and a E ~

sum

(4)

G(x,a)

Then if

M

~

(6)

I = q ~

ab~ L x(b)exp( 2 ~ i~-J b mod q

is even, an easy computation

(5)

THEOREM 2:

Let

character [15], § 13

M

shows

= G(~,a)

be a primitive

character mod q. Then

G(x,a ) = ~(a)G(x)

(a E ~ ) .

Here

a(x)

(7)

PROOF:

Let

(8)

See Landau [22],

=

a(x,1).

§ 126.

X1 be an even character mod q, form the row

Y E ~2);

and define the thetafunction

t E2,

t > 0

1 = (1,1), let

I

n

(9)

e(q,l,x1,Y,t)

I

= (Det y)4 t 2

x1(al)exp(- ~ Y[a]t).

a=(a 1 ,a 2) ' a 2 = 0 mod q

Put (10)

Q(Z) = ({~

0q) ,

v

(11)

Y

=

( Y E Q ( z ) ] ) -I

.

Then v

(12)

THEOREM

(13)

Det Y = q-2Det y-1

3:

Let

X1

e(q,l,x1,Y,t)

be an even primitive character mod q. Then

= G(X1)%(q,l,x1,Y,t

)

and

abs G(X1 ) = 1 .

(14)

PROOF: In (9) set a = b+qc gral column. Then (9) gives (15)

e(q,l,x1,Y,t)

with

1 = (Det(tY)) ~

b = (b10)'

~ X1(bl) blm°d q

Apply (3) with

and

c

an inte-

~ exp(-~(qYt)[c+b]). c EZ~2

n=2 , g=O, u=0, v= ~q , qYt instead of Y. Then

1 I ~(q,l,x1,Y,t) = (Det(tY))~(Det(qYt)) -~

(16)

~ ( ~ Xl(bl)exp(2wi ~ ) ) e x p ( - q c E ZZ2 blm°d q

x Y-licit-I).

From (4), (6), (12), (16) we get

(17)

I I e(q,l,xl,Y,t) = G(~l)(Det Y)~ t-~

V

m

L ~'(01 ) exp(- ~Y[Q(1)C]~I)= c E2Z2

V

G(x1)e(q,l,x1,Y,t -1) . --

v

Herewith one has formula (13). Inserting XS ,Y,t side of (13) and applying (13) once more one gets (18)

-I

in the left hand

G(xI)G(~I) = I

From (5), (7), (18) we deduce (14). Theorem 3 is proved.

According to MaaB [33], pages 210, 267 form the differential operator I 1 (19) D*(t) = t~ ~t t2 Tt t-~= - ~+ + 1t ~t t2 ~dtd2 Using (20)

d _t-2 dt =

d -7 dt

one see' s (21)

D*(t) : D*(t-1),

Put Ii D*(t) (22)

D(q,t) =

(q>

Then also

(23)

D(q,t) = D(q,t -I)

Put (24)

"e(q,l,x1,Y,t)

THEOREM ¢: (25)

PROOF:

Let

XI

= D(q,t)~(q,l,xI,Y,t)

be an even primitive character mod q. Then

e(q,l,x!,Y,t)

= G(X1)e(q,l,~l,Y,t -I) .

Apply Theorem 3 and formula (23).

From (9), (19) one obtains I I : (Det y)4 t 2 m

(26)

e(1,1,1,Y,t)

THEOREM 5:

(27)

L{(2uY[a]t)-(NY[ajt)21exp(-NY[a]t). a 6~ 2 a ~ o

For all even characters

abs e(q,l,x1,Y,t)

XlmOd q

I I <_ c1(Det Y)~ t~

one has

exp(-~q Y[aJt), a EZZ 2 a~O

with a constant

c I > I.

PROOF: For q = 1, X1 = 1 this follows from (26). For it follows from (9), (22) and XI(0) = O.

q > I

THEOREM 6: Let R 2 be a non-singular rational 2x2 matrix, XS an arbitrary even character mod q and j(Y) a positive number with (28)

Y ~ j(Y)E .

Then there exists a real number (29)

abs e(q,l,x1,Y[Re],t)

c 2 = c2(R2) ~ I with 1 1

~ c2J(Y)-l(Det Y)~ t -~exp(-c21j(Y)t).

PROOF: Let R 2 = rG with r 6 Q, r ~ 0 and G integral. If a is an integral column, so is b = Ga and if a ~ 0 then b ~ 0. Because of (27), (28) it suffices to prove for u > 0:

(3o)

e~(-2nub'b)

< dlu-le~(-d~lu)

~2

bE

b~O with a constant

dI ~ I .

Set (31)

U%(u) =

~ exp(-~v2u) V

=

•

--oo

Then from (3) we obtain 1

(32)

v~Cu) = u ~#(u -I) .

Furthermore (33)

t~(u) < ~(S)

(u > I).

Hence from (32), (33) I

(34)

From (33), (34) we get for (35)

(u<

~(u) <_ u ~ 0 1

I).

and some constant

~(u) <_ d2u ~exp(¢u)

(u > 0).

d2 = d 2 ( ¢ ) > 1

8

On the left hand side of (30) is with ¢ = I we get

exp(-2Nub'b)

b'b > 1. Hence from (31) and (35)

<_ exp(-~ru)~-2(u) <_ d2u-lexp(-(rr-2)u).

b 6

b~O Now (30) follows. Let (36)

3(1) = {z = x + iy E C; x,y E IR; y > Ol

be the upper half plane,

(37)

Sp(1, IR) = {M = (a b); a,b,c,d EIR; ad - Do = 1}

thesymplectic by

and

z-

group o f d e g r e e one. Sp(1, 1~) o p e r a t e s on

~(1)

M with

(38)

M(z>

= ~a z + b

.

The subgroup F(1) = Sp(1, 77) of Sp(1, IR) with integral the elliptic modular group. It eperates discontinuously on A fundamental

domain of f(1)

(39)

Sp(1, 2Z) is given by

= Iz ~ ~(I);

abs z > I; abs x <

Obviously

y>

(40) Let

(41)

z E }(I)

and set

Y =

Then (42)

Det Y = 1 .

(z ~

~(I)).

½}.

M is ~ (1).

Obviously (43)

y-1 = Y[I] ,

with

The column (45)

w = (i -i~)'

is an isotropic vector of is an isotropic vector of column. Then

Y. Because of (43) the column I-lw y-1. Let h = (hl,h2)' be an integral

Y[h] = 11h17 *

(46)

h212

and

(47) Let

(48)

w'Yh = hl~ + h 2 . r E I~, t E IR, t > 0 e*(r,z,t)

=

and set

I (hl~ + h2)2rexp(- ylhlZ + h212t) h E2Z 2

Then from (46), (47), (48) we get (49)

e*(r,z,t)

= t -2r

~ (h'tYw)2rexp(-~(tY)[h]). h E2Z 2

The application of (3) to (49) gives (50)

e*(r,z,t)

= (-1)rt-1-2r ~(h'w)2rexp(-wY-1[h]t -I). h ETz 2

Using (43) and writing

h

instead of

Ih

we get

"

10

(51)

e*(r,z,t) = t -1-2r

~ (-1)r(h'Iw)2rexp(-NY[hSt -I) . h E2E 2

An easy computation shows (52)

(-1)r(h'Iw) 2r = (hl~ + h2)2r .

Inserting this in (51) and using (46), (48) we get (53)

e*(r,z,t) = t-l-2re*(r,z,t -1).

An easy computation shows

(54) ~om

e*(r,M,t) = (c~ + d)-2re*(r,z,t) r > 0

(55)

(M ~ r ( 1 ) ) .

and (48) we deduce

abs e*(r,z,t) <_ (~)r

2~ r , t 2) ~(tlhl z + h21 ) e x p ~ - ~ hlZ + h21 , h E2E 2

h40 hence

(56)

abs ~*(r,z,t) _< c3(~) r

~ exp(-(N-¢)tlhlZ + h212), h E2Z 2

h40 with

0 < e < ~

and some constant

c3(¢) ~ 1.

THEOREM 7: Let M E r(1), z E ~(1). Then there exists a constant c 4 = c4(M) ~ 1 with t abs e*(r,M
(57) PROOF: by (54) (58)

For

z E ~(1), M E r(1)

we have

abs e*(r,M,t) ~ abs e*(r,z,t)

Icz + dl ~ 1 . Hence

(M E r(1)).

11 For

z E ~(1) we have 22

lhl z + h212 = Izi h I + 2xhlh 2 + h h 2I - lh111h21

>_.

+ h 2 = ~I h'h + ½ ( l h 1 I - l h 2 1 ) 2

>_ [I h'h.

Hence I t h l z + h212 >- 2 h ' h Using (56) with

¢ = ~

(z E # ( 1 ) )

we get _

abs e*(r,z,t) < c3(~t)

r

2

t

~ exp(- 1~y

h'

h).

h ~2 h~O Hence from (30), (58) we get (57). Theorem 7 is proved.

§ 2.

EPSTEIN'S ZETAFUNCTIONS AND L-SERIES

§ 2 contains the definition of Epstein's zetafunctions and Lseries. The functions are analytically continued and functional equations are proved.

Let D(2) be the group of unimodular 2x2 matrices, i. e., 0(2) consists of all integral 2x2 matrices U with abs U = 1. For q E form the subgroup

= IU = (ac ~) E ~ ( 2 ) ;

(59)

o m 0 mod qt

and set

(60)

A = tU : (.~1 +1) e ~ ( 2 ) } w

Let

i = (1,1),XI an even character mod I, Y E ~(2) and

complex variable. Define

w

a

12

(6~)

~(q,l '71'Y'~) = ~I

~ XI (hl) (y[h])-m

h=(h I ,h 2) 'E ZZ2 h 2 _= 0 mod q

h¢O ,¢

(62)

xI(a)((Y[U])I)-W

C*(q,I,xI ,Y,~) =

U = (ac ~) E 2/A H e r e (Y[U]) I means the upper left element of Y[U] as was already defined at the beginning of § I. It follows from Siegel [44], Chapter I, § 5 that the series (63) If

(61),

(62) converge absolutely for

Re w > 1 q = I (and hence

X

is the principal

called "Epstein's

zetafunction".

(62) an Epstein's

zetafunction.

For If

character)

q = I q > I

(61) is

we shall also call

we shall call (61),

"Epstein's L-series". The series

(62) may be written as

^c * ( q , I , × I , Y , ~ )

(64)

= ~1

~ x 1 ( a ) ( y [ ( ac ) ] ) -~

=1 c ~ 0 mod q In (61) we have to sum over those

Xl(hl) h

(65)

= 0

for

> 1. H e n c e it suffices

for which

= 1

Then one may set (66)

h = k(ca) ,

From (65) w e conclude follows that A

(6'/)

= I.

(ak,q> = 1, hence from

h 2 ~ O mod q it

c ~ 0 mod q . Therefore A

C(q,z,×1,Y,,~) = L(Xl,2~,)C*(q,l,×1,Y,~),

(62)

13 where

L(XI,...)

denotes Dirichlet's L-series.

With two complex variables

(68) (69)

~= ~(q,I,xI,Y,s)

S2-

Sl,S 2

set

St+

t ~ ,

= ~(q,l,xI,Y,w);~*(q,l,x1,Y,s)

=

A

= ~*(q,i,x1,Y,~)

Then

Sl-S 2-

(70)

~(q,I,xI,Y,s)

: ½

I x1(hl)(Y[h]) h = (h 1,h 2)'62Z 2 h 2 = 0 mod q

h#0 1

Sl-S2- ~

(7~)

~*(q,I,xI,Y,s)

If one sets absolutely

q~

Re s

xI(a)((Y[U])I )

=

(v = 1,2), the series (70),

(71) converge

for

I

(72)

~2 - ~ 1 > 5

"

From (67) we get (73)

THEOREM 8:

~(q,l,x1,Y,s)

= L(xI,2s 2 - 2s I + 1)~*(q,l,x,Y,s)

Let ~uI

(74)

U= 3

Then

u2

u4/

EY

•

14

(75)

;*(q,I,×I,Y[U],s)

(76)

~(q,I,xI,Y[U],s)

PROOF :

(77)

= X~I(ul)~*(q,I,xI,Y,s), = X~1(Ul)~(q,l,Y1,Y,s)

From (71) we deduce 1

s 1- s 2 -

c*(q,I,x1,Y[U],s)

=

xI(a)((Y[UV])I ) v

=

:

-I

×1 (ul)

~ x1(ul v ~ e/~

a)((Y[UV])

I

1 )s l - s 2 - ~

With V also UV runs over W/A and (UV)I ~ ula mod q. Hence (75) follows from (77). From (73) and (75) we get (76). Theorem 8 is proved. Set (78)

1 = (1,1) and f(l,Y,s)

(79)

1

= (Det Y) s 2 + ~ ( Y 1 ) W =

(81)

(0 1) ,

Sl = -s2;

THEOREM 9: The function degree s I + s 2 and

s2 = -si; ~ = (sI'82)

f(1,Y,s)

is homogeneous

f(z,Y,s) = f(1,Y,s)

(82)

(83)

1

= (y[w]) -I = y-I[w],

(80)

Let

s1-s2-

e = (1,1)

and

a E C. Then

(Det Y)af(1,Y,s)

= f(1,Y,ae + s)

"

in

Y

of

•

15

Let

1

D =

be a real upper tringular matrix. Then d f(1,Y[O],s) = f(1,Y,s)(abs dl)

(84)

2si- ~I

2s2+ I (abs d2)

Especially

(85)

f(1,Y[V],s) = f(l,Y,s)

PROOF:

(V 6 A)

It suffices to prove (82), the rest is trivial.

Set (86)

Y =

~112

Y2/Yl~I

Then

(87)

f(l,Y,s) = (net Y) Yl

s2+ ¼

_

Yl

Sl-s 2-

1

\

(88)

\- Y 1 2

Y2

Hence f(l,Y,s)=((Det Y)-I) -st+¼

s2+ ¼ Sl-S 2Yl (D-'gr-g) -s2+sl-~ =(Det Y) Yl

= f(l,Y,s)

This proves (82). Theorem 9 is proved. For (89)

U 6 ~

set = WU'-Iw 6 Y •

I

16 If

U

runs over

~/A

also

13

(90)

U

does. Put

V

,

U

=

then (91)

UlU 4 m U4U 1 ~ 1 mod q .

Set

(92)

A(q,I,xI,Y,s)

= (Det Y)

A(q,l,xi,Y,s)

=

s2+

I :C*(q,I,x1,Y,s).

Then (93)

I, XI (a)f(l'Y[U]'s)" u =

From (75), (92) (94)

THEOREM

(96)

deduce

A(q,I,xI,Y[U],s)

= X~I(ul)A(q,I,xI,Y,s)

A(q,t,x1,Y,s) = A ( q , I , x t , Y , s ) From

ulu 4 ~ ! I mod q

and (91)

we get

Ul ~ ~ ulm°d q' ~ 4 ~ ! u4m°d q -

Use

(97)

(u ~ ~).

10:

(95)

PROOF :

we

(~ ~) ~ ~/~

Y[U]

=

Y[U]

Then because of (82), (96), (97)

1"/

A(q,I,xI,Y,s) =

XI(Ul)f(!,Y[U] ,~') = u ~ ~/A

~'XI(Ul)f(I,Y[U],s) = A(q,I,x1,Y,s). u ~

~/~

Theorem 10 is proved.

THEOREM 11: (98)

Set

&(q,m)

: I(~(1-~) I (q=1) 1 (q> I)

Then the function W

(99)

~(q,l,×1,Y,~) = 5(q,m)(~) -m F(m)(Det Y)~ ~(q,l,xI,Y,m)

is homogeneous in (100)

Y

of degree

0 and w I ~ 1 dt I(q,I,xI,Y,~) = (Det Y)~- ~ A2 ~ e(q,l,xi,Y,t)t ~- ~ T " o

Furthermore

(lOl) PROOF :

6(q,I-~) = ~(q,~)

Let

q = I, X1 = 1. Then from (26) we deduce 1

~

1

(Det Y)g- ~ gI ~ e(1,!,1,Y,t)t ~ - g Tdt o (Det y)2 1

=

ZI (2wY[a]) ~ t~+lexp(_wy[ a ] t ) ~ a E~ 2 o a~O

(~Y[a]) 2 ~ t~+2exp(-~Y[a]t)~ o

1

18 W

(Det y ) 2 ~ -~ 21

l(Y[a])-~)(2F(w+l) a

- F(w+2))

E~ 2

a~o (Det y)2 - ~

Hence for

Let

q>

~(l,w)r(w)~(t,Z,l,Y,~)

= X(1,Z,I,Y,~)

q = I the theorem is proved.

1. Then from (9), (24) we get I ~ - ~ dt -~- = (Det y)2-[ !2 f e(q,l,x 1,Y,t)t '~'

1

0 oo

(Det y)2

~. X (a I) ~ tWexp (- ~ Y[a]t)~ a=(ala 2) ' o a2 = 0

mod

q

a~O W I

A

(Det y)2 (~)-~F(w)~(q,l,x1,y,w) = ~(q,l,x I,Y,~)This proves theorem 11. We introduce the functions (Io2) and for (103)

where

~(X1,w) = (~)-wr(w)L(×1,2~), q = I: F(w)

= ~(I,~)~(I,w)

L(1,w) = C(~)

=

~(I-~)~ -~ r(~)~(ew)

is Riemann's zeta function (q= 1)

(104)

F*(q,x1,w) = 8(q,w)g(X1,~) =

IF(~)

~(xl ,w)

( q > I)

19

From (67), (99), (I02), (I03), (104)we get (105)

w_ ~(q,l,xI,Y,~ ) = F*(q,x1,~)(Det y)2 ~.(q,l,x1,Y,w)

Put (106)

~(q,l,x1,Y,s) = I(q,I,xI,Y,~)

Then (107)

1 t n sl-s2- ~ 1) x X(q,l,Xl,Y,s) = 6(q,s2-s I + 2)(q) r(s2-s I + ~(s2_sl) + I (Det y)2 ~ c(q,l,x1,Y,s )

(lo8)

x(q,z,Xl,Y,s)

= d

F*(q,×l,s2

- sl + ½)(Pet ¥) ~ ( s 2 - ~ )

+~1 C (q,l,X1 ,Y,s) "

From (92), (108) we deduce (109)

k(q,l,xI,Y,s) = F*(q,Xl,S 2 - s I + ½)(Det Y) - ~(Sl+S2)A(q,i,xI,Y,s )

The formulas (80), (81), (95) give us (110)

~(q,l,x1,Y,s) = 1(q,l,×1,Y,s)

From (100) we get co

(111)

1(q,l,x1,Y,s) = (Det Y) (s2-sl) 2I j~ e ( q ' l ' × l ,y,t)ts2-sl -~dt o

THEOREM 12:

Let

m,p,¢ E lq; m,¢ > 0 oo

(112)

I(m,p) = J uPexp(-u)~ m

and set

20 Then there exists a constant (113)

c 5 = c5(P,e ) ~ I with

I(m,p) ~ c5(m° + m p-¢ + m p~s)

PROOF:

Obviously

(114)

I(m,p)

< I(1,p)

(115)

I(m,p)

<_ I ( 1 , p )

(m > 1) + ¢(m,p)

(0 < m < 1)

with I ~(m,p) = ]' uP-ldu

(116)

(0 < m < I)

m

If

p~O ¢(m,p) =

(117) For

aDs P ~

ruP] 1

< 1 mp p Jm - abs p + abs p

obviously

(118)

~(I + m p)

~(m,p)

(abs p ~ ~)

.

Now

(1~9)

1 S ~I( u

-C

+

U ~

) ,

and therefore

(120)

~(m,p) ~ ~(@(m,p-¢)

But for aDs p ~ ~ we deduce (121)

we have

+ $(m,p+¢)) abs(p ± e) ~ ~ • So from (118),

~(m,p) ~ ~(2 + m p-¢ + m p+c)

This holds for

(0 < m < 1)

abs p ~ ~ . But because of (118),

(119)

(120)

21 it also hold'sfor abs p ~ ~, i. e. for all p. From

(114), (115),

(121) we deduce (113). Theorem 12 is proved. THEOREM

(122)

13:

Let

~1

X1(q,l,xI,Y,s)

be a primitive

even character mod q and

= (Det Y) (s2-sl) ~I ~ e(q,l,xI,Y,t)ts2-sl

dt

T

I The integral of the right-hand side of (122) converges absolutely and is a holomorphic function for all s E ~2. Let R 2 be a nonsingular rational 2x2 matrix, j(Y) a positive number with Y ~ j(Y)E and Ac C 2 a compact domain. Then there exists a real number c 6 = c6(R2,~) ~ I and three linear functions

(123)

I(,,,o)

=

~(~2-al) + j($) or

(: = 1,2,3)

j($) with rational

(124)

j($) ($ = 1,2,3),

such that for

s ~ ~

the inequality

abs k1(q,l,~1,Y[R2],s ) 1 1 3 c6(Det y)~(a2-~1 )+T ~ (j(y))S($,o) $=I

holds. By the formula Sl-S 2 (125)

x(q,l,xI,Y,s ) = ~1(q,l,x1,Y,s)

+ G(XI) q

v X1(q,l,~1,Y,-s )

the function k(q,l,x1,Y,s ) is holomorphically continued to C 2. There is a real number c 7 = c7(R2,~) ~ 1 such that for s E one has (126)

abs ~(q,l,x1,Y[R2],s ) ~

1_(c2_Cl)+ ¼ 3 1 3 c6(Det y)2 ~(j(y))~($,d)+ c7(DetY)~(~2-dl)-~ ~(j(~)~(''-(~! : =1 ~=1 Finally one has the functional

equation

22 (127)

k(q,l,x1,Y,s) = G(x1)q sl-s2 k(q,l,~1,~,-s )

PROOF:

From theorem 6 and (122) we deduce

abs xI(q,I,xI,Y[R27,s) ~ dlJ(Y)-1(Det Y) ~(c2-~1)+¼

x

I

t~2-q1-~exp(-c21j(y)t)~ with a constant we get

1 d I ~ 1 . Using

as a new variable

t* = c~lj(Y)t I q1-~2 - ~

abs Xl(q,l,~l,Y[R2],s) ~ dlJ(Y)

I

I(c210(Y),o2-~I-~)

(Det

I I y)~(q2-Cl ) +

x

•

From this and theorem 12 we get the first part of theorem 13 till formula (124). From

(111)

we conclude

k(q'l'Xl'Y's)

= k l ( q ' l ' x 1 'Y's)+(DetY

~2_sI )½~I e ~ ' l ' x 1 'Y't~s2-sld-~tt o

Applying (12), (25) we get k (q,I,xI,Y,s)= kl(q,l,x1 ,Y, s)+G(x1)q

Sl-S 2

x

v -1 )ts2-sl (Det Y) 1(s1-s21 ~t e(q,i,x1,Y,t o Making the substitution

t ~ t -1

d_~t t

gives (125).

(126), (127) follow from (18), (124), (125). Theorem 13 is proved. From (68), (106), (127) we obtain I (128) k(q'l'x1'Y'w) = G ( X I ) J - •

^k(q,l,~1,Y,l-~). v

23 A

k(q,I,xI,Y,~) THEOREM 14:

is holomorphic for Let

q > 1

and

mod q. Then c*(q,I,xI,Y,s) in the domain

(129)

w E C.

XI an even primitive character and

A(q,I,x1,Y,s)

are holomorphic

e2 - al ~ O

C(q,I,x1,Y,s)

is holomorphic for

s E C2

PROOF: From q > 1 and (104) we deduce F (q,Xl,~) = g(X1,W). It follows from Landau [22], § 128 that g(X,~) has no zeros for Re ~ ~ ~ . Hence the first part of the theorem follows from (108) (109). The last part is a consequence of (107). Theorem 14 is proved.

THEOREM 15: Let q > I and XI be an even primitive character mod q. Then ~ (q,l,xI,Y,w) is holomorphic in the half-plane Re ~ ~ ~ . The function ~(q,l,xI,Y,~) is holomorphic for all w

E C

PROOF:

Apply theorem 14. 1

Set

(130)

P(1) = q2 WQ-I(1) =

Then (131)

abs P(1) = I

and (132)

p2(1) = E .

From (11), (80) we get V

(133)

Y = q-Iy[P(1)]

24 By theorem 11 and (106) k(q,l,x1,Y,s ) is homogeneous degree O. Therefore (110), (127) give us

in

Y

of

s I -s 2 (134)

X(q,l,x1,Y,(Sl,S2))=

This is equivalent

G(xI)q

k(q,!,~IY[P(1)],(s2,sl))

to I

(135)

~(q,l,×l,Y,~)

= G(×I)~-

~(q,l,~l,Y[p(1)~,l

- ~)

Let q = I . Then Y = O(2) and W,Q(1), P(1) E ~(2). Then from (68), theorem 8, (lo7), (108), (110), (127), (128), (134), (135) we deduce

(136)

k(1,1,1,Y,s)

= ;k(1,1,1,Y,~)

= ~.(1,1,1,Y,(s2,sl))

k(1,1,1,Y,-s)= (137)

§ 3.

&(I,I,I,Y,~)

= X(I,I,I,Y,I-~)

ELEMENTARY EISTF~STEIN

X(I,I,I,y-I,s), = ~(1,I,I,y-I,w)

SERIES

§ 3. contains the definition, analytic continuation equation of elementary Eisenstein series.

Let

z ~ 3(1). Like in (41) put

(138)

Y = Y x

Then (139)

Det Y = I

From (46) we deduce

(14o)

=

I Y[h] = ~lhlZ + h212

and functional

25

Let

I ' ( I ) = Sp(1, 7Z)

(141) For

and

F1(1) = {M = (±~ ~ ) M = (~ ~) £ Sp(1, ~ )

(142)

let,

6 F(t)}

l i k e in (38),

M - aZcz + db = XM + iy M

with real xM,Y M. Then

(143) Furthermore

put

(144) With

Y icz+dl2

YM =

M{z} r C~

(145)

U 0

and

6(1,r,z,s)

=

s 6 ~ =

cz + d set ~ (M{~})2r YM

s+~+r

M c rl(1)\r(1) Then from

(64), (140) we deduce

(146)

~(1,o,z,s)

Because

of (105),

(147) From ( 1 3 7 ) ,

(148)

= ~*(1,1,1,Y,s (139),

X(l,l,l,Y,s (147)

a real number

½)

(146) we get

) = F(s +

)~(1,0,z,s)

we o b t a i n

~(1,1,1,Y,s

THEOREM 16: Set 2×2 matrix, ~ c C

+

+

+ ~)

= ~(1,1,1,Y,-s

+ ~) L

.

c = Re s. Let R 2 be a non-singular rational compact, s E ~ , z 6 ~(I). Then there exists

c 8 = c8(R2,~ ) ~ I

and six linear functions

26

± o + j(~)

I

(~,a) =

(149)

(~ = 1,2,3,4,5,6).

or

J(~)

with

6 (150)

abe ~(I,I,1,Y[R2],s + 1) _< c 8

~[ y-~(l,a) I=I

PROOF: Because of z E ~(I) and ( 4 3 ) we may apply theorem 13 with j(Y) = j(Y) = dy -I with some constant d. Now theorem 16 follows from (42) and theorem 13. It is well known from the theory of elliptic modular forms that the series (145) converges absolutely for a > ~. The case r = 0 was just treated. Now let

r E~

• Put

(151)

~(r,s) = n-Sr(s+r)~(2s)

THEOREM 17 :

Let

a > I

(152)

and set

~*(1,r,z,s)

= ~(r,s + ~) ~(1,r,z,s)

Then dt x*(1,r,z,s) = gI ~ r ~ e,(r,z,t)t s + ½ + r ~-

(153)

o PROOF :

From (48) we obtain 1 mr ~

s + ~ + r dt T = o oo 1 I r ~ h2)2r f s + ~ + r exp(- ~lhl z + h2J2t) ~ ~ (h1~ + t e*(r,z,t)t

=

h EZZ 2 F(s +

+ r)w

=

o -s - ~ I ~

Z

(hl~ + h ~ ~2

h2)2r(

Y

lhl z + h21

2)

s +

+r

=

27

~(r,s

+

1 1 ~)~

~(cE

+

d)2r(

=1 Theorem

Y

Ioz+dl

S + 1+

r = X*(1,r,z,s)

2)

17 is proved.

From (145) we deduce (154)

~(1,r,M
: (MIEI)-2r ~(1,r,z,s)

(~ ~ r(1)).

Hence by (152) (155)

THEOREM

(156)

~*(1,r,M
Let

18:

s

j. e.(r,z,t)t s +

I and satisfies

k~(1,r,M
+ r Tdt

the equation

= (Ml~l)-2r~ (1,r,z,s) 1

(M 6 F(1))

and three functions

There are a constant c 9 = c9(M) > I ~(~,a) of type (149) with

(15a)

(M (r(1))

r e E. The function

k..( 1,r,z,s) = ~1 ~ r

is integral in (157)

= (Ml~l)-2rk*(1,r,z,s)

3 abs kl(1,r,M,s ) <_ c 9 I y~(~,o) ~=I

(z c if(1), M ~ r(1)).

By (159)

f(1,r,~,s) X~(1,r,~,s) ~(1,r,~,-s) =

+

the function k *( 1,r,z,s) is holomorphically it satisfies the functional equation

(16o)

~*(1,r,z,s) = ~*(1,r,z,-s)

Furthermore (161)

continued to ~ and

there is an estimation 6

abs •*(1,r,M,s)

<_ c 9 I y1(~,a) ~=1

(z £~(11, M ~ F(1)).

28 with functions

S(~,o)

of type (149).

(157) follows from (54). Let z e ~(I), M ~ r(1). From theorems 7 and 12 we get I ~ I o + ~ + r ~ ~ o - ~ _c~1 ~dty, t abs k~(1,r,M(z>,s) S c9Y ( ) exp( I I ~ +~+ r (y-1 = c9Y I ,o - )

PROOF:

Now the estimate

(158) follows from theorem 12.

Because of (158) the function From (53),

k1(1,r,z,s)

is integral in

s

(153) we get s+l+rdt

k*(1,r,z,s)

= kl(1,r,z,s ) + 1 r

~ ~ * (r,z,t)t

t

O

= x~(1,r,z,s)

I 1 r ~ e*(r,z,t-1)t s + ~ ~

I 2

r T dt =

0

Xl(1,r,z,s)

+ 1 mr ~ ~*(r,z,t)t -s +

+ r dt T

=

1

Xl(1,r,z,s)

+ Xl(1,r,z,-s)

Herewith we have (159). From (158), Theorem 18 is proved.

CHAPTER II. PREPARATIONAL

(159) we obtain (160),

(161).

MATERIAL

In the following two chapters we develop the theory of Selberg's zetafunctions and L-series. In the present chapter we collect some material which will be needed later. We prove some results on systems of Dirichletcharacters. We investigate matrices and subgroups of the unimodular group. We collect basic results of the space of positive matrices and on thetafunctions.

28 with functions

S(~,o)

of type (149).

(157) follows from (54). Let z e ~(I), M ~ r(1). From theorems 7 and 12 we get I ~ I o + ~ + r ~ ~ o - ~ _c~1 ~dty, t abs k~(1,r,M(z>,s) S c9Y ( ) exp( I I ~ +~+ r (y-1 = c9Y I ,o - )

PROOF:

Now the estimate

(158) follows from theorem 12.

Because of (158) the function From (53),

k1(1,r,z,s)

is integral in

s

(153) we get s+l+rdt

k*(1,r,z,s)

= kl(1,r,z,s ) + 1 r

~ ~ * (r,z,t)t

t

O

= x~(1,r,z,s)

I 1 r ~ e*(r,z,t-1)t s + ~ ~

I 2

r T dt =

0

Xl(1,r,z,s)

+ 1 mr ~ ~*(r,z,t)t -s +

+ r dt T

=

1

Xl(1,r,z,s)

+ Xl(1,r,z,-s)

Herewith we have (159). From (158), Theorem 18 is proved.

CHAPTER II. PREPARATIONAL

(159) we obtain (160),

(161).

MATERIAL

In the following two chapters we develop the theory of Selberg's zetafunctions and L-series. In the present chapter we collect some material which will be needed later. We prove some results on systems of Dirichletcharacters. We investigate matrices and subgroups of the unimodular group. We collect basic results of the space of positive matrices and on thetafunctions.

29

§ 4. S Y S T E M S

OF P R I M I T I V E

In this p a r a g r a p h X1,...,X m

denote

CHARACTERS

we i n v e s t i g a t e m

characters

are all p r o d u c t s

XU

... X v

In this p a r a g r a p h

we

shall

m,q

E ~

×1''"

, q > 1. U n d e r ,X m m o d q

Especially

Xl'''''×m

even.

THEOREM

19:

Consider

(163)

problem. with

supposition

Let

assumptions

primitive~

the f o l l o w i n g

problem.

exist m c h a r a c t e r s

the p r o d u c t s

= X~'''Xv

are p r i m i t i v e ~ are

(I ~ u ~ v ~ m)

which

Xv ~°

q. U n d e r

investigate

such that

(162)

the f o l l o w i n g mod

(1 <_ ~ _< v <_ m)

we are i n t e r e s t e d

in the

case

that

the d e c o m p o s i t i o n

q = ql...qr

with (164) For

a E ~,



(I = 1,...,r) (165) Let

(~

a X

m a mod

q~ a

X (~)

the r e s i d u e c l a s s e s

a mod q

~

mod

~

I m o d -g-

(~ = 1,...,r)

q~

q and define

the

characters

X (~)

by

X

(166)

(167)

define

~,a = 1,...,r).

by

be a c h a r a c t e r

= 1,..,r)

Then

= I

(~ ~ ~;

=

(~)( a)

= x(a

)

(~ = 1 , . .

i s a c h a r a c t e r mod q~ (~ = 1 , . . . , r ) a m a I ... a r m o d

q ,

,r) and one has

Let

3O

(168)

x(a) = X(al)

The formulas X

... X(ar)

= X(1)(a)

... x(r)(a)

•

(166) and (168) define a bijective map between

and the system

(169)

X

X(1),...,

is a p r i m i t i v e

primitive

×(r)

character mod q if and only if

character mod q~ (~ = 1,...,r).

X (~)

Furthermore

is a X

if and only if there is an even number of odd characters X(1) .... , y~r). ~

Especially

X

is even if all

is even among

X(1),...,M (r) are

even. Let

XI .... ,X m

According = 1,

•

(170)

.

be characters mod q and

(166) define the characters ,r)

•

fX(~~ )

~ X(~)

defined by (162). ' ~v~Z(~) (I _< U _< v _ ~ m;

Then = X~(~)

..- X v(1)

(1 _ < ~ _< v < _ m;

~ = 1 , . . . ,.r )

The assertion about the even characters

PROOF:

the rest from H a s s e [15], § 13,6,

follows from (168)

especially pages 210, 211.

By theorem 19 our problem is reduced to the case that

q

is a

power of a prime-number. THEOREM

20:

Let

p

(171) For

be a prime-number, q = p

p = 2, k = I

p = 2, k ~ 2

there exists no p r i m i t i v e

c

respectively X

X mod 2 k d

x = x 4 ×2k X4, X2 k

is a p r i m i t i v e

For

k

character.

For

may be u n i q u e l y w r i t t e n as

(c mod 2, d mod 2 k-2)

are "basischaracters" 2k

and

k

each character

(172) Here

k E ~

w i t h the conductors

4

2 the factor d does not appear. = X2k character mod 2 k, if and only if

31

(173)

c @ 0 mod 2

(k = 2),

(174)

d ~ 0 rood 2

(k > 2 ) .

It is

(175)

x4(-1) = -1

(176)

X2k(-1)= 1

Hence

X

is even if and only if

(177)

c m 0 mod 2 p ~ 3, k ~ 1

For

each character

X mod pk

may be u n i q u e l y

written as

(178) Here

c

x Xp

and

p respectively

X k

=

d

xp Xpk

(c mod p-l, d mod pk-1)

are "basischaracters"

with the conductors

~k.

d the factor X k does not appear. X k P character m o d p if and only if

For

k = 1

(179)

c @ 0 mod p-1

(k = 1) ,

(180)

d @ 0 mod p

(k > 1)

It is

(181)

xp(-1)

= -1

(182)

X k (-1) = I P

X

is even if and only if (177) is true.

is a p r i m i t i v e

32

PROOF:

All assertions

found in Hasse formulas

[15],

(181),

except the formulas

(181),

(182) may be

§ 13.6, page 212. For the proof of the

(182) we use the representation O~'

(i83)

a - w

(l+p)~"mod

p

k

with (184)

0 S ~'

from Hasse [15],

< p-l,

0 S ~"

< pk-1

p a g e 212 t h i r d

(P ~ 3)

row from above.

We s e t

a = -1

and take the square of (183). Then .2~'

(185) i.

1 ~ w

(l+p)

2-"

~ mod p

k

,

eo,

(186) From

2~' ~ 0 mod p-l, 2~" -= 0 rood p (184),

k-1

(186) we get

(187)

, =

~,,

p-1 2

'

=

0

.

Hence (188)

-I - w ~'l+p)°mod pk

Because of (188) the formulas (181), (182) follow from the left box in Hasse [15], middle of page 212. Theorem 20 is proved. Choose characters (189)

~1,...,~m+ 1

with

-I

¢v+1 ~v = Xv

(V = 1,...,m)

~ v- 1+ l

(1

Th en (190)

~

= Xo v u

-< u <- v -< m ) .

33

THEOREM

21:

For

(191)

the k = I

q = 2k

~ (1 -< U _< v _< m) can be p r i m i t i v e only for m=1 Xv~ t h e r e is no p r i m i t i v e character, for k = 2 there

one odd p r i m i t i v e primitive

character,

for

k > 2

there

are

. For is o n l y

even and odd p

characters.

PROOF:

For

2

apply

k ~

(k ~ I)

k = 1

the a s s e r t i o n

(190)

and

~

By t h e o r e m

20 the

My ~

from

theorem

20.

For

set c

(192)

follows

d

= X4 ~'t X2~

(U = 1 , . . . , m + 1 )

(I <_ u <_ v <_ m)

are p r i m i t i v e

if and

o n l y if (193)

c

@ c

(194)

d

@ d

holds.

X~

(~ = I .... ,m)

(195)

From

mod 2

(1S

mod

(I S U < v S m+l)

v

v

2

is even,

if

cH+ 1 = c lmod 2

this

THEOREM

theorem

22:

Let

21 f o l l o w s

p

u < v S m+1)

(u = 1,...,m)

immediately.

be an odd prime,

k E ~

and

q = pk

(196)

@

If t h e r e

are

characters

(1 ~ ~ ~ V ~ m)

Xl'''''Wm

are primitive,

such

that

then

(197)

m < p-1

(k = I),

(198)

m < p

(k >

I)

all

My u

(k = 2), (k > 2)

34

N o w let (197) respectively arbitrary index w i t h

(198) be fulfilled and let ~ be an

1 ~ ~

~ m. Furthermore

character m o d q. Then there are characters all

Xv~°

(1 _< U _< v _< m)

(199) holds,

let

X

be a primitive

XI'''''Xm'

such that

are p r i m i t i v e and that furthermore

Xa = X in case

k > I

moreover

each character

X~ (~ @ ~; H = 1,...,m)

m a y be chosen even or odd. If one demands that all characters X1''"

'Ym

are even the same assertion

is true if the condition

(197) is replaced by (200)

PROOF:

m < ~

Set c ¢~ = ×pU

(201)

d ×

(~ = I, . . . . m + 1 ) . P

By theorem 20 the

Xv ° ~ ( 1 _ ~ v _~ m _)

are primitive,

if and only

if (202)

c

@ c mod p-1 v

(I S ~J < v S m+1)

(k = I)

(203)

d

@ dvmod p

(I < ~ < v < m+l)

(k > 1)

holds.

The

X~

(W = I, .... ,m)

(204)

PROOF:

cU+ I ~ c~mod 2

Consider first the case

there are exactly inequality

(2o5)

are even if

p-1

(~ = 1,...,m)

k = 1. Let (202) be true.

different r e s i d u e c l a s s e s mod p-1

(197) follows. Now let (197) be true and set

w

b

= xa X k

P

Then (202) may be fulfilled with the additional (206)

c +I - c

~ a mod p-1

condition

Since

the

35 Q

H e n c e the

XvW

(I ~ W ~ v ~ m)

To d e m a n d that all c h a r a c t e r s to d e m a n d that all

~I,...,¢m+ I

(207)

c

the case

only if (198) holds. obtain

(206)

(199) holds.

THEOREM

is true one may in a d d i t i o n

23:

y~

(U ~ ~)

if and to (203)

2tpl q . . . . p1,...,p r

are d i f f e r e n t

may be ~J even or odd.

Ps+1 < P s + 2 <

(212)

0 <_ t ,

(213)

ts+l,...,t r ~ 2

If there are c h a r a c t e r s

= X~ " ' " X~

t _ 2, m = 1

tr "'" Pr

odd primes,

(211)

~u

m a y be c h o s e n

ts+1 PsPs+I

Pl < P2 <

products

(206) the c

Let

(210)

(214)

of (197).

(203) may be f u l f i l l e d

E x c e p t of the c o n d i t i o n

(209)

either

instead

d + I - d~ m b mod p

chosen arbitrary. H e n c e T h e o r e m 22 is proved.

Here

(200)

and

(208) Hence

Then

If (198)

i. e.,

(u = I,...,m+I)

is true w i t h k > 1.

and (199) holds.

are even is the same as

are even,

m 0 mod 2

T h e n the same as b e f o r e Now consider

are p r i m i t i v e Y1'''''Xm

"'" < Ps

Xl'''''~m

;

"'" < Pr

'

mod q, such that all the

(1 _< k, -< ~ _< m)

are primitive,

or

t = 0, m < m i n ( P 1 - 1 , P s + 1 )

then

36

N o w let mod

(214)

q and

~

characters

be true,

X1,...,Xm

(1 ~ ~ ~ v ~ m)

X

mod

be an a r b i t r a r y

index with q,

index with

character

m o d q. T h e n

such that

all p r o d u c t s

furthermore

I ~ ~ ~ m

there

all p r o d u c t s

and m o r e o v e r

t = O, m < m i n ( P ~

an a r b i t r a r y

are

primitive

character

1 ~ ~ ~ m. T h e n t h e r e

such t h a t

are p r i m i t i v e

(215)

PROOF:

let

an a r b i t r a r y

...Tv

7a = X" Let

1 ' Ps + I) and

My = X

are

X

an even p r i m i t i v e

even c h a r a c t e r s

0

Xv u (1 ~ U ~ v ~ m)

,

X I , . . . , X m m o d q,

are p r i m i t i v e

and

X~ = X"

Combine

theorems

19,

21,

22

§ 5. M A T R I C E S

In this p a r a g r a p h coefficients

Let (216)

n,w,q

we p r o v e

and we

results

consider

on m a t r i c e s

subgroups

with

E ~,

n >

I, W >

I, q > 1 , D

(217)

k o = O, k

(218)

0 = ko < kI <

(219)

i ~) = k ~ - k V-I

(220)

k=

£~

(t = I,..

,w)

... < k w = n

(~1,...,kw),

k

1

t=1

,

,

(v = I ,.. ,w) , I = (li,.

, 1 w)

Then (221)

integral

of the u n i m o d u l a r

(v = 1 , . . . , w )

group

37 Especially

let

(222) with

k

n

times

=

(I ,2 .... ,n)

, i = (I,..

,1)

1. S e t

= lw+l -v

(223)

(v = 1,...,w~

(224)

k°

(225)

= O,

=

(v = 1 , . . . , W ) , ,~=1

(226)

k = (k1,...,k

w)

Then

(227)

DEFINITION

k

1:

Let

= k w - k w _ v = n - k W-~

~ = 1,...,n

and

,w)

(v = O,

v = 1,...,w.

Then

set

(228) for

(229)

kv_1<

~
Furthermore set (230)

~(~)

=

for

(231)

kv_1<

~ < B k

~

Obviously

(232)

~(,)

= w+1

- ®(n+1-~)

(~ ----1 , . . . , n )

38

Set

(233)

w*

= w-l,

(234)

k*

=

n*

= kw_ I

(k I .... , k w . )

,

, I

(11

=

. . . . .

lw.)

Then

(235)

n = n*

+ Iw

.

Let v I

(236)

V

(237)

=

lw_ v

(v =

v

v

ko

,1 w)

,

,

V

= 0

= Iw

V

1 = (11 . . . .

(238)

v Iw

I,...,w-I),

v

, k %) =

~ i

(%) = 1 , . . . , w ) ,

~=I V

(239)

k = V

(240) Let

and

V

,~)

,

~

~

l = (1 ,lw), denote

(241)

V

(k 1 ....

the

cyclic

k = (k*,k w)

permutation

)i = ( l w , l I , .

,lw_ 1),

set

(242)

}!

(243)

11

= (2~.,11,.

A

"'1--1)

'

A

= lw.,

1

= 1 %)-1 (v = 2, " .,w*) ' 1 W = I w "

Then

(244) (245)

A

A

... , w ) = ( A

A

k ° = 0,

k

=

%)

A

~

i

t=1

} I ,1 w) (~

=

,

1,...,w)

39

DEFINITION

~ 0

If

2:

n ~ m

and

G

is an integral n × m matrix,

is the greatest common devisor of all

of G. If

= 1

w e say that

G

m × m subdeterminants

is "primitive" 1

If

A

k8 x ky m a t r i x

is a

the splitting of Set

1 A

1 = A

~(n)

in

1

x 1

let 1

submatrices

(v = 1 , . . , m i n ( ~-~, y ) )--. a

~K

of

A

with

A = (Avp)

A

H

those elements Let

A

(1 ~ ~, y ~ w)

Obviously

be

belonging to v~ 1 A

~(:) = ~, w(~)

consits of @

be the group of u n i m o d u l a r n×n matrices U, i. e. of

the integral matrices subgroups of

U

with

abs U = I. Define the following

D(n)

(246)

1 Y1(1) : {U : ( U )

1 6 o(n); Uvp ~ 0 mod q

(t

(247)

I AI(1) = {U = ( U )

I E n(n); Uvu

(1 <_it < v < w ) }

(248)

Y2(1) = ~1(1),

(249)

~(n,q)

D(n,q)

is called the "principal

0

< 1 , < v <_w)},

o

A2(1) = A l ( 1 )

0 Yl(1)

,

= {U 6 D(n); U ~ E mod q}

congruence subgroup"

of O(n)

of

level q

T H E O R E M 24 :

(250)

(251)

It is

1 1 (Det Ul)...(Det Uw) = Det U = + I mod q

Ul...u n ~ Det U = ~ I mod q

1 (252)

PROOF:

1.

Det U % ) = i+ 1

Clear.

(U 6 ~I(I)),

(U E ~2(I)),

(,, = 1,...,w) (u ~ A I ( 1 ) ) .

,

40 For

I < ~ <

matrices Set

t < w

1 A = (A

let )

651(i,~,, ) 1 A

with

(~2(1;t,~) : ~ l ( l ; k $ , k

be the set of i n t e g r a l

~ 0 mod q

(I _< :~ _< a; I _< ~ < ~ _<

= 1,Rk A = k~l

(~ = 1,2),

(~

For

q > I, the c o n d i t i o n

~).

),

~e(1;~,~)=lAE C~o((1;$,~);<,q>

(253)

k x k

Rk A = k

in (253) f o l l o w s

= 1,2).

from

<,q> = I .

THEOREM

25 :

It is

(255)

RkA

PROOF:

~1(I;~'~) 1 with A = 0 ~1( l" , k $~," k )'

Set

k :w

(A E ~ ( 1 ; ~ , ~ ) )

•

Clear.

Let

(257)

=

be the set of i n t e g r a l (I ~ ~, ~ ~; I ~ ~ < ~ ~

k~ x k~ m a t r i c e s ~). Set

~2(1;t,~)

£~(i~ ,,,~) = }~(l~ ~,,~) n/o~(I~ ,,,~) A

= (a U )

(~ = 1 , . . . , k t; ~? = 1,...,k

~->(I; ~,~ ),] ~>(i~ denote those

1 A = (Art j) =

(~ = 1,2) .

) . Then

>-(1; ~,. ),j<>-(1;~,~ ),

,,~ ), }.>(1 ~,,..),£>(I ~,,~ )

subsets

of

~2(i;,,~),

]2(I;,,~),~2(I;,,x),

for

41

which

a

> 0

respectively

a

> 0

~*(1;$,~,--~*(1;$,~),~*(1;~,~

~>(i;~,~,),~>(i;$,,), (258)

(1

k -I k -2 a2~ al~ ...

(259)

matrices

PROOF:

A

with prescribed

Let

Thereexists

a matrix

(260)

P

P

(261)

<

'v <

k )

there are

exactly

1 a k -1

diagonalelements

al,...,a k

.

be an integral U E O(n,q)

nxm matrix with

I ~ m ~ n.

with

is primitive

and

P ~ (~) mod q .

E

PROOF:

U

U = (P *)

if and only if

Here

<:

Clear.

THEOREM 27:

denotes the

See Christian

THEOREM 28:

an

< a

~*(1;~,~,), ~*(1;~,~),~*(1;$,~)

In

of ~ > ( 1 ; ~ , ~ ) ,

for which

0 < a

THEOREM 26:

(v = 1 , . . . , k ). Finally let

be those subsets

U E ~ (1) _ _

(262) if and only if

Let

P

mxm unit matrix.

[7], page 24, Satz 3.6.

be a

n x k

matrix

with

u = (P *) P E ~(l;w,~)

(~ = 1,2).

(I ~ ~ ~ w). There exists

42 PROOF: Now let

From U = (P *) E ¥ (1) follows immediately P E ~ P E~(1;w,~). We have to show that there exists a

(l;w,~).

U E V (1) with (262). If P E ~ l ( 1 ; w , ~ ) there is a V E AI(1) with VP E ~2(l;w,~). Therefore it suffices to prove the theorem for = 2, i. e. for ~1(1) respectively ~l(1;n,m). Now let be 0

(263)

P E ~1(1;n,m)

•

For n = m the assertion is true. Now let we consider the case m = 1. Then we have (264)

1 < m < n. At first

P ~ (PlO...O)'mod q •

Chose an

(265)

a E~

with

Vl =

aPl ~ 1 mod q

1-1

and set

Pl

E ~I(1)

0

•

E (n-

Then (266)

VIP ~ ( 1 0 . . . O ) ' m o d

q . 0

Because of theorem 27 there is a V2V1P = (1 0 ... 0)' . Set (267)

V 2 E o(n,q) c ~I(i)

with

V = V2V I E ~1(1) • Then

VP = (1 0...0)'

.

Now let m > I. We make the induction assumption that for each P I E ~1(1;n,m-1) there exists a V I E W1(1) such that o

(268)

o

VIP I =

(E(m01))

. o

From (263) it follows P = (P1 *) with P 1 E is a VI E ~ 1 ( 1 ) w i t h (269)

(:(m-l)

~l(1,n,m-1). d)

VIP =

, P2

Hence there

43 1 P2 6 ~1(l,n+1-m,1)

(270)

1 1 = (I,...,I)

Here

. a

with

.

n+1-m times I. As already proved there is

I

V2 E g l ( t )

with

(271)

V2 P2 -- (I 0...0)'

.

With

(272)

V =

I

E m-l)

-d

0 .....

/E (m-1)

O~

o 0

I vl ~ ~I(l )

E (n

V2

follows (273)

VP = (E(m)) 0

.

o

o

For each P 6 i1(l;n,m) there is a (262) follows with U = V -1 .

V 6 ~1(I)

with (273). Now

Theorem 28 is proved.

Let

~(1)

be an arbitrary but fixed system of representatives of

the cosets ~2(1)/A2(1) and assume E £ L(1). Then ,~(1) can be also considered as a complete set of representatives of the cosets ~I(I)/~I(1).

THEOREM 29: A complete set of representatives of the cosets ~(1;w'w*)/A~(l*) (~ = 1,2) is given by the products (274 )

B

=

UD

with (275 )

1 U = (U

) 6 .~(1) ,

44 1 @ D = (D~u) 6 j (l;w,w*)

(276) Furthermore 1 B

(277)

1 - U

1 D

mod q

(~

= 1,...,w*)

.

PROOF: If B E ~ l ( 1 ; w , w * ) there exists a V E At(l*) with BV 6 ~ 2 ( 1 ; w , w * ) . Hence it suffices to prove the theorem for ~ = 2. Now

(278)

~2(1;w,w*)

=~l(1;m,n*)

°

First we prove that each (279)

B E%1(1;n,m)

may be written as (280)

B = UI DI

with (281)

U 1E

(282)

D1

Yl(i)

,

o

At first let

m = 1

E ~l(1;n,m) and

,

a = .

From (253), (279) it follows I. We make the induction assumption that for each (1;n,m-1 B E ~1(1;n,m-1) there exists a V E Wl(1) with VB o j1 From (279) follows B = (B I *) with B 1 E ~1(1;n,m-1). Hence there exists a V I E W1(1) with

)-

45

\ *

\

I (283)

VIB =

' am-1 ,0 \\.

Here 1 (284)

bI E %1(1,n+1-m,1)

Hence there is a

I V 2 E TI(1)

I V2b I E J 1(l,n+1-m,1)

with

IEim-1) (285)

.

i~

°

v =

v I ~ ~I(1)

V follows o

(286) Setting

D 1 = VB E ~ l ( l ; n , m )

U1 = V-1

•

we obtain (280), (281), (282).

Set (287)

W = W(n) =

\ 1 " 0

Obviously each matrix (288)

B2

6 ;62(l,n,n)

may be written as (289) with

B2 = D2 U2

"

. With

46

(290)

U 2 6 ~1(]°),

(291)

D 2 6 #l(l;n,n)

o

,

because i f we set B I = wB~W,

the conditions (289), (290), (282).

u I = wu~w,

D I = WD~W

,

(291) are equivalent to (280), (281),

Now proceeding with (280) we set (292)

U 1 = UV

with (275) and (293)

V

£ A2(1)__

e

Then (294)

B

=

UV D 1

and (295)

V D I ~#2(l,w,w*)

By applying (289), (290), (291) with we easily get the existence of a (296)

. 11,12,...,i w

V 6 A2(1 ) ,

such that (297)

V D I -- DV

with (276) holds. From (294), (297) we obtain

instead of

n

47 (298)

with

B = UDV

(275), (276), (296) .

Hence each coset of o~2(l;w,w*)/A2(l*) B of type (274) with (275), (276).

possesses a representative

The formula (277) is a trivial consequence of (274). Finally we have to show that each coset of -~(l;w,w*)/Ae(l*) (~ = 1,2) possesses exactly one representative (274) with A ~A V~ (275), (276). Suppose B = UD and B = UD with (299)

U,U E ~ l ) ,

(3oo)

D,D E ~ (1;w,w*)

represent the same coset of ~ ( l ; w , w w ) / A ~ ( l * )

(3oi)

(~ = 1,2). Then

= UDV

with

(302)

(~

V E As(l*)

=

1,2)

•

But %(1) E W2(1) and J * (l;w,w*) c J2(1;W,W*). Hence ~A ~V UD,UD E~2(1;w,w*) and from (301), (302) we get (303)

V E A2(1") •

From (301) we get V

(3o4)

A~

u IUD

=

DV

.

Then (300), (303),

(304) show us

(305)

-IG

2(1)

48 and because of (299) it is v

(306)

=

The formulas

(301),

U .

(306) give

(307)

=

From (300),

DV

.

(307) we deduce /k

(3o8)

V = E

and

v

o

--

o

.

Herewith it is shown that the cosets ~_ (_ l ; w , w * ) / A ~ ( l * ) (~ = 1,2) have exactly one representative of the described form. Theorem 29 is proved.

THEOREM 30: The groups ~ ( I ) , ~ (! i ) may be obtained by transposing all elements of ~ (I), &~(1). Then (309)

W(n)~(1)W(n)

= ~ (I), W(n)A~(1)W(n)

= ~(I)

(e

=

1,2)

•

1 Let

U = (Uvu) E ~ ( i )

and set

1 (310)

U = (U

) = W(n)U'-Iw(n)

E ~ (I)

(e = 1,2)

Then I (311)

PROOF :

~ I

mod q

(v = 1,...,w)

Clear.

THEOREM 31 : Then (}12)

1

(Det Uv)(Det Uw+l_v)

Let

Av

be integral

B

AvAv_q

k + I × kv matrices

A~j (I ~

_<

(v=1,...,w-1).

< w-l)

49 are integral kv+ 1 x k bility properties hold

matrices.

Furthermore the following divisi-

(313)

I (B _1,v_i>

(314)

(Bv,p_I>I ( 1 < u < v

(315)

I

If

A

(v

(316)

A

2 .... ,w-l).

*

xO i

x I

P

matrices

A

~u

(V = 1,...,W-1)

I • Avv

=

0

0/t

(p = 1,...,v),

(317)

(abs

(318)

(Bv_1,p)(abs

(319)

(abs Avl) =

PROOF:

=

then

Avv ) =

(v = 2,...,w-I)

Avp) =

(1
(v = 2,...,W-1)

With

(320)

U

6 0 ( k ~+1 )

(v = 1 '''" , w - l )

set (321)

A *1 = U 1 A1, A * = U A U-1 v v v v-1

(v = 2,

"'"

,w-l)

Then (322)

<w),

is of type Avl

with

(v=2,...,w-1),

* Bvp = Av*

and therefore

... A * = U

B V

V ~Jr

U-I ~- 1

(1 _< tl _< v <_ w - l )

,

, <w),

50 =
(323)

>

(1S

~ Sv

Sw-1)

By suitable choice of U I we first bring A I into the form (316)o If Uv_ 1 is already fixed we choose U such that A obtains the form (316). Hence we may assume that all A are of type (316~ Hence it suffices to prove (313), (314), (315) for matrices of type (316). But in this case (313), (314), (315) follow from (317), (318), (319). Hence it suffices to prove the latter formulas.

From (312), (316) we get

B vU

(324)

=

t B'~U'BI" o

(I S u S

*I

v

<_w-l)

with B vp~ = A v~ Av_ 1 ,~ . . . It follwo~ (325)

(326)

=

A

(1 <

. 1,. < . u .<

~i~

11 abs B j~ = t=l

II

v < w-l).

(I S u S v Sw-1).

11 abs A

k=U t = l

From this (317), (318), (319) follow immediately. Theorem 31 is proved. Put (327)

~M(n*) Q(1) = 0

qE (I

0 (328)

qn

P(1) = qnw(n)Q(1)-I = lw E(n. )

W iw

II

I

PO v

¥

+

c+

(D

H

o

Im

0

I~.

c+

5::::: el-

~o 0

r.o

Qo

o o

I

PO

II

g-

~:i-,

II

If

O", v

(D

d

v

hJ

v

II

II

v

k~

I

C rT~

v

II

<)4 k~ ck

(D

!

II

__%

hJ v

v

v

~J~ k~4

l-J.

k-4 c+

t~ ix.)

O

t~

v

II

v II

I-J

v

I

F-J v

G~

f

v

fT~

f~ ~,0 v

0

52 let all elements of the the numbers 0,I,...,q-I

Iw x (11+...+ lw_ 2) matrix L and form the r(1) matrices

I(i*) (338)

o

K (i) = qI

(11+ .... +lw-2)

/

iN. n* E (z

0

/

run over

E

i, 0

°L

"

E(

Then (339)

abs K (1) 0

(340)

~1(1;w,w*)p-1(1 *) =

lwkw_ 2 n* = q

U

K (1)~1(l;w,w*)

~=1

P

with A

(341)

KpSYt(1;w,w*)

A

r) K % ~ 1 ( i ; w , w * )

= ¢i (O ~ ;k; ~ , p = 1 , . . . , r )

Furthermore let

(342)

1 A = (A u) i~l(1;w,w*)

(343)

A*= (i'u) 6 ~1(1;w,w*)

,

and

(344)

Ap-I(1 *) = K A*

for certain p

(1 < p < r).

Then h

(345)

I. 1 A I = W(lw.)Aw.W(lw.),

!. 1 A v = A~-I

(~

=

2,...,W*)

.

53

PROOF:

An elementary computation

shows

(345),

A

(346)

1. A

(347)

I. Awl - 0 rood q ,

(348)

- 0 rood q

(I < ~ < ~ < w*)

,

= LB + q-1(Awl,...,Aw,w_2) (A w 2 ... Aww.) *

with ''"

, ,w-2

(349) ~Aw-2,1

"'"

-2,w-

But

(350)

= 1 .

Hence there is exactly one

(35~)

L

with

LB + q-l(Awl,...,Aw,w_2)

~ 0 mod q,

that means (352)

Awv = 0 mod q

(~, =

2,...,W*)

.

From this theorem 33 follows.

§ 6. THE RIEMANNIAN SPACE OF P O S I T I V E MATRICES The space of all real p o s i t i v e nxn matrices

Y

forms a "weakly

symmetric R i e m a n n i a n space" in the sense of Selberg F40]. We p r o v e some basic results on this space w h i c h are about identical w i t h the results in Maa~ [33~, § 6.

54 The s p a c e ~ ( n ) o f all real symmetric nxn matrices the real d~mension d(n)

(353)

= ~.t~

Y > 0

has

•

By the substitution

Y[A]

Y-

(354)

it is mapped onto itself. Here A is a non-singular real nxn matrix. Let [dY] denote the Euclidean volumelement in ~(n). Then n+l (355)

dVy = (Det Y ) - - ~ -

[dY]

is invariant under (354) and under (356)

y . y-1.

The proof can be found in Maa~ [33], § 6. Furthermore the proof of the following results stands in Christian [7], IV.I. Each matrix (357)

Y E~(n)

may be uniquely written as Y = RrOl

•

Here (358)

R = [rl,...,rn]

is a diagonalmatrix and /I (359)

D =

an upper unipotent matrix. One has (360)

rl = Yl

55 (361)

0 < r

~ y~

(~ = 1,...,n)

.

The relation (357) is called "Jacobi's transformation". For

u > 0

the "Siegeldomain"

~(n, u) c ~(n)

(362)

rI r2 rn_ 1 r~ ' ~3 ''''' rn

(363)

-u < d

There exists a constant (364)

is defined by

(1

< tl

ci0 = clo(n,u) > I

<

'. <

~ < n).

such that for

Y g ~(n,u)

we have (365)

I ~ y-~ r ~ c10

(v = 1,...,n)

(366)

c~

(367)

Yl Y2 Yn-1 Y2 ' ''''' Yn

CIo

Y -

(u 6 n(n))

,

Dg Y ~ Y ~ o10 Og Y ,

By (368)

YFU]

the group o(n) operates discontinuously on ~(n). A fundamental domain is given by "Minkowski's pyramid"/Y~(n). This is a convex pyramid with the cusp in the origin which is bounded by finitely many hyperplanes of (d(n)-l) real dimensions. There exists a constant (369)

c11 = c11(n) > 0

with

T/~Cn) c T(n,c11)

Therefore the inequalities especially true in ~ n ) .

(362),

.

(363), (365) till (367) are

56 Let (370)

h

= h (I) = [Q(n) : ~ (z)l_

(~ = 1 , 2 )

and

~1,"', ~

(371)

~ p(n)

(e

= 1,2)

with h

(372)

n(n) =

o~

U

•~=1

F

,"z,,.,

~ (I) n,

(~ = 1 , 2 )

.

Then h

(373)

~ ~(l) =

(¢ = 1,2)

~=1

is a fundamental

domain of

(374 )

7~I(n)

~ (1). Put = IY ¢ ~ n ) ;

(375)

#~1(I)

= {Y E ~ ( 1 ) ; D e t

(376)

~'ell(1)

=

Det Y ~ I} ,

Y > 1}

(~ = 1,2),

h

THEOREM

34:

is continuous (377)

U ,- .-. ~ ~ ' ~ ( n-) [ F v ] ,o=1

= 1,2)

Let ~(y) denote a complex-valued function which in 0 < y < ~. Then for 0 < Yl < Y2 one has Y2 ~ ~(Det Y)dvy = _v ~n n + 1

97~n) Yl < Det Y < Y2

Vn =

~ Yl

with

( 378 )

(e

] rdY] <~ q/L(n) Det Y < I

~(y)d_~y

57

PROOF:

See Maa~ [33q, page 145, Lemma 2.

THEOREM

35:

Consider the g e n e r a l i z e d Jacobi's t r a n s f o r m a t i o n

(379)

Y : RFD]

(380)

R :

,

R2 . t

0

E (381)

1 xl

Rw

D12 . . - D l w ~

:

D

.

with

"

0

E

\'\

matrices

R

/

(v = l,...,w)

and

1 xl

matrices

D

(I ~ ~ < u ~ w). Set W

(382)

[~!

: -IT

[dR.]

(383)

[an]

:

,, -FI< u ~ wrd

v= 1 1 !

o u]

Then w-1

(384) PROOF:

pages

n-k

[dY] = - I T (Det R ) v:l For 193,

w = 2

the theorem was p r o v e d in Christian

[7],

194. From this the t h e o r e m follows by induction w i t h

respect to w. From (355),

VrdR][dD ].

(384) we obtain

58 w ½(n_k _k _I) dVy = -]-[-{(Det R ) dv R }[dD]

(385)

v=l

For

v

v

Y ~ ~(n) set = (Y[W(n)]) -1 = y-1[W(n)]

(386)

,

t v

Y

=

Y

=

(Y[Q(1)]) -1

(388)

Y

=

Y

=

Y

(389)

Det Y = (Det y)-1,

(390)

Det Y = q

(387)

=

y-l[Q(1)-l]

Then ,

-21 w

m

(391)

Det Y = q

(Det y)-1 ,

2kw_21 w n*

Det Y[K (1)]

I shall now give a record about results which can in detail be found in Maa~ [33], §§ 5,6. One also may see in Maa~ [28], [29] and in Selberg [40]. Let

dY

be the differential

of Y. By

ds 2 = Tr(y-ldy) 2

(392)

a positive definite metric is defined on ~ ( n ) which is invariant under the substitutions (354) and (356). By this metric ~ ( n ) becomes a "weakly s ~ e t r i c R i e m a ~ i a n space" in the sense of Maa~ [33], § 5 and Selberg [40]. The differential operators, which are invariant under (354) form a commutative ring. Set

(393)

euv =

~1

U ~ v

(~,v

=

1,...,n)

,

59 (394)

-~(e ~ ) ~Y = Uv bYu~

THEOREM 36:

.

The differential operators

(395)

(~ = 1,...,n)

Tr((Y~-~y) ~ )

form an algebraically independent basis of the commutative ring of invariant differential operators. PROOF:

See Maa8 [33], page 64, theorem.

Each invariant differential operator L(Y) to the metric (392) exactly one adjoint L (396) where

L(Y) =

Z(y-l)

possesses with respect and

,

means the conjugate complex. Furthermore A A

(397)

L

=

L .

For the proof see Maa~ ~33], pages 58 till 60, page 68 formula ) = L(Y" ~_~~ , Y~ = y-1 5Y

L(Y

A

and page 78 formula

L = L . The adjoint has the property

f(tg)dvy --

(398)

dvy yn)

provided that L~

are

finitely

(L~f)(L*g)

vanish on the boundary of

many d i f f e r e n t i a l

operators

is the generalization of partial integration. A special invariant differential operator is (399)

M n = (Det Y)(Det ~-~y)

@

derived

~n). f r o m L.

Here (398)

60 See Maa8 [33], page 67. For rential operator

k E ~ we form the invariant diffe-

A

(400)

Pk = (Det y)-k Mn(Det y)k Mn

•

Then A

(4oi)

--

A

A

Pk = (Det y)kMn(Det Y) kM n = Mn(Det Y)kMn(Det yl-k

(402)

Pk = (Det y)k Pk(Det y)-k

,

•

See Maa8 [33], page 210 . Put k k D*(k) = D (k,Y) = (Det Y)~ Pk (Det Y)- T

(403) Since

D*(k)

is a real operator we obtain from (396), (402), (403)

(404)

D*(k) = D*(k) ,

i. e., D*(k) is self-adjoint. Furthermore D*(k) is an invariant operator. Hence from (386), (387) we obtain

(4o5)

D*(k,Y) = D*(k,Y -I) = D*(k,Y) -- D (k,Y).

This operator is the generalization of

-D*(t) defined in (19).

With the help of the metric (392) a distance ~(Y,Y*) can be defined between any two points Y,Y E ~ n ) . As before let Y be the left upper ~x~ submatrix of Y. THEOREM 37 (406)

PROOF:

It is o(Y~,Y;) ~ 0(Y,Y*)

See Maa~ r33], page 145, Lemma I.

(v

=

1,...,n)

.

61

Let

(407) be

a row

s = (Sl,...,s of

complex

variables.

w)

Set

= Re

(v=1,...,w)

(408)

s

s

(409)

o~ =

(41o)

s* = (Sl,...,Sw.)

(411)

s ~) = - S w + l _ v

(412)

S

(413)

s ~;

( Crl~ . .. ,crw ) = R e

=

(S"I~...

s

,

, (~

=

1,...,w)

,

~ W ) V

v

(414)

,

= -Sw-~

=

(v =

(~S l , . . .

,

1,...,w-1),

sw

=

-s w

,

[w )

Then

(415)

s = (s*, -s w) ,

(416)

s = s = s

%/ v

•

Put

(417)

~s

(418)

"~s

(419)

A

-~

s I = sw.,

s

(420)

s = (Sl,...,s w) ,

A

A

(421)

=

(Sw,Sl,...,Sw_

=

1)

,

(Sw_l,Sl,...,Sw_2)

A

/k

= s~-1 -~k

•

s -- ( ~ s

,

W

,s w) .

(v = 2 , . . . , w * ) , s

w = s w,

62

Set W

(422)

{l,s}

1

= ~. ~ I s

.

~=1 Then A

I .,sl = f l, st

(423)

~.u

= Ii',s

~

V

=-Iz,sl

THEOREM

38:

The function

(424)

n* w-1 f(l,Y,s) = (Det Y) sw+ T ~ T ( D e t

ibm+l+ 1 V 4 Yk )sv-sv+1 -

v= 1

M

is homogeneous in Y of degree n{l,sl, and it is an eigenfunction of all invariant differential operators. Let (425)

e(w) = ( 1 , . . . , 1 )

with elements 1 and

a E ~. Then

(426)

(Det Y)af(1,Y,s) = f(1,Y,ae(w) s

(427)

n*

+

be a real matrix with

f(1,YKD],s)

1 D

= 0 (1 ~ u < v ~ w). ~U

W

(428)

.

-4- f(l ,Yn.,S* -(s w + ~)e(w*))

f(l,Y,s) = (Det Y) w

1 Let D = (D ) MU Then

+ s) ,

= ('T1-(abs

H=I

2s D )

+ ~(k + k U

u

- n) U-1

)f(l,Y,s).

U

Especially (429)

Let

f(I,Y[VU,s) = f(l,Y,s) m(t)

(V 6 &~(1); ~ = 1 , 2 )

.

be given by definition I and set 1 + k (~) + km(,)_l -

(430)

p~ = s (~) +

(: = 1,...,n),

4

.

63

(431)

p = (pl,...,pn)

•

Then

(432)

f(l,Y,s)

= f({,Y,p)

.

PROOF: The formulas follow from an easy computation. That f(1,Y,s) is an eigenfunction of all invariant differential operators follows from (432) and Maa6 [33], page 69.

THEOREM 39 :

It is

(433) PROOF:

f(l,Y,s)

= f(1,Y,s)

From the theory of determinates we get the equation

(434)

Det(Y )

From (223),

(227),

=

(Det Y)Det( ( )Yn _ v )

(411),

(424),

= (Det Y) sw* ~

=

D ~ )s~-s~+1 -[]-( et (Y)n_k

,,)=1

_

. . . ,n) .

w-1

+ ~i

(sv-sv+l

v=l w-1

1,

(434) we get

kw-1

f(1,Y,s)

(v

-

1V+ 1 + I ~ 4

)

x

1~+1 + I V 4

v .%,

_~ (Det Y)

kw-1

w - "-~

w-1

)Sw-v

-~- (Det(Y)~ ~=1

_ ~ _ lw-~)+l+ lw- V Sw-~+1 4

W--~

aj

~

. kw-1

(Det Y) w

w-1

--"4"- -rT" ~=I

f(l,Y,s)

.

_

(Det(Y)~) s~-s~+l

IV+ I + I 3~ 4

64 Theorem 39 is proved.

Let

THEOREM 40:

(435)

X 6 ~(n)

J (l,X,s) =

and put

/ f(l,Y,s)exp(-Tr(X-Iy))dVy Y E~n)

Then W (436)

1 -1

1 -n +

J (l,X,s) = ~ ~ v=l

-

$=o

PROOF: Because of (432) it suffices to consider the case and w = n. Then we have to prove

(437)

°

J(1,X,s)

~

n

UTr(s

=~

n-1

°

--=-).f(1,x,s)

~=S Put (438)

X = T'T

with an upper triangular matrix

tl (439)

t12 "'" !In 1

T =

• tn/

O Then (440)

Det(X ) = t~ ... t 2

The substitution

Y ~ Y[T]

o

(441)

J (1,X,s) =

.

(v = I .... ,n)

gives o

,I f(1,Y[T],s)exp(-Tr Y)dVy . Y ~ ~(n)

i={

65

From (428) we get

° f(I,Y[T],s)

(442)

=

n 2s +~ (-~-t u L,

n+l -

o )f(1,Y,s)

"T"

.

u=l

Applying

(440) gives us n

(443)

2s

qTt U

n+l

o

u+u "2-

= f(l,x,s)

u=l

Hence o

(444)

o

o

f(1,YrTl,s ) = f(1,X,s)f(1,Y,s)

Inserting this into

(441)

gives

o

~

J(l,X,s)

(445)

.

o

= J(l,E,s)f(l,X,s)

Therefore it suffices to prove n o

(446)

J (1,E,s)

UT r ( s

= rr

-

-~-)

.

%)=']

Perform the Jacobi t r a n s f o r m a t i o n

(379),

this is similar to (438), formula

(443) gives

n (447)

°

f(l,Y,s) = q T r

s

i. e.

Y = RID]. Because

+ ~ _ n+l -4-

uu

tl=l

Furthermore n

(448) u=l

1

<

tl

and from (385) n

(449)

dvy =

n+l T

Wl-(r tj.= 1

u

-

tl ~v r

)[~o] u

.

<

iJ t~',) ,~, < n

66 Hence o

(450)

n

J(I,E,s)

=

~

S

_ ~+

(-IT r u

n+l'

T)

x

t!

r 1,...,rn> o

u=l n

exp(-

~ rj u=l

Performing the substitution

(451)

u

-

~ rud2 v) dVrl...dVrn[dD] 1 < u < ~) < n

= #~-'

Uv

gives

d

tJ

U%)

n co n-1 S (-T]- Sr~ tl T exp(_ru)dv r ) .

~-([,E,s)=( [ exp(-u2)duJ

U=I

,-,oo

O

U

This gives (446). Theorem 40 is proved. Perform the variable transformation (452)

u ~ = S v + l - s v * "-

(453)

t

~

-

(~,

=

I,...,W-1)

,

U = (Ul,...,Uw_ I) •

Set

(454)

f(l,Y,u,s w +

) = f(1,Y,s) .

Hence ~..

(455)

§ 7

f(1,Y,u,a)

w-1 --U = (Det y)a -~- (Oet Yk )

THETA FUNCTIONS

In this paragraph we deduce results on thetafunctions which will be needed in the next chapter in order to prove analytic continuation and functional equations of Selberg's zetafunctions and L-series.

67

T H E O R E M 41 :

Let Y 6 ~(n),

(456) and

U,V

T E ~(m)

complex n × n matrices.

(457)

Then

~ exp(-~ Tr(Y[A+V~T + 2~i A'U)) = A m

exp(-2wi Tr(U'V))(Det

n

Y)- ~ ( m e t

×

T) - ~

exp(-w T r ( y - I [ A - U ] T -1 + 2wi A'V)) . A A runs over all integral

PROOF:

n × m matrices.

For a n x n matrix

B = _(b )

b l lC

(458)

)

C = (c

and a m x m m a t r i x

form the m n × mn m a t r i x

b12C

1

~ (C,B) =

•

.

.

.

J

.

.

.

.

XbmlC

If one takes all elements of

.

.

o

o

e

.

bm2C

C

and

blmC

....

o

o

.

m

o

e

e

e

...

B

e

e

e

e

.

b~

as indeterminates

one has

decompositions (459)

C = CIC 2 , B = BIB 2

with C 1

=

, C2

_ _

(460)

B1

"t

, B2=

=

P

68

Then ~(C1,B I) is a lower triangular matrix and upper triangular matrix and one has

(461)

~ (C,B) = ~ ( C l , B 1 ) ~ C 2 , B 2 )

@(C2,B 2)

an

.

From this one easily sees (462)

Det ~(C,B) = (Det c)m(Det B) n ,

(463)

( ~ ( C , B ) ) -1 = 7 ( C - l , B - 1 )

•

Since this holds for indeterminate C,B, the formula (462) holds for all complex C,B; formula (463) holds for complex non-singular C,B. Let C,B be real symmetric and positive. Then one may take C 1 = C~, B 1 = B~ . Then (461) becomes

(464) Hence

~(C,B) ~(C,B)

Now decompose (465)

= ~(C2,B2)'~/~(C2,B 2) .

is symmetric and positive. A,U,V

in n-rowed colums

A = (al,...,am); U = (Ul,...,Um); V = (Vl,...,v m)

and form the ran-rowed columns

Then (457) is identical with

(467)

~ exp(-~(~(Y,T))[a+v]

+ 2~i a'u) =

a 1 exp(-2~iu'v)(Det

~(Y,T)) - ~ ~ e x p ( - ~ ( ~ - l ( y , T ) ) [ a - u ] + 2~ia'v) . a

But this follows from (3). Theorem 41 is proved.

69

DEFINITION

2:

the Gaussian

For

m E ~

and an even character

form

~ mod q

sum m2

(468)

G(m,x,C ) = q--~-

~ x(Det D)exp(2q-~ Tr(C'D))

,

D mod q where

D

runs over all integral m x m matrices mod q and set G(m,7)

(469) THEOREM 42:

(470)

Let

X

G(m,7,C)

=

G(m,7,E)

.

be an even primitive

character

mod q. Then

= ~(Det C)G(m,x)

for all integral m x m matrices

C.

Furthermore

(471) PROOF: Andrianov

abs G(m,y)

Formula

= 1 .

(470) is mentioned without proof already in

[2], page 41. For

m = 1

the theorem is proved

in Lan-

dau [22], pages 484 - 485. The following proof is a generalization of Landau's proof. First let (472)

= 1 . @

Then with

D

also

D

= C'D

runs over all residue

classes mod q.

Hence from (468) m2 G(m,~,C)

= ~(Det C)q --~-

~ 7(Det(C'D))exp(~

Tr(C'D))

D mod q m2

T ~(Det C)q

,2~i ~, ~(Det u )exp~-~- Tr D*) D'rood q

=

70 [(Det C)G(m,%) Now let

(473)

(Det C,q) > 1 .

We have to prove

(474)

G(m,?,C) = 0 .

Let U,V ~ O(m). Then with classes mod q. Hence (475)

G(m,~,UCV)

D

also

= G(m,%,C)

U' DV' runs over all residue

(U,V ~ O(m))

.

Therefore by the theorem of elementary divisors we may take (476)

C = [c1,...,Cm~

with

(477)

Cl,...,C m ~ 0

and (478)

c11c21 . - -

I

Cm

Then from (473), (478) we deduce

(479)

(Cm,q> = t > 1 .

Set

(480)

c m = tb, q = tr, D =

(481)

ID~mil'm ~

•

71

(482)

d=kr+n

with

(483)

k = (kl,...,k m) , n = (nl,...,n m) .

Then

(484)

=

G(m,M,C) -

m2 T

Cldl+'''+Cm-ldm-1

q

I x(Det D)exp(2Ni

bnm + --~- ) •

q

DlmOd q n mod r k mod t

We shall show, that there exists an integer (485)

= I ,

(486)

a e 1 mod r ,

(487)

x(a)

Let ~ of

number

it follows that

a 6 ~

(488) Then let (489)

al,a 2

~

=

(a ~ ~)

1

=

and

(490)

a 1 ~ a 2 mod

Because of (489) there is a

(486). Because

suppose it would be

two numbers with



(485),

~ . We show that there exists a

with (487). Because

~(a)

with

~ 1 .

be the set of integers which satisfy

1 ~ ~

a

c

r

.

with

= 1

.

72

(49~)

alc ~ a2mod q ,

hence (492)

alc-

F r o m (489) till

(492)

a2mod

it follows

i.e.,

c E ~.

From

r .

= 1, c ~ 1 mod r ,

(488),

(491) we get

x(a 2) = ~(a I )~(c) = ~(a I) Hence f r o m

(489),

(490)

(493)

x(a 2) = ~(a I) .

But t h e n

X

w o u l d be induced

not be primitive. a E ~ with With

it w o u l d f o l l o w

d

to each

~

is p r i m i t i v e

m o d r, i. e., ~ w o u l d there must be a n u m b e r

(487).

also d

But since

by a c h a r a c t e r

ad

runs over all r e s i d u e - c l a s s e s

there belongs

(494)

a decomposition

ad = k r + n

N o w it follows

from

m o d q. Hence

(482),

.

(486),

(494)

n _= d = ad = n*mod r , hence

(495) N o w let

n

k

m

n*mod

r

.

run over all r e s i d u e c l a s s e s

mod t. T h e n

73

(496)

a(kr +

n)

= k r + n

runs over t m different residueclasses mod q. For these (495) must hold. Hence k runs exactly over all residueclasses mod t. Let A = [1,...,I,a]

(497)

.

Then it follows x(a)G(m,M,C) m2 -~-

q

[ x(Det(

= G(m,7,AC) =

DI Cld1÷'''+Cm-ldm-1 a(kr+n) ))exp(2~i q

bnm + --{-) =

D1mod q n mod r k mod t m

q

m2 "~-

~ ~(Det(

DlmOd q n mod r k mod t

D1 ))exp(2~i Cldl+" "'+Cm-ldm-1 ham k*r+n q + ~ ) =

Q(m,~,c)

.

Hence (498)

M(a)G(m,x,C)

= G(m,xC)

.

From (487), (498) we deduce (474) . Formula (471) was proved in Gri6enko [12], page 607 but it also follows from the thetatransformation formula that will be proved later (Theorem 45). Theorem 42 is provedo

74 DEFINITION 3:

Let

(499)

x = (Xl,'--,Xw~)

be a row of even characters mod q and put W*

(5o0)

~I( I ,x) = -FF a(1u,x ~)

,

U=1 W* l T2(1 * ,X) = ~ T G(1,X~) u o ~=I

(5oi)

DEFINITION &: mod q and

Let

(502)

X = (Xl,---,Xw*)

Y 6 ~(n),

be a row of even characters

T G q~.(n*)

.

Set

(503)

n* ~ (I,x,Y,T) = (Det Y ) T ( D e t

n T) ~

x

1 1 I X1 (oet A1) "'" Xw *(Det Aw*)exp(- ~ Tr(Y[AW(n*)]T)) 1 A = (A tt) E % ( l ; w , w * ) (~

=

1,2)

Set (5O4) with

(505)

1 = (I,...,1,1 w) n* times I and X = (~I ,''',Xn* ~ )

with (506)

X'~ = X~(:)

(: = 1,...,n*)

.

.

75 Then V

(507)

~

rl

e2(I,x,Y,T) = eI(I,~,Y,T) •

THEOREM

Let

43:

q > I. Then

n* n e (1,M,Y,T) = (Det Y)T(Det T) ~

V

(508)

x

1 1 ~xI(Det A1) ... Xw.(Det Aw.)exp(- ~ Tr(Y[AW(n*)]T)) 1 A = (Avu) E ~(l;w,w*) (~ = 1 , 2 )

PROOF:

Apply

THEOREM 44:

x(a) = 0

for

> 1 .

Set

(509)

(U = 1,...,W*)

x~ = Xw-~

and for

.

define

U E ~(l*)

U

by (310) with

I*

instead of

1 .

Then v

e (1,x,Y,T[U ]) =

(51o)

w* I* w* (-~-~v(Det U ))~ (1,~,Y,T) = ~ - ~ ~=1

PROOF:

i* (Det UL))e (1,~,Y,T) (~=1,2)

~=1

From (310), (503) one gets n*

n

e (1,x,Y,T[U]) = (Det Y)T(Det T) T x 1 1 ~Xl(Det A1)..- Xw.(Det Aw.)exp(- ~ Tr(Y[AU-Iw(n*)] T))" 1 A = (Avu) 6 ~(l;w,w*)

76

Substituting AU instead of A we get the first of the formulas (510). Using (311) and (509) we get the second. Theorem 44 is proved. Now we prove the thetafransformation formula from wich (471) follows. THEOREM 45: Then

Let

XI,...,Xw.

V

(511)

PROOF:

be primitive even characters mod q.

~

V

V

~

V

e (I,x,Y,T) = ~ (I ,X)e (1,x,Y,T)

(e = 1,2) .

Because of (507) it suffices to consider the case

Starting with (503) we set; (512)

A = B + qCW(n*)

with 1 (513)

B --

(B',n~ ) ~ ~,~ 1 (l;w,w*)

and integral C. From ( 5 0 3 ) f o l l o w s n*

(514)

eI(I,x,Y,T)

= (Det Y ) T ( D e t

n

T) ~ x

I i ~ x1(Det B 1) ... Xw.(Det Bw.) x B mod q }i exp(-n Tr((qY)[C + BW(n*)]T)) q C To the inner sum we apply (457) with V = BW(n*~ q

Then

, U = O, m = n* .

qY

instead of

Y,

~ = I.

77

(515)

n* v8 1 (1,~,Y,T) = (Det Y)- T ( D e t

n T)- ~ q

1

nn* "-2--

x

1

C,BW(n.))) x

~ ( L x1(Det BI) ... Xw.(Det Bw.)exp(2wi Tr " C B mod q exp(- q~ Tr(y-I[c~T-I))

q

.

Here the summation runs over all integral C. Set A = Q(1)C

(516) with

1 A = (A u)

and v

I Awp =- 0 rood q

(517)

(u =

1,...,w*)

.

Then from (386), (387), (389), (390), (515), (516) we get n*

(518)

~I(I,x,Y,T) = (Det ~ ) T

n

(Det T)~

x

~(A)exp(- ~ Tr(Y[AW(n*)]T)) A

with n .2 (519) 1 MI (Det B1)... B mod q

2

~(A) = q

×

1

Xw.(Det Bw.)exp(2wi Tr(

A,Q(1)-IBw(n.,,), ))

Let V

(52O) Then

1 Q(1)-IBW(n *) : O : (O

) .

q

78 v

1 D%)U

(521)

(~, = w; u = 1,...,w*

= 0

and I ~ ~ < u _< w*)~

V

1 Det B

(522)

U

1 = Det Dw_ u

(p = 1,...,w*) .

Hence V

Tr(A'Q(1)-IBW(n*)) q

(523)

=

v

I 1 w* "' D ~ Tr(Aq ~ ) t,=l

~

%/

1 1 A' D ~ ) ~ Tr( V~q

+

.

1 _< ~-< v < W ~

From (509), (519), (522), (523) we deduce

(524)

I <

n* w* A' D ~(A) = q- -~- "~-( ~, ~u(Det Du)exp(2~i T r ( ~ ) ) u=1 D mod q U A' D -~~ exp(2~i Tr ~u q ~u) . p < ~, <_ w * D modq 9u v

x

v

1 1 The sum over D is 0 for A ~ O mod q and it is q ~ u for A - 0 mod q. Hence because of (468), (517) it follows V~J w* v ~ v -~- G(lu,xu,At ) (A £ 0~l(1;w,w*)) , (525) *(A)=

ll=l V

0

CA ~ ~1(l;w,w*))

and because of theorem 42 and definition 3 3.

(526)

)(A) = x1(Det A 1) .

Xw.( et Aw.)

T1(1 ,X)

(A ~ 0~l(1;w,w*))

0

(A }O~1(1;w,w*))

v

From (518), (526) we get

×

79

n*

(527)

el(1,x,Y,T)

n

= (Det Y)~- (Det T) ~ ~1(1 ,X) V

~ gl(Det A1 ) . . . ~w*(Det ~ . ) a ~ ~l(i;w,~) exp(- ~ Tr(Y[AW(n*)]T)

,X)

= T1(±

×

I(I,x,Y, T) •

But

(528)

"rl(1 ,X) = "rl(1 ,X) •

From (527), (528) we get (511). Theorem 45 is proved. Let

D (k,T)

be defined by (403) and set O (n,T)

(529)

(q

=

11

D(q,T) =

1) l

(q >

"

This is a generalization of (22). From (405) we get (530)

D(q,T)

= O(q,T -1)

=

D(q,~)

.

Set (531)

@u(I,M,Y,T) = D(q,T)e (I,x,Y,T)

THEOREM 46: (532) W*

Let

(e

=

1,2)

U E ~ (I*). Then

e~(I,~,Y,T[U 7) = 1"

w*

1-

(-]'[- X (Det U ))e (I,x,Y,T) = (-[[-g (Det U~))8 (I,x,Y,T)

v=l PROOF:

•

v=l Apply theorem 44 and the invariance of

D(q,T) .

(e = 1,2).

80 THEOREM 47: Then

Let

X1,...,Xw.

be primitive

V

(533)

e (I,x,Y,T) = T (l*,x)e

PROOF:

Apply theorem 45 and (530).

THEOREM 48: (534)

Let

C

V ~

(1,x,Y,T)

(e

=

1,2)

.

be a n* x n* matrix. Then

(Det ~-~T)exp(Tr(CT))

PROOF:

~

even characters mod q.

= (Det C)exp(Tr(CT))

@

Easy computation.

THEOREM 49: (535)

Let

q = I. Then

~2(1,%,Y,T)

= el(1,x,Y,T)

•

n* = (Det Y ) T ( D e t

P n (T)exp(-N

n T) ~

Tr(Y[AW(n*)]T))

×

•

A 6 ~l(1;w,w*) PROOF: For q = equality follows. is the sum on the From (399), (400)

I, ~l(1;w,w*) =~2(1;w,w*) . Hence the first From (403), (503) we see that 81(1,x,Y,T) right-hand-side of (535) taken over ~1(1;w,w*). we see that Pn(T) . . . . (Det ~ ) . Hence because

of (534) all summands with Rk A < n* are zero. So it suffices to take the summation over the A with Rk A = n*. Theorem 49 is proved. In [33], page 79 Maa8 considers the operator and then

computes

L. A p p l y i n g

this

with

n*+l A Mn.(T) = (-1)n*(Det T ) m ( D e t

(536) Combining

L = (Det Y)h(Det ~-~T)h

h = 1

we g e t

l-n* ~)(Det

T)T

(399), (400), (536) we obtain

n*+l -k

(537)

.

Pk(T) = (-1)n*(Det T) - T

~+k (Det ~-~T)(Det T)

(Det ~ )

.

81 THEOREM 50:

Let

(538)

X g ~(n) .

Then (539)

(Det ~-~x)(f(l,X-1,s)exp(-Tr X)) = f(l,x-l,s+e(w))exp(-TrX)R(X),

where R(X) is a polynomial of degree with w times 1.

~ n

and

e(w) = (S,...,S)

Because of (432) it suffices to prove the theorem for 1 = 1. Suppose at first

PROOF:

~T > Tn-1

(540)

(v = 1,...,n),

and set

n~ (541)

~(s) -- ~

n -U1-r(s - .~!) %)=1

Then because of (437) ° 1 ° f(l,X,s) = Y-I(s)#(I,X-I,s)

(542) because of (435) (543)

f([,X-1,s)exp(-Tr X) = ?-1(s)

[ f(1,Y,s)exp(-Tr(X(Y+E)))dVy. Y 6 ~n)

Hence because of (534) o

(544)

(Det

)(f(1,X-1,s)exp(-Tr X)) =

(-1)ny-l(s)

~ f(1,Y,s)Det(Y+E)exp(-Tr(X(Y+E)))dVy = Y ~ ~(n)

(-l~y-1(s)exp(-Tr X) r f(1,Y,s)Det(Y+E)exp(-Tr(XY))dvy Y g ~(n)

82 X = T-1T '-1

With T of type (439) put Y - Y[T]. Then (Det ~)(f(1,x-S,s)exp(-Tr

f f({,Y[T],s)Det(Y[T~ Y ~(n) Apply (426),

(545)

X))

and make the substitution

= (-1)ny-l(s)exp(-Tr X)

x

+ E)exp(-Tr Y)dvy .

(444). Then

(Det ~)(f({,x-S,s)exp(-Tr

X))=f({,x-l,s+e(n))exp(-TrX)

R~s,~

with

(546)

R (s,X) = ( - 1 ) n y - l ( s )

f f(1,Y,s)Det(Y+X)exp(-Tr Y)dVy . Y E ?(n)

Obviously

R*(s,X)

is a polynomial

in

X

of degree

~ n.

The function

(547)

R(X)

-- R ( s , X )

=

f - ( 1 , x - l , s + e(n))exp(Tr X)(Det is holomorphic For (548)

in

s

sEcn

.

= R*(s,X)

.

for all

X))

(540) we have R(s,X)

Let ~ be a partial derivative of (540) we get from (548) (549)

)(f(1,x-l,s)exp(-Tr

X

of order

> n. Then for

~R(s,X) = 0

identically in s and X. By the principle of analytic continuation this must be true for all s E C n. Hence R(s,X) is a polynomial

83 in

X

Let and

of degree

~ n. Theorem 50 is proved.

a E C. If we apply theorem 50 for the special case T g ~(n*) instead of X E "~n) we get for a E

w = 1

(550)

(Det ~ ) ( ( D e t T)aexp(-Tr T)) = (Det Ta-lexp(-Tr T)R(T),

where

R(T)

is a polynomial of degree

From (534), (537), (550)

(551) Here

we deduce

Pk(T)exp(-Tr T) = (Det T)R(T)ex~(-Tr T).

R(T)

THEOREM 51:

(552)

is a polynomial of degree ~ n*.

Let S E ~n*)

Then there exists a constant (553)

PROOF:

(554)

~ n* .

• c12 = c12(k,n*) ~ 1 with

Pk(T)exp(-Tr(ST)) ! c12exp(- ½ Tr(ST)).

Because of (551) there is a constant

c12

with

Pk(T)exp(-Tr T) ~ c12 exp(- ~ Tr T) .

Let A be a real non-singular n* ×n* matrix. Since invariant operator (554) gives us

Pk(T)

Pk(T) exp(-Tr(T~A])) ~ c12exp(- ½ Tr(T~A])) . Hence (555)

Pk(T) exp(-Tr(AA'T)) ~ c12 exp(- ~ Tr(AA'T)) .

is an

84

Since S may be written as S = AA' with suitable A the assert~tion (553) follows from (555). Theorem 51 is proved.

THEOREM 5 2 :

Let 2

(q = I)

q

(q>

(556)

There exists a constant

I) with

c13 = c13(1) ~ 1

n

n*

(557)

abs e (1,x,Y,T) ~ c13(Det Y ) T ( D e t

RkA

exp(- ~. Tr(Y[A]T) = n*

(e

=

x

T) T 1,2)

•

The summation is over all integral n x n* matrices

A

with

Rk A = n* . PROOF: Apply theorems 25, 43, 51, formula (535) and write instead of AW(n*) . THEOREM 53:

Let

A

Rn,Rn. non-singular rational n x n respectively

n* × n* matrices, XI,...,Xw. arbitrary even characters mod q, Y E~(n) and j(Y) a positive number with

(558)

Y a j(Y)E ,

furthermore (559)

T ~

~(n*,~)

.

Then there exists a real number

(560) with

c14 = c 1 4 ( n , n * , w , R n , R n . )

~ 1

85 (561)

abs e (1,x,Y[Rn~,T[Rn. ]) nn* n *~ n 2 (Det Y ) T ( D e t T)- T c14J(Y)

PROOF:

-1 e~(-c14

j(Y)Tr T)

(e = 1 , 2 ) .

From (366), (559) we deduce

(562)

c:~ Dg T

~

T

:

ci0 Dg T .

Set R n = gn I G n , Rn. = gnZ Gn. with gn,gn. # ~ and integral Gn,Gn.. Then from theorem 52 and the inequalities (558), (562) we obtain n* n (563) abs e (1,x,Y[Rn],T[Rn. ]) ~ c13(Det Y ) T ( D e t T) T x exp(-2~ Tr( ~(Y) (Dg T)[Gn.A'G~]) 2 2 Rk A = n* 2q* gn gn* c10 Form the diagonal matrix (564)

S =

J(Y)D~ 2 2T 2q* gn gn* c10

= Is1 .... 'Sn*~

and put (565)

$(n,n*;s) = RkB

~ exp(-2w Tr(S[B])) = n*

,

where B runs over all integral n* x n matrices of Rk B = n*. Then from (563) we get n* n (566) abs e (1,x,Y[Rn~,T[Rn.~) ~ 013(Det Y ) T ( D e t T)~(n,n*;S). If we can prove (567)

~(n,n*;S) ~ d1(Det S)

n ~ exp(-d: 1 T r

S)

with some constant d I ~ 1, the assertion (561) follows from (562),

(564), (566), (56?).

86

Since

S ~ a diagonal matrix n* $(n,n*;S) ~ - ~ -

(568)

$(n,S,s )

L1= 1

Let

u > 0

and

~ (u)

be defined

by

(31).

Then

{(n,l,u) < n(~(2u)-l)~n-1(2u).

(569) Applying (570)

t~(2u) - I ~ exp(-~u)~(u)

and

(571)

.~(2u) ~ ( u )

we get #(n,l,u) < n exp(-wu)~gn(u)

(572)

Let

~ > 0

(573)

•

be given. From (35), (572) we get n ~(n,l,u) ! d3u 2 ex'p(-(~ - n¢)u)

with some constant d 3 = d3(¢) ~ 1 . Now (567) follows from (568), (573). Theorem 53 is proved. Let

THEOREM 54: and set

(574)

(575)

X1,...,Xw.

A

/%

X1 = X w e '

X,~ = X,~_ 1 A

be primitive even characters mod q

(~

/ .A

A

X = ~X I'''''Xw*)

Then (576)

= 1,...,W*-1)

el(I,~,Y,T[p'I(I*)])

=

%

"

;

87

kw_2 iw I

r

2

q

eI(1,x,Y[K (1)]) ,T) , 0=1 v~ v . BI(I,x,Y,T[P(I )]) =q

(577)

kw-2 lw 2

r v ~ ^~ v ~ el(I,x,Y[K0(1) ]) T) ~=1

Let q = I. Then r = 1 , KI(1) ~ o(n), P(I*) 6 o(n*). Then (576) is true because from (535) we deduce that %I(1,x,Y,T) is invariant under Y - YFU], T - T[V] with U E Q(n), V 6 o(n*).

PROOF:

Now let

q > I. From (330), (332), (340), (344), (345), (391),

(508), (529), (531), (574) fonows n* n ~I(I,~,Y,T[P'I(I*)]) = (Det Y)-4-(Det T) ~

(578)

x

~ Tr(Y[AW(n*)p'-I(I*)W(n*)W(n*~T)) I XI (Det ~I )" . "Mw *(Det A~)exp(u i 1 A = ( A ) 6~1(l;w,w*) n* n -- (Det Y)T(Det T)~ x 1 1 Xl (Det A1)... Xw.(Det Aw.)exp(- q Tr(Y[Ap-I(I*)W(n*)] T)) 1 A=(A t,) E~l(1;w,w*) kw-21w

r n* n I (Det Y[ K 0 ] )-4-(Det T )~ x ~--I^ ^ I. ^ I. I XI (Det At)''" Xw*(Det Aw*)exp(- q Tr(YFKoA*W(n*)]T))

= q

A*=(A* u) 6~l(1,W,W*)

u

= q

kw-2 lw 2 ,

r I ~I(I,x,Y[Ko], ^ ^ T) • D=I

88 This proves (576). From (244), (533), (576) we deduce (579)

~F1(1 ,X)O1(I,x,Y,T[p-1(I*)]) kw-2 lw

q

2

= "FI(~ 1 ,X)

×

r A

~ Gl(1,~,(y[Kp(1)i/' ~) . ~=1

From (242), (471), (500), (574), (575) we get

(58o)

T I( ~*l* ,x) A

= 71(1 * ,x) ~

o.

Formula (332) gives (581)

T[p-I(I*)] = TIP(I*)]

Put (581) in the left-hand-side of (579), write T instead of and use (580). Then (577) follows. Theorem 54 is proved.

T

CHAPTER III.

SELBERG'S ZETA- AND L-SERIES

The Selberg's zetafunction in this chapter are identical with the zetafunctions considered in Maa~ [33], § 17 and Terras [45], [46]. Selberg's L-series are associated to these zetafunctions in the same way as Dirichlet's L-series are associated with Riemann's zetafunction.

We prove analytic continuation and functional equations

of these functions.

The methods are similar to those of Maa~ [33],

§ 17.

§ 8.

DESCENDING CHAINS

A descending chain is a system of matrices with integral elements such that the number of rows and columns become smaller by going down the chain.

It is

defined when two descending chains are

equivalent and equivalence class invariants are defined. The theory of descending chains is developed so far as it is needed for Selberg's

DEFINITION 5:

(582)

zetafunctions

Let

and L-series.

= 1,2. A matrix system

= {Aw_I,...,A1}

with

(~

(583)

= 1,...,W-1),

is called a "descending chain" of type a. Two descedning chains ~and

(584)

/#/.* = IAw_I,...,All

are called "equivalent", (585)

if there exist

U# ~ ~ ( ( 1 1 , . . . , i # ) )

(~ = 1 , . . . , W - 1 )

90 with (586)

Aw_ 1 = Aw_ I Uw_ I ,

(587)

A s = U~+ 1 Ap U 8

*

-1

(~

=

The equivalence class to which ~ belongs is

1,...,W-2)

.

{~}.

Set

THEOREM 55 : (588)

i . I. A8 = (A~,vU), A s = (As,vU .)

(589)

1 Bey = A8 A13_1... Ay = (B~y,v u) E ~(l;~+1,y)

(13 = 1,...,w-1),

(1 _< y < 8 < w - l )

(590)

* Bf3 v

=

,

A~ A~_ * 1 ... Ay* = (1;V,vU). 6 ~" (1;p+1,y) (I <_y _< p <_ w-l) .

Then (591)

Bw_1, Y = Bw_l, Y U Y

(592)

BeY

*

-1

=

U

U8 +1BSv

(y = I,...,w-1)

,

(I _< y <_13 <_w-2) .

V

The following natural numbers are invariant under equivalence: (593)

(1 < y < 8 <w-l)



(594)

h11 = h11(~) =

(595)

h~1 = h~1(~) = =

(596) (597)

,



(13 = 2 , . . . , W - 1 )

,

- I > -

=

h~v = hSy(Y%)

,


=

(13 = 2 , . . . , W - 1 )

(2!y
,

.

91

Set C Bw_l, Y = (.Y)

(598) with a k

Y

x k

Y

A

(599) (600)

matrix

C . Then also

A

h

= h (~) = abs C V v Y A v v h hy = hy(,e~,) = Y

(y = I,...,w-1), (y = 1,...,w-1)

are class invariants lying in ~. All those class invariants are coprime to q. Hence by .

(601)

one can define (602)

V

v

.

v

h I = hlmOd q; hy = hy h v_Imod q

(y = 2,... ,w-1 )

w-1 class invariants

(h$ mod q) = (h:(x~)mod q)

(y = 1,...,W-1) •

Furthermore = -['[-h~v ~=I

(603)

(604)

i BSy,~-

(606)

Det C

V - "~y U=I

=

1,...,W-1)

,

(I iyi~<_w-1),

= "T'l- 77- h u=1 ~=y

(605)

(~

i I AS, ... AV, ~ modq(1 < v < y < ~ <_ w-l),

w-1 1 -~- Det A mod q vu v=y

(y = 1,...,w-1).

PROOF: Formula (605) follows from (589). From (583), (605) one easily gets BSV 6 ~ u (l;~+l,y). Herewith (589) is completely proved. Obviously also (590) is true. The formulas (591), (592) follow from (586), (587), (589), (590). The formulas (591), (592) show that are equivalence-class invariants. Because of

92 (583) is ~ O, hence E ~. Hence the hsy ( l ~ y ~ w - 1 ) are positive equivalence class invariants. Because of theorem 31 they ly in ~. From (591), (598) one sees that also (599) is an equivalence class invariant lying in ~. ~ is a positive equivalence Y class invariant. From the definition of follows labs Cy

hence

h V ~ ~.

From the definition of ~ ( 1 ; ~ + l , y )

it follows that all those class

invariants are coprime to q. From (599) follows h y = -+ Det

(607)

Cy

(V = 1 , . . . , w - I )

.

From (589), (598) follows Y Det C V - -~- Det Bw-1 ,~, t mod q :=1

(608)

and from this and (605) we get (606). Finally (603) is a special case of (604). Hence it suffices to prove (604). For ~ = V = I follows (604) from (594). Now let (604) is identical with

(609)



=

v = 1. Then

"]T h~l ~=1

This follows from (595) by induction with respect to 6. Now let 1 < v ~ w-1 and the formula (604) already proved for all v* < V and ~ arbitrary in the interval v* ~ ~ ~ w-1. We shall prove (604) for V and all ~ with y ~ ~ ~ w-1. First let From (596) and the induction assumption follows = hyv v-1

y

-I

y-1

h,,,,('Tr Tr" h,)('Tr- h;1_1, ,) -~=1 y-1 h

YY

~=y-1

"TT" h ,.=1

:=1 y

Y~

= "TT" h t=l

Y~

~ = V.

93 Y < 6 ~ w-1 and assume v ~ 6 < 8. From (597)

Herewith one has (604) for 6 = Y. Now let that the assertion is true for all 6 with and the induction assumption we get

< s13,,,> = hlgy
>-1

=

6-1 v-1 [8 v-1 6-1 hov(-]'[- -TT" h ) ( - ] T "]'T h, t)(-rT" ~ h -1),~ = :=1 ~=y ~,=1 ~=v-1 ,,=1 ~=y-1 Y

6

y-1

y-1

6

v-1

6-I

6~ )(-FF -R-h,~ )(-FF T F h -I:) = (-FF "T'Fh t)(-FF h -1 ¢=1 ~=y

:=1

~=1 ~=y-1

V !8 V-1 !8 V-1 ('TF-[T"h~,,~)('TT- "TT" h ) ( T r " ~,=1 ~=Y ~ t= 1 R=V_ 1 t ~=1

v

:=1 ~=v-1

Tr'lh~l,,)

=

~=V-

6

"tT" - F F h ~ t t=l ~=V

•

Herewith (604) and theorem 55 are proved. DEFINITION 6:

(61o)

A descending chain is called "special" if (6 = I,...,w-1)

A6 E /(1;6*1,6)

.

The set of special descending chains may be denoted by ~(1). descending chain is called "reduced" if there exists a (611)

A

U E ~(I),

such that

(612)

IU-1Aw_ 1 , A w _ 2 , . . . , A l l

E ~(1)

.

The set of reduced descending chains may be denoted with

~(1).

94 THEOREM

56 :

If

(613)

~=

{UDw_I,Dw_2,...,DII

6 ~(1)

with 1 (614)

U = (U

) £ ~
(615)

1 D~ = (Ds,vu) £#*(1;~+1,8)

(8 = 1 , . . . , w - 1 ) ,

we have (616)

1 h~y(/x-) = Det D~,?

(617)

v 1 w-1 1 hv(z~) ~ --+-~-(Det Uu -~- Det D~U mod) q U=I v

(618)

(619)

If x ~ (620)

¥

h (~) ~ Z - ~ - ( D e t v p=1

(1 _< v _< ~ < w-l),

(v=1,...,w-1),

~=y l

U )mod q p

1 h*(~) ~ + Det U y -V mod q

(V=1,...,w-1),

(v=l,...,w-1).

~(1) one may instead of (618), (619) even say ^ V w-1 I hv (~) = -~- -~- (Det D U) (y=l,...,w-1), u=l ~=y %/

(621)

h (/..~) V

=

1

(v=l,...,w-1).

The formulas (616) follow from theorem 31 and the equations (594) till (597). Formula (617) follows from (599), (606), (614) (615). Formula (618) is a consequence of (600), (615), (617). Formula (619) follows from (601),(618). Formulas (620), (621) follow from (599), (600) and (615). Theorem 56 is proved.

PROOF:

95 THEOREM 57: Each descending chain of type 2 is also of type 1. Each descending chain of type I is equivalent to a descending chain of type 2. Each reduced descending chain is of type 2. Each equivalence class of type ~ (~=1,2) contains exactly one reduced descending chain. Formaly one has the Cartesian product decomposition

£(i)

(622)

:

~i) × ~(i).

Because of

PROOF:

(625)

W)

and (582) one sees that each descending chain of type 2 is also of type I. From (624)

~ ( i ) E ~2(I)

and

(1 <_ u <_ ~_< w)

(625)

it follows that each reduced descending chain is of type 2. If we can show that each descending chain of type I is equivalent to a reduced descending chain, we have also proved that each descending chain of type I is equivalent to a descending chain of type 2. Let

~ = 1,2

(626)

and **

X~

**

**

= {Aw_I,...,A I }

be a descending chain of type ~. By (627)

Aw_ I = Aw_ I Vw_ I ,

(628)

A~ = V6+ I A~

*-1

**

*

V~

(6 = I,...,w-2) ,

96 ¢e

(629)

VI3 E '}' ((11,...,li9))

(~ =

1,...,w-1)

an equivalent descending chain (630)

~*

= {Aw_I,...,A 1}

is d e f i n e d . Let V1 = E . By t h e o r e m 29 t h e r e exists V2 s u c h t h a t A 1 E ~ ( I ; 2 , 1 ) . Now let 1 < ~ < w-2. Suppose one has already

(631)

A.L E ,~(1;~+1,~) ,X-

(~ = 1,...,~-'1)

*

and let V I,...,V~ be already fixed. By theorem 29 there exists a V~+ 1 with A~ E ~(11~+1,~). Hence (632)

A# E ~ ( I ; # + 1 , # )

(# = I,...,w-2)

.

Applying theorem 29 once more one sees that there is a

(633)

u
with

(634)

U

-1 *

Aw_ 1 E ~ ( l ; w , w * ) .

Set (635)

Aw_ 1 = Aw_ I Vw_1,

(636)

A~ = V~+ I A~ V~

(~ = I , . . . , w - 2 ) ,

(637)

V~ E A~((ll,...,l~))

(~ = 1 , . . . , w - 1 ) .

Like in the proof of theorem 29 one sees the following. First one can choose Vw_ I such that

(638)

U-1A~_ 1 E

J*(t;w,w*).

Then one can successively fix Vw_2,Vw_3,...,VI (639) Hence

A~ E ~*(1;~+1,~)

such that

(~ = I,...,w-2).

97

(640)

&~= {Aw_ I,Aw_2,...,AI}

is a reduced descending chain which is equivalent to ~ and consequently to ~**. Therefore each equivalence class contains a reduced descending chain. Now let us assume that two reduced descending chains

(641)

Z~= {U Aw_I,Aw_2,...,A1},

(642)

= {U Aw_ 1 ,Aw_ 2, • • • ,All

~

with (643)

U,U*

(644)

As,A ~ 6

~(1),

-/

(1;~+1,~)

(8 = 1 , . . . , W - 1 ) ,

are equivalent. Then *

-1

(645)

A6 = V#+ I A6 V~

(~ = 1,...,W-1),

(646)

V8 6 ~((11,...,18))

(~

(647)

V

= U*-1U

=

1,...,w)

.

W

Obviously V 1 ~ ~ (11) = A~(ll). Now let ~ ~ 1 and assume V B £ Ae((ll,...,ls)). Then AsV6 6 J (1;~+I,~) and because of (644) (645) it follows V#+ 1 £ A e ( ( l l , . . ~ # + l ) ) . Hence (648)

v~ ~ n~((ll,...,l~))

(6=I,...,w; ~=1,2).

Now ~(1) ~ ~2(1) and because of (643), (647), (648) we have Vw 6 A2(1) , hence by (643) (649) and

U V w = E. Assume

= U

~ < w

and

V~+ 1 = E. From (644), (645) follows

98 VG = E. Hence

V~ = E (6 = I, .... w).

Therefore (650)

A; = A8

(8 = 1,...,w) .

Hence each equivalence class contains exactly one reduced descending chain. Let U run over ~ ( I ) and {Aw,Aw_I,...,A1} run independently over ~ ( 1 ) . Then {UAw,Aw_I,...,A1} runs exactly once over ~ ( 1 ) . This gives the formula (622). Theorem 57 is proved.

§ 9.

CHARACTERS

In this paragraph we consider systems of Dirichlet characters. Their theory is developed to that extend that is needed for Selberg's Lseries.

DEFINITION 7: Let the characterrow

(651) and the

~l,...,Xw_ 1

~

=

be even characters mod q and form

(xl,...,~w_l)

d(w-1) = ~ p r o d u c t s O

(652)

7vU = Xu . . .

is called "primitive" if all

7v

o

(1 ~ u ~ v ~ w - l )

Y,,U (1 ~ ~j ~ ~ ~ w-l)

.

are primitive.

Theorem 23 tells us under which circumstances primitive characterrows do exist. Choose a row

(653)

~ = (~l,...,~w)

of even characters such that

99 -1 $v+1 Sv = X'v

(654)

(v = 1,...,w-1).

Then $ is determined by X up to an even character as common factor. $ is called "primitive" if and only if X is primitive, i. e., if all products °

(655)

-1

(1 < u < ~ < w - 1 )

Xvu = $v+1 @u

are primitive. Set

¢~ = $ w + 1 - ~

(657) (658)

(v

-1

(656)

= ($1,"',$w)

-1 (v=l

$ V

= Sw-v

(659)

'"

$*

1,...,w),

;

..,w-l); v 1 $w = Sw ;

($1,...,$w);

=

(660)

=

=

($1,

...

,$w,)

;

Then (661)

= ($*'$w) $

(662)

'

= $ =$ .

Let (663)

~$ = ($w' $1'''''$w-I ) ;

(664)

~*$* = ($w_1,$1,...,$w_2) A

(665) (666) Then

;

A

'$'1 = $w*'$~ = $~-I (~=2'''''w*)'~W = Sw ; : ($1' .... Sw ) "

100

(667)

$

=

(~*~*,~w)

•

If one furthermore puts

~(3.,¢)

(668)

-- ,,r (3.*,,~,1¢*)

(~ = 1,2)

,

(e = 1,2)

.

one has v

(669)

be a characterrow and ~

Let X Then the expression

THEDREM

v

~qoz(l,@)qcz(l,¢) = 1

58:

w-1

(670)

~(z,x,,~,)

A

= q" F % ( h

(~))

V=1 depends only on the equivalence class of ~ . w-1

v

be a descending chain.

Furthermore

w-1

v

(671)

(672)

®(i,~,~) = (-[T" % ( h ,j=l

TF-R-

h

)) ,

:=1 ~=v

w-1 o ~x : ( h ~ ) ) ~ ( 1 , X , ~ ) = (-~-- Xw_1 ,~(h~))( "~~=I 1<~<~<w-1 A

Since h (~) (v = I, .... w-l) are equivalence-invariant the same is true for ~(I,x,~). From (600), (604) it follows w-1 A V (673) h = h -~- -~- h

PROOF:

and hence (671). From (601), (671) one obtains w-1 ~ w-1 W-1 v V

~ ( 1 , X , ~ ) = -~- X (h v - ~ - - ~ v=l ~=I ~=v (

h t)=-~- X ~=I

w-1

(-~-(h; -~- h :)) = ~=1 ~=~

-~xv(h;) )( -~xv(h ~ t )) 1 _< I _<-j_<w-1 1 <_ ~ <,j <_ x_< w-1

101

Now (672) follows from (652). Theorem 58 is proved. Let

U E ~ (I)

(a = 1,2)

and set

w 1 0(I,$,U) =-[]-@v(Det U ) ~=1

(674)

THEOREM 59:

It is

(675)

w-1 1 ° (Det U . ) ~(l,$,U) = -~- ~w-l,. ,,)=1

(676)

~(l,$,U)

PROOF:

=

(u E ~ (1))

(U E ¥ (1))(~

~(l,t,U)

=

,

1,2).

From (250) we deduce

~(I,$,U)

w-1 1 w-1 1 (-77- $ (Det U ))*$;1(-~ - Det U~) ~=1

w-1

,,,=1

1

w-1

1

o

~ - ( ¢ ; l ¢ v ) ( D e t Uv) = ~ - Mw-l,v(Det Uv). v=l

v=l

This gives (675). Formula (676) follows from (223), (674). Theorem 59 is proved. THEOREM60:

(311), (656),

Let

(677)

= IAw_I,...,A1}

be a descending chain, 1 A~ -- (A~,~u) = ( a ~ , ~ )

(678) where

a~ P

are the scalar elements of

(~ = 1,...,w-1) A~. p Then

,

102

(679)

w-1 v w-1 1 • (1,X,~) = -~- x,,,(-rT" -~-(Det A~,:)) ~=1 :=1 ~=v

(680)

~(1,X,~) =

"~-

,

1 A~t) ,

~(Det

I <_ ~_< ~<_w-1 w-1

'v

w-1

1-1

a~,k _,~ ,

~(1,~,,~.) = Tr" %CTT "IT " ~

(68~)

v=l

:=1 ~=v ~=o 1 -1

~(l,X,~)

(682)

= 1 _< ~ _< 13 _< w-1

PROOF:

(683)

,,=o

From (599), (606) follows ^ h

V w-1 1 - + "~- -~- Det A mod q Y vU H.=1 ~=V

(v

=

1,...,w-1)

•

From (670), (683) follows (679). This gives

~(1,X,C~) = -~1 < ~, < v < f 3 < w - 1

1 xv(Det A~,:) =

-~- { ~ ( D e t 1 < :<_~3<w-1

1 A~:)

Herewith one has (680). From (679), (680) one immediately gets (681) 7 (682). Theorem 60 is proved.

§ 10.

SELBERG'S ZETA AND L-SERIES

In this paragraph Selberg's zeta- and L-series are defined and the convergence is investigated. Some elementary relations and functional equations are proved.

DEFINITION 8:

(684)

Let

u = (ul,... ,Uw_ I)

be a row of complex variables, ~ a characterrow and

103 (685)

Y 6 ~(n) .

Define "Selberg's L-series" by w-1

(686)

.~(1,~(,Y,u) = ~ ~(1,M,~)-]T(Det(Y[Bw_ 1 ,SJ)) I,,~t 13=1

(687)

~*(1, X,Y,u) = U ~

(1)

-u~

w-1 1 )XOet((Y[U~k~ -u~. -~-(({w-1,8(Det U~ )) 6=1

In (686) the n x k 6 matrices Bw_ 1 ~ are connected with the descending chain ~ by (589). In (6861 we sum over all classes of equivalent descending chains of type ~ = 1,2. But by theorem 57 the right-hand-side of (686) does not depend on ~. In the right hand side of (687) the set ~(i) represents W (1)/&~(1), but also in this case the right-hand-side of (687) does not depend on ~. As remarked earlier, (YFUJ)k~ means the left upper k~ × k B submatrix of Y[U]. In case q = 1 the Selberg's L-series is also called a "Selberg's zetafunction". DEFINITION 9: (688)

In

Aw-1

let ~ (I)

be the set of all real rows

v = (Vl,...,Vw_l)

with

(689)

v

1 > ~(i + 1 + 1 ) = ½(kv+l

kv_l )

(,,

=

1 ....

Let (690)

u = (Ul, .... Uw_l)

be a complex row. For ~ (691) Set

c ~w-1

let

V~+ i ~ w-1 = lu ~ ~w-1; Re u E 7~}

•

,w-l).

104

(692)

u(1)

T H E O R E M 61: represents

In

~(i)

the series

a holomorphic

compact subset of

= ~(1)

+ i mw-1

(687) converges absolutely and

function in u. Let

~(1).

.

e > 0

and D0 a

For

(693)

u 6 ~ 0 + ilq w-1

(694)

Y > eE

,

i

the series

(687) converges a b s o l u t e l y u n i f o r m l y w i t h respect to

u and Y. PROOF:

It suffices to show that the series

lutely u n i f o r m l y in (693), (695)

(694),

(687) converges abso-

Set

v = (Vl, .... Vw_ 1) = Re u

.

T h e n the "absolut series" belonging to (687) is m a J o r i z e d by w-1

(696)

~*(t,l,cE,v)

Because of

v £ ~

= ¢ ~=1

^* ~ (I, I ,E,v) w-1

it suffices to show that

(697) convergences

k~ v~

the expression

^*

C (l,1,E,v)= u n i f o r m l y for

is bounded.

Hence

6=I

;(l,u,u,v,O) u ~ ~(1) v 6 ~

• Here

is defined by (455).

Let

(698)

b = (bl, .... bw_l) g ~ ( i )

be another real row and (699)

b %) --< v %)

(v 6 ~ )

(%) = 1,...,w-I)

•

105

Since

U

is integral we have

(700)

Det((U'U)k

(v

) ~ 1

=

I,...,w-1)

.

Hence A

(7oi)

A

f(l,U,U,v,O) ~ f(l,u'u,b,O)

(v ~ )

.

Therefore the series (697) is majorized by (702)

C*(I,I,E,b)

Hence i t

.

s u f f i c e s to prove the convergence o f (702) f o r

(703)

b E ~0(I)

.

We may confine to the case q = 1, because this series majorizes the series of the case q > I since it has more summands. We proceed like in Maa~ [33], § 10. Like in § 6 let D(Y,Y*) be the invariant metric in ~ (n) defined with the help of (392). Let ~o be a positive number and define the ball

~ = (Y E ~ ( n )

(704)

I p(Y,E) < ~o )

Here and in the rest of the proof let 1 which depend on 1,Oo,b .

ds,d2,...

Let A be a real non singular nxn matrix and the existence of a d I with 1 Det((A'A)v) d-V~ Det((Y[A])v

(705) Let

(706)

(A'A)

= C'C

denote constants

1 ~ ~ ~ n. We show


(Y ~-~)

with a non-singular ~xv matrix C. Then

Det((A'A) V) Det((Y[Aj)v)

1 =

Det(((Y[A])

)[C-1])

.

106

From the invariance

~(((Y[A])J[C-1],E)

of

~

and theorem 37 we get

= p((Y[A])9,(A'A)

) < p(Y[A],A'A)

= 0(Y,E) < Po "

Hence (705) follows from (706). From (455),

(705) we deduce

(707)

f(1,A'A,b,O)

A < d 2 f(l,Y[A],b,O)

(Y 6 ~ )

•

Hence

f(l,A'A,b,O)I

dVy <_

d2 ~ f(l,Y[A],b,O)dVy

This shows (708)

f(1,A'A,b,O)

< d3

From (697),

(708) we see that

f(1,Y[A],b,O)dVy

~*(l,l,E,b)

.

is up to

d3

majorized

by ~ f(l,Y[U],b,O)dVy

(709)

•

U 6 o(n)/A(1) Apply (705) with

A = U

~ = n

we have

Det(U'U) I

(711)

~

From Minkowski's

for

(700). Then we see

I ~11 < Det((YFU])

(710) For

and observe

N

matrices

)

(Y 6 3)

(~ : 1,...,n)

.

= 1. Hence (705) shows

< Det Y[U] <_ d I

(Y E-J~) •

reduction theory we see that

U 6 fl(n)

at the most. Here

~[U]

is the image

107 of t%under U. Let { be a fundamental domain of A(1) in Then we may assume that there are V1,...,V N 6 A(1) with

~n).

N

(713)

~[U]

6

U {[V %)=1

]

(U 6 cCn)/A(1))

.

Then (709) is equal to I j" f(l,Y,b,O)dVy U E O(n)/A(1) e[U]

~

N GI

with A

(714)

G1 = ] f(1,Y,b,O) dVy .

g (, = 1,...,w-1)

d~l ~ Det(Y k ) 1 TII ~ Det Y ~ d 1

Therefore it suffices to prove the convergence of (714). (379), (380), (381)

Perform the generalized Jacobitransformation i.e., (715)

Y = RID] ,

(716)

R =

(717)

D =

(: ,1 '.

E

D12

...

,

ilwl

0 Then by (385)

(718)

w l(n_k _k _ 1 ) dVy = -]-['{(Det R ) dv R }[dm] . "0----1

"0

108 The fundamentaldomain ~ of A(1) may be described by R %) E ~ ( I V ) (~ = 1,...,w) and the condition that all elements of D are between 0 and 1. Substituting this into (714) and immediately integrating over D gives (719)

w G 1 = # #(I,R,b,O)('TT-(Det ~=1 R %) e /;O(z %) )

~(n_k_k%)_1 ) R v)

)dVR1...dv R w

(,~ = I, .... w)

"0

< -~-Det R d I -U

(v = 1,...,w-11

u=l W

1 ~1 D < e~t T_

Ru -< dl

U=I

From (455) we get w-1

(720)

f(1,R,v,0)

= " ~ - (Det R ) v=l

-b -... -bw_ 1

')

Set

(721)

gv = ½(kv+ kv-1 - n) + b%)+...+ bw_ 1

(722)

(~, = 1 , . . . , W - 1 ) ,

gw = ~ kw-1 "

Then (723)

Inserting

gl > g2 > "'" > g w > 0 (720),

(721), (722) into (719) gives W

(724)

GI =

~ (-~-(Det R%))-g%))dVR1... R%) 6 ~6(1%)) %)=1 %)

1~ dI 1

< --~- Det R -u.

(%) = 1,...,w-1)

U=I w

d"7 _< T F tl=l

pet

_< d 1

.

dVRw .

109

Apply (377), i. e., Y2 (725)

~ ~(Det Y)dVy = "n+l ~" vn ~

~(n)

{(Y) dv y

Yl

Yl ~ Det Y ~ Y2 From this we deduce (726)

~ (Det Y)-gdvy

=

n+1 Vn Yl-g -2~

(g > 0)

~
(,~ = 1

~ •

.. ,w-l)

w

~1 < - ~ -

r

L~

< d1

m

u.---1

Substitute (728)

t

v

=-~-

r

(v = 1,...,w) .

U=I

Then

b(tl,...,t w)

~(r~,...,I")

= Det

11°:t!)tl I

-- tl... tw I_ •

110

Hence b(rl,.--,r w) _ b(t 1, ,tw) = tll ... tw~ 1 •

(729) Furthermore

(730)

w w-1 -~- r-l-guu = (-~l.t=l

-1-g tgv+l-gvv )tw

~=1

Therefore (731)

w-1 G2 = (-~-

~ [

dl tg~+l-gv-lv dtv ) [

,a=l

1

"1

1 -gw

dt w •

This converges because of (723). Theorem 61 is proved.

THEOREM 62: In M(1) the series (686) converges absolutely and represents a holomorphic function in u. Let ~ be a compact subset o f ~ ( 1 ) . For (732)

u E ~+

i]R w-1

the series (686) converges absolutely uniformly with respect to u. Let L(X,z) den@te the ordinary Dirichlet L-series for X and put I -S (733) ^L(I,3(,u) = ~ L('`ju,2(uu÷...+u )-k +k - ~ ). ~J U 1 <_ U _<' J _ < w - 1

~=o

Then A

(734)

~(1,x,Y,u) = ~.(1,X, u ) ~*(1,x,Y,u)

.

PROOF: First we prove (734) by a formal computation. Afterwards we shall see that all steps are justified. Then the convergence properties of (686) will follow from theorem 61.

111

Because of theorem 57 one may in (686) sum over all reduced descending chains

(735)

~=

{UDw_I,Dw_2,-..,Dll

U =

1 (U u) E ~(i),

with (736)

1 D 6 = (n~,vU) = (d~,t~,) 6 ~*(I;6+1,6)

(737)

(6

I,...,w-1).

=

Then

(738)

= 1,...,W-1) .

(6

Bw_I, 6 = UDw_I... D~

Hence w-1

6

I

Det(Y[Bw_I, ~ ])= Det((Y[U])k )-]-[-- ~ ( D e t

(739)

Dx,u)2

(~=1,... ,w-1 ).

9=6 U=I

From (616), (619), (672) we deduce w-1 1 (740) ~(I'x'~)=(-~[- ~w-1, (Det Uv)X ~=1 1 _< ~ _ ~<_ (741)

~(I'x'Y'u)=

1 <w-1

W-1 I ~ ( "]q-(~(w_l,6(Detu6llDet((Y[U]lk6)-u6) ×

u ~ ~(1) 13=1 o i (( "]-[- X~u(Oet D ) 13u3~iw-1

c•i

))( I~

-[]-

I

(Det D p)

-2u6)

)

6~2w-1

Hence

(742)

^~(I,x,Y,u) = A 2*( l,~,Y,u)

with

(745)

^=

~(

T7

~(1) I_~<~<~<w-l_ -

1 1 -2(uu+...+uv) ) ~1(Det Ovt,)(Det D U)

112

Formula

(743) gives

(744)

A -- ~ (

]]-

~/(1)

I _< ~ <_ .

I-_~

-2(u +...+u )

(~.u (d. k -~)d.,k~, < w-1

~=0

) •

'

Hence b e c a u s e o f t h e o r e m 26 1 -I

(745)

^=

7T

~

~Fg

~(d~,k-,

l<_u_
-2(u +...+u )-k +k +~ )d ~ , k - u. "J U

u

and because of (733) A

(746)

h = L(I,x,u)

From (742),

(746) follows

•

(734) provided that all substitutions

are allowed. We shall prove that all substitutions Let

Xo

be the principal

(747)

are allowed for

u 6 ~(1).

character mod q and

Xp = (×o' .... Xo)

with w-~ times

Xo. Let

v E ~ (I). From wellknown properties of

Dirichlet's L-series (see for instance Landau [221) and from theorem 61 it follows that all series on the right-hand-side of (734) converge absolutely for

u E ~(1)

absolutely uniformly for (732). Now let

and that they converge X = ~p

and

u = v F ~ (i).

Then all terms in the preceding computation are ~ O. Hence (734) must be true for

u = v, X = Xp. If

u E ~(1)

and

~

are arbitrary

the absolute values of all terms are majorized by the terms of the case

Re u, Xp- Hence (734) must hold for all

Then the convergence properties of

~(1,x,Y,u)

u 6 ~(I)

and all X"

follow from theorem

61. Theorem 62 is proved.

THEOREM 63:

The functions

(686),

degree w-1 (748)

- ~

k

up

~=I For

U E ~(I)

(~ = 1,2)

we have

(687) are homogeneous

in

Y

of

113

w-1 I o C (I,x,Y[U],u)=(-~-$w_I,u( Det Uu))-I ~*(1,x,Y,u) , ~=1 w-1 i ~(I ,x,Y[U],u)=(TT Xw-1,~, ° (Det u ) )-I C (l,x,Y,u) u=l

A@

(749)

(75o)

PROOF: It suffices to prove (749). It is (~ = 1,2). Hence (687) may be written

= ~

(1)/Aa(1)

w-1 i -u ~*(l'7'Y'u)= ~ -[T[(~w-I,~ (Det V#XDet((Y[UV])k))~). V 6 ~ (1)/A~(1) 13=I

(75~)

With

~(i)

V

also

UV

runs over

^. (l,y,Y,u)=

[

~ (1)/A~(1). Hence

w-1 1 1 -u -IT(( °Xw_1 ,~(Det Ui3Det v~)XDet((Y[UV~k ~) 13)

v ~ ~ (1)/A~(1)_ ~=I w-1 1 = -~- (° ~(Det U~)) ~*(1,x,Y[UI. ,u) Xw-1, ~=I

•

This proves (749). Formula (750) follows then from (734). Theorem 63 is proved. Let the variable row s = (sl,...,s w) and u be connected by (452). Let ¢ and 7 be connected by (654). Set A

(752)

¢(l,¢,Y,s) = ~(l,x,Y,u),

(753)

~*(l,¢,Y,s) = ~*(l,7,Y,u) ,

(754)

L(1,¢,s) = £(1,v,u)

.

Then by (734) (755)

~(I,$,Y,s) = L(I,$,s)~*(I,$,Y,s)

.

114

Furthermore because of (655), (680) C(l,~,X,s) =

(756)

-1

(

{~t

1

Iv+l+l y 4 ) ;

w-1

A#$)X-I-F-(Det(Y[Bw_I,v] )Cv-s~*l-

-[T- (@#+It~:)(Det

,~=1

1<_ ~ <_ ~_< w-1

C (1,$,Y,s)=

(757)

w-1

q-F u E Y (l)/A~(a)

(D tC(z[u]

1v+l+l v 4

))%-Sv+l-

~=I 1 -I I

t

(758)

)+ ~(Iv+l+ l~l)-X) • 1_< #<_v < v _ 1

THEOREM 64: degree

The functions (756), (757) are homogeneous in

(759)

n({1,s} - sw - ~ kw_1) .

Furthermore for *(

U E W (1); ~ = 1,2

(760)

~

(761)

C(I,$,Y[U],s)

PROOF:

~=o

1,,,Y[U],s)

= 0-1(l,~,u)¢

*

(1,~,Y,s)

= #-l(l,$,U)C(l,$,Y,s)

Apply theorem 63. One has w-1 w-1 w-1 -~ k u u w = - ~ kL1 sl,+1 + ~ k H s ~ t T=I

u=l

~=1

,

.

Y

of

115 w-1 1

w

w

u=l

u=l

T~I ~(~.÷~÷~ ) = ~ ~s+ w

w-1

~ ~s ~w~w ~ ~ ~#~.+~~ ~)= u=l

w-1

~is U U

-

1

nSw - T

p=l

p=l w-1

~ kuku+l u=l

+ ¼ ~ ku-I u=l

w-1

ku

w-2

n{l,s} - ns w - ~[

k

p=l n(il,s} - sw - ~ kw_1)

p

k

Li+ 1

+

k

u=o

U

k

U+ 1

.

This proves (759). Formulas (760), (761) follow from (675), (749), (750). Theorem 64 is proved.

THEOREM

65:

Set n ~

(762)

A(1,$,Y,s) = (Dot Y)

Sw + T

*

C (i,~,X,s)

.

The series (756), (757), (762) converge absolutely for (763)

~v +1 " ~v >

1

%,+1 4

and are there holomorphic.

+I

V

(v = 1 , . . . , w - 1 )

The domain (763) is invariant under

the substitution (764)

1-

1 ,

s-

s .

The function (762) is homogeneous in (765) One has

nil,s}.

Y

of degree

116

(766)

A(I,~,Y,s)

(767)

A(1,$,Y[U],s)

(768)

=

~(l,,,u)f(l,Y[U],s) u ~ Y (I)/A~(Z)

= O-I(1,$,U)A(I,$,y,s)

A(I,~,Y,s) = A(I,$,Y,s)

,

(U E ~ (1)),

,

n ~

(769)

(Det Y) sw+ ~- ~(1,$,Y,s) = L(1,,,S)A(1,,,Y,s).

The assertion about convergence and holomorphy follows from theorems 61, 62. The invariance of the domain (763) under the substitution (764) is easily computed. (766) follows easily from (424), (757), (762). The degree (765) follows from theorem 38 and also from theorem 64. Formula (767) follows from (760), (762). Formula (769) is a consequence of (755) and (762). Finally (768) follows from (310), (433), (676), (766). Theorem 65 is proved.

PROOF:

§ 11.

ANALYTIC CONTINUATION

In this paragraph the analytic continuation of Selberg's zeta- and L-series is proved and an important functional equation is derived. The method is similar to Maafl [33], § 17. From the functional equation we derive a result about Gaussian sums. We show that Selberg's L-series are holomorphic in certain domains.

THEOREM 66: Let F(...) be the F-function, a even character mod q and

s

a complex variable,

-S

(770)

~(X,S) = (q)

F(s)L(×,2s).

For q > 1 the functions g(X,S) and L(X,2s ) are holomorphic in ~. For q = 1 the functions s(~ - s)~(X,S) and (3- s)L(x,2s) are holomorphic in ¢. Let X be primitive. Then all zeros of

117

g(x,s)

lie in the strip

(771)

O
PROOF :

s<~.

1

See Landau [22], § 128.

T H E O R E M 67:

a E ¢,

Let

(772)

Y E ~(n),

T E ~(n*)

.

Then the functions (773)

e (1,~ I

(774)

e (1,,; I @ ,Y,T)(Det T) a A(l*,@*,T,s*) ( w ~ 3 ;

'1

,Y,T)(Det T) a

(w = 2; ~ = 1,2) ~ = 1,2)

is invariant under the substitution (775)

T

PROOF:

-

(U E ~a(1); a = 1 , 2 ) .

TEU]

Apply theorem 46 and formula (767).

Define

v

(776)

n*(n*-l)

~(l,,,s)

= q

w* 1 -1

4

v=l w*l

(777)

-1

~(1,~,s)

½

- s~ +

-

) ,

:=o i +I

~(~,s) = -TT - n - i ( % v=l

(778)

v

~

"TI--FT-~(,w1,~,Sw

sw + ~

l+l

- {9(Sw-%+

:=o 6(l,s)g(1,,,s)

(q = 1)

~(1,~,s)

(q > 1)

=

~

-

½)I,

118 THEOREM 68 :

In the domain

> lv+l+ i v

(779)

°v+l =

one has for

-

°v

(v =

4

1,...,w-1)

1,2:

(780)

~(1,$,s)(Det

½(Det

Y)s2-il's}

[

Y)-il'S}A(1,¢,Y,s)

e(1,~l~l

,Y,T)(Det

=

T) s2-sl

T E~e(~)

(w = 2),

~(1,$,s)(Det y ) - t l , s }

(781) 1,s} ~

PROOF:

of theorem 67 the integrals

Because

A(1,$,Y,s) =

e (1,$wl¢*,Y,T)(Det T) s wA~± ''~

{(Det Y) sw-{

depend on the choice of the fundamental

~J ~J T ,$*,T,s*)dv

in (780),

domain

_~ e ( l ~ ) .

(782)

Like at the

First let

sub-

w = 2. We put

E = ½(Det Y) s2 ;e (1,$21¢1,Y,T)(Det T) s2-sl

We have to prove (783)

(w ~ 3).

(781) do not

end of the proof of theorem 62 one sees that the following stitutions are allowed. First let

dv T

~ = ~(1,$,S)A(1,$,Y,s) q = 1. From theorem 49 one deduces n @

= (Det Y) s2+ -4A E~l(llw,w*)

.

dv T

119 n

1

~ (Det T) 82 s1+TPn(T)exp(-~ Tr(Y[AW(n*)]T)) dv T

Hence by (399), (400), (534) n ~

= (Det Y) s2+ ~

~

Det(-~Y[A])

A E j~l(l;w,w*)/~l(l*)

(Det T) T E ~(n*)

s2_s1_ ~ a n+ 1 (Mn.((Det T) ~ exp(-~ Tr(Y[AW(n*)]T))dv T .

Therefore n ~

= (Det Y) 82+ ~

(784)

~,

Det(-~Y[A])

x

A E ~l(1;w,w*)/~1(l*)

T

For (785)

E

(Mn.(Det T) ~n*)

Y E~n)

n s2-s 1- ~

consider the integral F(n,Y,s) =

which is a special case of Theorem 40 shows (786)

n +1 exp(-N Tr(Y[AW(n*)]T))dv T • )(Det T) ~

~ (Det v)Sexp(-Tr(YV)) dv V , V C ~n)

J(l,X,s)

considered in theorem 40.

(Det y)S =

with (787)

FCn,E,s) = ~

-~- C(s - ~) . ~=o

120

Therefore by (399), (534) Mn(X)(De t y)S = F-1(n,E,_s)

~ (Det v)-SMn(Y)exp(-Tr(YV))dvv V 6 ~(n)

=

F-1(n,E,-s)(Det Y)(-1)nF(n,Y,1-s) = (Det Y)S(-1)n F(n,E,1-s) F(n,E,-s) '

Hence (788)

n-1 Mn(Y)(De t y)S = ( - ] T ( s + ~))(Det y)S k;=O

From (784), (788) we obtain (789)

n*-1 n ~ S2+T E = (-1)n*-FT-(s2-sl - ~n+ ~XDet Y) ~;=O

~ (Det(~Y[A]))

A E ~l(l;w,w*)/~l(l*)

~(Det T) s2-s1+ ~ + 1exp(-w Tr(Y[AW(n*)]T))dv T • T 6

From (309), (785), (789) we get n*-1 (790) E = -[T-(Sl-S2+~-~XDet

Y)

n* s2+ T

Det(wY[A])F(n* ,~Y[AW(n*)],s2-s1+~+

I)

A 6~1(1;w,w*)/71(l*) and by (786), (787) n*(n*-l) n*-I (791) ~ = ~ 4 -IT { (st-s2+ ~- ~Xs2-s1+ ~- ~) } ~=O

n*-I

"IT F(s2-sl+~-~XDet v=O

Y)

s2+ ~*

n

(Det (wY[ A] ) )

A 6 ~1(1;w,w*)/~1(l*)

Sl-S2- T

x

121 From (756) with

X

the principal character,

n*(n*-l) + n.(sl_s2 - ~)

(792)

~ =,T

4

(777) and (791) we get

n*-I

46(1,s)(T~£(s2-s1+~-~))

x

n* (Det Y) s2+ ~

C(I,$,Y,s)

and because of (769) n*(n;-1) + n.(s1_ s _n)

(793)

~ = ~

n*-1

2 T6(1,s)(T~r(s2_sl+~_~))

L(1,~,s)A(1,$,Y,s)

.

From (758), (770), (776), (778), (793) we get (783). This concludes the case w = 2 , q = 1 . Now let

w = 2,q > I. From theorem 43, (529), (531), (782) we get

~1

n* = (Det Y) s2+T

(794)

~ ($

A 1

n #(Det T)S2-Sl+Texp(-

i $1)(Det At)

x

~(l~w,w*) ~ Tr(Y[AW(n*)~T)

dv T

Formulas (309), (794) give us

(795 )

n~ = (Det Y) s2+ T

1

(~l~l)(Det A1)F(n*,~Z[AW(n*)],s2-sl+~).

A ~ ~(l~w,w*)/~

(l*)

From (786), (787) we deduce

n* -(796)

~ = (Det Y) s2+ 4 ~

n* (n*-I

4

) ÷ n*(s 1-s 2- ~) X

122 n*-I ('l'FF(s2-sl+~-~)

~=o

[ ($21@1TM

A ~ -~(1;w'w*)/~(l*)

Now (783) follows like in the case complete. For

w

>

3

1 A1)(Det y[A]) sl-s2-~

q = 1. The case

w = 2

is

we set

= ~(Det Y) sw ~ e (1,@wl~*,Y,T)(Det T) WA(1 , ~ * , T , s * ) d v T .

(797)

T ~ U 4~(Z

)

Again we have to prove (783). Set

(798)

Pn

(q = 1)

I

(q > 1)

S --

Then from theorems 43, 49 and (531) we deduce n* n 1 1 (799) @~(1,x,Y,T)~(Det Y)~ (Det T) ~ ~ Xl(Det A1)...Xw.(Det Aw.) A £~41;w,w*) S exp(- ~ Tr(Y[AW(n*)]T)) Inserting 4766), 4799) into 4797) gives n* (800) ~ = (Det Y) sw+~ ~ U 6 ~ (i)/A~(I

)

.

~ A 6

~(l;w,~)

1 1 ~ (~wl@1)(Det A1)-.O($wl~w.)(Det Aw.)~(l*,@*,U) I

X

x

A/

~ f(Z*,T[Ul,S*+(Sw+ T ~ ~(I*)

~) e(w*))S exp(- ~ Tr4Y[AW4n*)]T)) dv T.

123

Keep U E ~ (i)/A (i) fixed. Then because of (309) we have W(n*)U'W(n*) E ~ (1). Set (801)

A = A*W(n*)U'W(n*) .

Then with A also A (311), (674) we deduce (802)

runs over ~ (l;w,w*). From (309), (310),

1 (~w1~l)(Det A 1)

...

(~w1~w.)(Det Aw.)P(l* ~*,U)

(~w1~1) (De t 1;)

• ..

(~w1$w.)(Det Aw.) •

1.

Substitute (801), (802) into (800) and write Then

(8o3)

A

instead of

A .

n* 1 1 = (Det Y) sw+~ ~($wl~lXDet A1)...(~wl~w.)(Det Aw. L ~ A E ~(1;w,w*)

UE ~ ~*~Ae(1 )

I ~ f(l * ,T[U],s*+(Sw+ ~ ~) e(w*))S exp(- ~ Tr(Y[AW(n.)]T[U])dVT"

If

V

runs over

(804)

As(l*)

then

W(n*)V'W(n*)

runs over (805)

As(l*). Hence n* E = (Det Y) sw+~

1 1 ~ ($~l~l)(Det A1)...(~;l~w.)(Det Aw.) x

A E~(I;w,w*)/A~(I*)

f(l*,T,s*+(Sw+ ~) e(w*))S exp(- ~ Tr(Y[AW(n*)]T)dv T • T E ~(n*)

124

Set

(806)

q

Y[AW(n*)] = c-lc -1'

with an upper triangular matrix cI (807)

C = 0

'Io n

Then (808)

~q Y[A]

From (432), (444), (808) (809)

=

C'C

.

we deduce

Y(l ,T,s) = f(I*,T[C-1],s)f(I

,~ Y[A],s)

In the integral of (805) make the substitution (810)

T - T[C]. Then

E = E I ( S~'~ + ( s w + ~ ) e ( w * ) ) ~

2

with (811)

(812)

~1(s*) =

22

=

(Det Y) sw+ ~*

% ~ f(1 ,T,s*)S exp(-Tr T)dv T , T 6 ~(n*) x

I 1 ~ ~ ~ l(~wl~1)(Det A1)...(~wl~w.)(Det Aw.)f(1 ,~ Y[A],s*-(Sw+~) e(w*)). A ¢ ~(llw,w*)/A~(l*) From (433) we deduce (813)

* n s + n* 22 = (q) n*(~l's*}-sw-T) (Det Y) w 2[

x

1 1 I(~w1¢1 )(Det A 1 )...(~wl~w.)(Det Aw.)f(l*,Y[A ],s*-(sw+ ~) e(w*) ).

125

From theorem 29 and formulas (427), (428), (674), (766) we obtain (814)

"~2 = "z3 ^(1,~,Y,s) *

1

1

E3=(q~)n*(Ii's*}-Sw-~)l(,;l~l)(Det D1)...(,;1$w.)(Det DW.)

(815)

x

D E ~*(l;w,w*) w* 1 2(Su-Sw)+~(kp+k p 1-n*-n) "]T(Det D ) U=I

.

Application of theorem 26 gives * w* 1-1 (816) ~3=(~)n*(ll's*l-Sw-~)~T ~TL(,;1,.,2(Sw-S

- ~)

)+ ~

..,=1 :=o

In order to prove (783) we have to show (817)

Let

~l(S"~ + (S

w

* ~)e(w*))~ 3 = ~(1,$,s)

•

q > 1. Then by theorem 40 n*(~*-l)

(818)

~l(S*) = J(l*,~.,s*)=~ ~

w*

1-1

1-n*

-IT TF r(%+ ~ .=1

- ½) ,

~,=o

hence (819)

n*(n;-1) w* 1~-I ~l(S* + (sw + ~)e(w*)) = ~ 7 1 - - I T r(sw % * .=I %=0

-~)

From (816), (819) one gets (817) by an elementary computation. Finally let get

q = 1. From (798), (811) and

(820)

F~I(S*) =

1

instead of

~ f(l*,T,s*)Pnexp(-Tr T) dvT T E ~(n*)

.

1

we

126 Then (817) is equivalent to n*(~*-l) (821) ~l(S*) =

w* 1-1

1 -n*

q'f- TI- r(%+1. - % - - - ~ ) x ~=1 t=O

W* i-I m-

(8 -

-fl-

1 -n* s

+ -m-

-

~)

•

v=l ~,=o Hence it is left to prove (821). We proceed like in Maa~ [33], page 83. From theorems 39, 40 we obtain for X E ~ n ) Mn(X)f(1,X,s) = N~(X-1)f(I,X,s) = 1

f(l,Y,S)Mn(X " )exp(-Tr(X-ly) )dry

Y ~ ?(n) (-1)n(Det X)~(l'E's+e(w~)~) f(l,X,

)

~1,E,s) n

~/

(-I) }(l,E,s+e(w)) f(1,X,s)

.

J(~,E,[) Hence by (436) (822)

w i~-I n- 1 Mn(X)f(1,X,s)=(- ~- -~-(s v + - - ~ ~=I

+ ½))f(l,X,~)

t=O

From (400), (820), (822) we deduce _n^ n+ 1 Hl(S*)= ~ f(l*,T,s*)(Det T) ~Mn.(Det T) ~ (Det ~ exp (- Tr T))dvT = T E ~(n*) (-1) n*

I(Mn.f(l*,T,s*- 9(w*))(Det T) ~ + 1 exp(-Tr T)dv T

r ~ ?(n*)

=

127

W

I-1

n*S M

1

- ~ + --~

+ ~)) J(1,E,s*+e(w*)) .

,,,=1 I=o Now (821) follows from (436). Theorem 68 is proved. Set max(l~'lu)'1

v

(823)

-FT

~(l,,,s) =

TF

l
(825)

61(~,s)=

½)

9

~=o

max(1 ~ - 1 (824)

I +I

~(~I~,%-s~+_~ -

TT

-FF

1_
~=o

I +I

i +I

~(s-~+ -~v-~-½×s-s+-h--~-~)l,

61(i,s)~(i,~,s)

(q = I)

~( i,~ ,s)

(q > I)

~(i,~,s) =

Then (826)

~(i,~,s)

= ~(i,$,s); ~(i,~,s) = R(i,~,s).

Set (827)

(828)

v

V

v

W

%(i,¢,s)

= g(i,,,s)~(i

,**,s*)

%(i,,,s)

= g(l,,,s)~(i*,**,s*)

%(i,~,s)

= 6(l,S)~l(i*,s*)~o(i,~,s)

Then (829)

V

(830) Put

%(l,~,s)

= %(i,~,s)

(q=

I) ,

(q>l)

128 n*(n*-l) w* max(lw'l,~)-I %/

(83~)

{(1,¢,s)=

4

q

7r v=l

max( lw, 1 v)-1

W~

(832)

I=I

62(1,s)=T~ v=l

1 +l

l=l V

62(1,s)~(1,¢,s )

(833)

~(z,~,s)

(q = 1)

= {(l,,,s)

(q > 1)

Then

%(l,~,s)~(l,~,s)

(834)

~(z,~,s)

=

(835)

~(z,~,s)

= %(1,~,s){(z,¢,s)

THEOREM 69: (836)

The function k(1,~,Y,s) = @o(1,~,s)(Det

is homogeneous in (837)

PROOF:

of degree

O. Furthermore

~(l,¢,s)k(l,~,Y,s)

= ~(l,~,s)k(l,~,Y,s)

Apply (765), (768), (826), (835).

THEOREM 70: = I ,2 and

(838)

Y

y)-{1,sl A(1,~,Y,s)

Let

q > I

and

¢

be a primitive character row,

Xel(1,$,Y,s)= l ( D e t Y) s 2 - { l ' s }

~ e (1,,21,1,Y,TXDet T)

s2-s 1 dv T

(w = 2) ,

129 (839)

k~l(1,~,Y,s)

=

½

~ ~ ~ Sw-{l,s} ~(1 ,$*,s*)(Det Y)

j" Q (1,~wl~*,Y,T)(Det T)

Sw-{l*,,*t

~ (I*, ~*,T, s*)dVT

X

(w Z 3).

T ~ ~=I(1 )

The integrals on the right-hand-side of (838), (839) converge absolutely and are holomorphic for all s £ Cw. Let R n be a non-singular rational nxn matrix and (840)

Y = (y%~) £ [(n,u)

•

Let ~ c Cw be a compact domain. Then there exists a real number c15 = c15(1,U,Rn, ~ ) > 1 and finitely many (say g(1)) linear functions w (~ : I ...,g(1)) (841) ,S (1, %,O) = j ( l , ti~)a + 3(1,~)

v=l with rational j(l,~,~), 3(l,t) (~ = 1, .... g(1); ~ = 1,...,w), such that for s £ ~ the inequality n* (842) abs ~I(I'~'Y[Rn ]'s) _< 015(Det y)aw_{l,~l+~ ~(1)ylZ(l't'a) ~=1 holds. By the formula

(843)

~(1,~,Y,s)= ~I(1, $,Y,s)+

~ ( l , ~ ) q 21w({l~s}-sw)km~l,$,Y,s)/v ~ v

the function X(I,~,Y,s) is holomorphically continued to Cw. For Y £ ~ (n,u) and s £ ~ there is a real number o16 = o16(l,c,Rn,~)>1 with

g(1) (844)

abs k(l,~,Y[Rn],S) ~ c15(Det Y) °w-{l'°}+~*

~ yl~(I'%'0) %=1 n* g(1) v v + 016(De t y)Ow-ll'al-T ~ y~0g(l,~,a) t=l

130 Finally there is the functional

equation

)q21w(il,si-Sw)x(v (845)

k(1,~,Y,s)

= D~(1,$

v v 1,~,Y,;)

Hence

(846) For

~l(z,~) q = I

= ~2(z,~)

the same is true for

(847)

11 . . . . .

.

w = 2

or

w ~ 3

and

iw

If (847) does not hold, we can only prove that ~I(I,~,Y,s) and k(1,$,Y,s) are meromorphic in C w. They become holomorphic if one multiplies them by a finite product PROOF:

of linear functions

Because of (840) there is a constant

(848)

dI ~ I

in s.

with

d;1(Dg Y) ~ Y ~ d1(Dg Y).

Hence there exists a constant

d 2 = d2(n,~) >_ d I >_ 1

J(Y) = d 2 ly

(849)

I

v = d 21ynl , j(Y)

with

.

Finally we have (850)

(Y[Rn]) v -- Y[Rn~

with the non-singular

rational n×n matrix

Let (851)

Then by Minkowski's with

T = (t

) E ~(n*)

.

reduction theory there is a constant

d3 > 1

131 (852)

d ~lt I ~ t

_< d3tn.

(853)

abs t

(854)

d~l -1 en*-I tn* < Det T < d~tltn:-1 --D n

(855)

1 (Det T) ~

(~

=

1,...,n*)

(I ~ ~; t,~ = 1,...,n*),

~ d3tn.

d3tn. .

Furthermore (856) Now let

Tr T ~ tn.

.

T 6 97~I(n*) , i. e.,

(857)

Det T

>

1 .

Then by (854), (855) we obtain -I t1_n . n* 3

(858)

tI ~ d

(859)

tn. ~ d~ 1

From (852), (854), (858) we deduce (860)

d32 t n* l-n* -< t I -< d3tn.

(t . . .1,.

(861)

d-n*t 1-(n*-I)2 ~ Det T ~ d n* 3 n* 3 t~

,n* ) ,

.

Hence for all real numbers p (862)

tp~ --< d~ abs P(t~. + t (1-n*)p~n.~

(863)

(Det T) p < ~n*abs p(t~**p + t(1-(n*-l)2)p)) --

Now let

~3

n @

(~ = 1,...,n*),

"

w = 2. From (373), (838) we get with some constant d 4 > 1:

132

(864)

abs ~S(1,$,Y[Rn],S) ~ d4(Det Y) c2-11'ql

x

h ~ abs S (1,@2151,Y[Rn],T[Fev])(Det T) ~2-°1 dv T v=l

T E ~1(n*)

From theorem 53 and formulas (849), (856), (863) we get

(865)

d5(Det

abs k~l(1,@,YFRn],S ) y)~2 -~l'al+~*

nn* --~Yl ~

.[ tn*~1 (:'a) exp(-d61Yltn *)[dT] T E ~l(n*)

with finitely many linear functions 81(~,0). Accoraing to (853), (859), (860) we integrate over all t ~ exeept tn.. This gives n* (866) abs X~I(I,@,Y[Rn],S ) ~ d7(Det Y) c2-~I'°}+ ~ x nn ~

-7

co

t~n2(~'~)exp(-d6 S Yl tn*) dtn* d; 1

with finitely many linear functions

~2(:,o). Like in (112) set for

m > O; p E ~ :

(867)

I(m,p) = ~ uPexp(-u)~

.

m

Then theorem 12 gives for (868)

~ > 0

I(m,p) ~ c5(m° + mp-c + mp+¢)

.

Substitute in the integrals of (866) the variable Then each integral becomes of type (867). Applying (842). Hence X~l(1,$,Y,s ) is holomorphic in ~ . and Y E ~(n) there are ~ and p with s E ~ ,

u = d61yltn .. (868) we get But if s E cW Y E ~(n,p).

133

Hence for each in s.

Y E __ ~n)

From (828) we obtain (780),

(869)

the function

k~l(!,~,Y,s)

@o(1,~,s) = ~(1,~,s)

for

is integral

w = 2. Hence by

(836)

X(1,~,Y,s)= ~(Det Y) s2-11'sl

# e (I,,~I,1,Y,T)(Det T)

T

Because of (390),

(87o)

s2-s t

~v T

(w = 2 ) .

(413), (414), (423), (838), (869) therefore

k(1,~,Y,s) = k~1(1,~,Y,s) + V

V

V

V

~/

v q21w(ll's I-s2) (Det y)S2-11's21 ~ 8 (1,~2 I~I'Y'T)(Det T) Sl-S 2 dv T T E #~(I ) Det T < I

(w = 2).

Let T E ~(n*) and U E ~ ( i ) . Then T[U] = T[U]. Applying theorem 30 we see that the integration over T ~ { ~ ( 1 ); Det T <_ 1 I is the same as an integration over T E ~ i ( I ). Hence from theorem 47 and (870) we obtain (843). From (842), (843) we deduce (844), (845)~ (846). For w = 2 the theorem is proved. Let

w > 3

and assume that the theorem is true for

w-1 = w*°

Let q > 1. By theorem 66 the function ~(x,s) is holomorphic. Hence by (831) and (833) the function ~(1,~,s) is holomorphic. Let q = 1 and (847) hold. Then by (831) we have ~(1,$,s) = 1 and by (832), (833) the function ¢(1,~,m) is again holomorphic. Apply theorem 53 to (839) and estimate X ( I * , * * , T , ~ ) by (844) with w* instead of w. Since ~(1 ,~*, s* ) is holomorphic it is bounded in ~. Now we get (842) like in the case w = 2. From theorem 47 and formula (837) we get (843) and hence (844), (845), (846).

134

Now let q = I and suppose that (847) does not hold. By theerem I 66 the product s~(X,S ) has a pole of first order at s = ~ . Because of (831), (832), (833) the function ~(1,@,s) is not holomorphic but it may be made holomorphic by multiplying it with certain finitely many linear functions in s. Hence one can prove with the former method that X~l(1,~,Y,s) and k(1,e,Y,s) are meromorphic in Cw and become holomorphic by multiplication with certain finitely many linear functions in s. Theorem 70 is proved. THEOREM 71:

Let

m £ ~

and

~

be an even primitive character

mod q° Then (871)

PROOF:

G(m, X) = (G(1,X)) m

.

Apply (500), (501), (846).

In [33], page 220 upper part Maa8 considers a homogeneous polynomial wo(x) arbitrary and (872)

of degree g n with even g. Now let X a character mod q with

g 6 ~

be

X(-I) = (-I) g .

Instead of Maa~'s function

~o(...) (page 220) consider the sum

Wo(CA) (873)

~

x(Det A1)

1 A = (A u) £2~(1;2,1)/A~(1")

(~=1,2) s+ (Det Y[A])

with Y = C'C E ~(n). From theorem 29 it follows that (873) does not depend on ~. I conjecture the following. With a mixture of the methods of Maa~ [33] and this manuscript one sees that (873) has analytic continuation and satisfies a functional equation. We get an equation of type (871) for all primitive characters X mod q (also odd ones).

135 THEOREM 72:

Let

q >

(874)

1

~(1)

and = min(ll,...,1 w) ~ I.

In the domain

°v+l

(875)

the functions

_

OV -> 1¥+1+4 1

(v =

~(l,~,s), E(l,~,s),

I,...,W-I)

%(i,~,s), ~(l,~,s)

have no

zeros. PROOF:

Apply theorem 66.

Let

q > I. Then the functions A(I,~,Y,s), ~(l,~,Y,s), are meromorphic in ~w. They are holomorphic in the

THEOREM 73: ~*(l,#,Y,s) domain (875). PROOF:

Use theorems 70, 72.

THEOREM 74:

Let

q > I

(876) Then

11 = 12 . . . . A(I,~,Y,s)

(877)

PROOF:

(831),

and

~*(l,$,Y,s)

= Iw • are holomorphic in the domain

or+ I - ov --> 0

C(I,~,Y,s) (878)

and

is holomorphic in

(V = 1 , . . . , W - 1 ) .

C w. Furthermore

~(l,~,s) = q

.

The first part follows from theorem 73 and (878) from (833). From (769), (836) we deduce

136 n-X-

(879)

~(I,,,Y,s)

=

L(l'~'Sl(Det

Y) { l ' s } - s w - ~

X(I,@,Y,s)

~o(i,, ' In the case (876)

n*(n*-I ) (880)

%(1,~,s)

= q

4

11-1 "TT-

I<[I<~)<W

(881)

L(I,@,s)

=

-~-

l_
(880),

~.

-T'F ~(~]l~u,s,~-%+ ~=0

11-I -~-n(¢~lCu,2(sv-s u ) _ ~=o

(881) we see that

),

+ 11 - ~).

L (1,@,s) ~o(l,~,s )

is holomorphic

for s 6 C w. Because of theorem 70 and (879) the function ~(l,@,Y,s) is holomorphic for s E C w. Theorem 74 is proved. A

A~

THEOREM 75: Let q > 1. The functions C(I,x,Y,z), ~ (I,x,Y,z) are meromorphic for z £ ~w-1. They are holomorphic in the domain

(882)

PROOF:

Re z

> v --

1¥+1+ 2

I ~

-

1 ~(I)

(v

=

Apply theorem 73.

THEOREM 76: Let q > 1 and (876) be true. Then holomorphic in the domain 11 Re z v -"'2-" >

(883) A

~(l,M,Y,z)

PROOF:

1,...,W-1).

is holomorphic

Apply theorem 74.

in

C w-1

is

~*(1,x,Y,z)

(v = 1,...,w-1)

.

137

§ 12.

FUNCTIONAL EQUATIONS

For Selberg's zeta- and L-series functional-equations are proved like in Maa~ [33], § 17 and Tetras [¢5], [46]. For Selberg's Lseries the same types of functional equations are true as for Selberg's zetafunctions, but for L-series the expressions become more complicated.

From (845) we deduce (884)

k(1,@,Y,s)

= ~l(1,s)q

21w({l,sl-Sw)k(~,v v ~,Y,[)

From (837) we get

(885)

x(l,~,Y,s) =

X(1,@,Y,s)

.

~(1,~,s) Set

(886)

~Y = y[p(1) -I] •

Then 21w n v Y

(887) Since

(888)

k(l,@,Y,s)

is homogeneous

k(l,@,Y,s)

in

Y

of degree

0

= ?S(I,@,s)x(~I,~,~Y,~s)

with V

(889)

V

V

~(i,, ,s)

~I(i,~,s) = ~([,~,s) ~l(l'*)q

21w( {I, s l-s w)

we get

138

THEOREM 77:

Let

w i> 3

and set

{(I ,¢*,s*)

(89o)

V2(1,¢,s)

=

,}***,)*s*)

21wkw-2 n*

q

'tSw -

tl,st

n*

+ T )

Then r

kll(1,t,Y,s)

(891)

= Y2(1,~,s)

X11(I'$'Y[K 0 ~,s)

0=1

~',, ,-

~

XX

)v ×

(892)

~11(1'$'Y'~) = v2(l'¢'s) z~ XII(I,,,(Y[Ko~ 0=1 r

(893)

k(1,~,Y,s)

= Y2(1,~,s)

X(1,¢,

,s) ,

~ s)

0=1

PROOF:

Because of

= ~

~ and

-1y = Y[P(1)], we get

(8£4) From (887), (894) we deduce

(895) x(z ,~*,T,s*)= 6~1(~ -1 * , )*~*, )*s*)x(#*l*,}*~*,T[~(}*l

)],~*s*).

From (223), (242), (328) we deduce (896)

P(~*l*)

= P(l*)

.

From (333) we get

(897) Hence i f

p(}*l*)~l(}*l*)p-l(}*~*) T

runs over a fundamental

= ~1(1") domain o f

. ~I(1")

the matrix

139

(898)

T* = T[P(~*l*)] = T[P(l*)]

runs over a fundamental domain of

~1(~*1 ) .

Now insert (895) into (839), make the variable transformation T - T and apply (576). Then (891) follows. In the same way (892) follows from (577), Using

~(1,~)=

~(1,$) formula (893)

follows from (843), (891), (892). Theorem 77 is proved. Obviously the permutations

~ and

A generate the symmetric group

~ w " So from (888), (893) one may deduce the behaviour of k(1,@,Y,s) under all permutations.

§ 13.

RESIDUES OF SELBERG'S ZETAFUNCTIONS

Riemann's zetafunction has a pole at

s = 1. Dirichlet's L-series

are holomorphic everywhere. A similar situation is true for Selberg's zeta- and L-series. In § 11 we saw that Selberg's L-series have nice holomorphy properties. Selberg's zetafunction has more poles and one may compute residues. Details about this residue computation are given in Maa~ [33], § 17.

From now on we consider only the case q = I. It will turn out that then the functions A(1,$,Y,s), C(1,~,Y,s), ~ (1,~,Y,s) are not holomorphic in (875). For

q = 1

the only character is the

principal character. Hence we set

C*(1,Y,s ) = ~* (1,$,Y,s) ;

(899)

~(1,Y,s) = C(1,¢,Y,s);

(900)

A(1,Y,s) = A(1,$,Y,s '), X(1,Y,s) = k(1,@,Y,s)

(9oi)

L(1,s) : L(1,$,s); E(1,s) : :E(1,@,s) ;

(902)

%(1,s)

= %(1,e,s);

(903)

%(1,s)

= y (1,~,s)

~(Z,s)

= ~(l,~,s)

(v = 1,2); ~(l,s)

;

; = ~(1,~,s);

140

^ ^ g(l,Y,u) = ~(I,x,Y,u);

(904)

A

^. ~ (l,Y,u) = C (l,x,Y,u)

A

L(I,u) = L(I,x,u)

(905)

;

•

Then A

(906)

~(1,Y,u) =

~

w-1 --U~ -~-(Det(Y[Bw_1,p])) w-1 --U~ -~-(Det((Y[U])kp))

A@

(l,Y,u) =

(907)

~

U 6 o(n)/A(1) with

A(1) = A~(1)

(~ = 1 , 2 )

; 1-1

A

L(I,u) = -[']I < ~__<~ _< W-1

(908)

where

~

~(2(u~+...+ U v)- kv+ kp- ~{) , X=o

is Riemann's zetafunction; A

(909)

A

C(1,Y,u)

THEOREM78: degree (910)

~=I

The functions

-

= L(1,u)

~(1,Y,u)

.

(906), (907) are homogeneous

w-1 ~ kp U U U=I

They are invariant under (911) PROOF:

Y - Y[U]

(U 6 n(n))

.

Apply theorem 63.

We have (912)

~(1,Y,s) = L(1,s)~ (l,Y,s)

;

in Y of

141

(913)

w-1 IV+I+ Iv, C(1,Y,s) = ~ (T~(Det(Y[Bw_l,~]))sv-s~+l 4 )

(914)

~*(I,Y,s) =

;

w-1 I~+1+I N (-~-(Det((Y[U])k))sv-sv+l4 ) ;

[

U 6 n(n)/A(1) v=l

v

1-I (915)

L(Z,s) =

"T'F

~

l
where again THEOREM 79: degree

C

is Riemann's zetafunction.

The functions (913), (914) are homogeneous in

(916)

of

substitution (911 ).

Apply theorem 64.

THEOREM 80:

The function n* A(I,Y,s) = (Det Y) sw+ T~*(1,Y,s)

(917) is homogeneous in PROOF:

Y

n({l,s} - sw - } kw_l).

They are invariant under the PROOF:

c(2(s,~+l-S .) + ½(i,~+1+ :L) - ,,)

~.=o

Y

of degree

n{l,s I and invariant under (911).

Apply theorem 65.

One has (918)

^(1,Y,s) =

~ f(1,Y[u],s); u ~ n(n)/~(1)

(919)

^(1,Y,s) = ^(1,Y,s) ;

142 n..x.

(Det Y) sw + -4- G(1,Y,s)

(920)

Let

THEOREM 81 :

H(I,Y,...)

= L(1,S)A(l,Y,s)

.

be one of the functions

(899),

(900),

(904). Then H( I'Y- 1 , . . . )

(921)

PROOF:

For

q = 1

= H ( I , Y~, . . . )

= H ( I , Y ,v. . . )

.

we have

(922)

W(n),Q(1),P(1)

£ O(n).

Now (921) follows from (923)

H(I,Y[U], .... ) = H(I,Y,...)

(U £ D(n))

•

Instead of (770) we get (924) where

~(s) = ~ C

is Riemann's

THEOREM 82:

r(s)~(2s)

,

zetafunction.

The function

(925)

s(~ - s)~(s)

is holomorphic

in

lie in the strip PROOF:

--S

C

and invariant I

under

s - ~ - s . All zeros

0
See theorem 66 and Landau [22],

§ 128.

The function o

(926)

~(s) = s~(s)

is holomorphic (776),

(777),

in

C

except a pole of first order at

(778) we deduce

1 s = ~ . From

143 w* 1-1 (927)

~(I,~)=TF "U- i~(s~ -

i +i

~ + -~T ~ -

~)(~- ~w+

~-~- ~)I.

,j=1 i=O max(iv' lu )-1 (928) S(1,s) = "~l<~<~<w

~

i +I

"~- l~(s -su+ :=o

- ~)(sp-sv+ ~

- ~)I.

Then (929)

~(l,s) = ~(l,s) ;

(930)

~o(1,s) = g(1,s)~(l*,s*) w*

(931)

max(lw'l~)-I

o(~,~)=~

-rr ~ ~(Ow-V ~-~- ~(°~-~e ~

~=1 (932)

~(1,s)

THEOREM 83 : (933)

:

~=I = ~o(1,s)~(l,s)

•

The function l(l,Y,s) = ~o(l,s) - (Det Y)-{!'SIA(I,Y,s)

is homogeneous in

Y

of degree

O. Furthermore

(934)

@(l,s)k(l,Y,s) -- ~(l,s)k(l,X,s) .

(935)

k(l,Y,s) = ~(1,Y,s) .

V

PROOF:

V

V

Apply theorems 69, 70.

From (886), (888), (889), (890), (922), (923) we get (936)

k(l,Y,s) = yl(!,s)k(~l,Y,~s)

,

-~)I,

144 N

(937)

Yl(1,s)

,~2

= ~(l,s) ~(~,~1

'

#(1,s*)~(}*l ,}*s*) (938)

Y2(l,s) =

~.~ ~-~.~ ~.~.~. o(ylIs*)oS*

Applying (887) we get @(l*,s*) (939)

For

Y2(l,s) =

q = I

we have

(940)

r = 1

~(i ,s*)

and

K I E Q(n). Hence (893) gives

X (1,Y, s)Y2(1, s)x (I,Y, s)

THEOREM 84: function

For a given integer

(941)

S

in

1 ~ S ~ w* = w-1

the

(u D - ~(i + 1 + 1 ))C*(I,Y,s) U

is holomorphic in the domain defined by (942)

abs(u

Its value at up factor, equal to

- ~(ID.I+ 1 )) < ¼,

Re uv > n

is, up to a positive constant

P = ~(I+I + 10)

(:a < w*) (943)

{

~*(~'Y'~)

,I

~*(~,Y,~)(Det Y) -~lw~

(o =

w*)

and

(944)

= (ll,...,lw~);s

(l_
= (s 1, . . . . sw.) ,

145

I (945)

1

=

l

1 %)

(1 < ~ < O)

1 + 1

(v = p)

~ O+1 lv+ 1

(O < v <_ w*)

I

%)

(946)

%v =

SO+

i~+ 1

(v = ~)

sv+ 1 PROOF:

(~ < ~ < w*)

See Maa~ [33], page 287, theorem 3 or Terras [453,

From (917),

(933) we deduce

(947)

X(I,Y,s) = {o(l,s)(Det Y

An easy computation

[463 .

)Sw-{l.s}*~* .

~ (l,Y,s) .

shows

(948)

{l,s} = {l,s} +

1010+ 1 .

Then theorem 84 gives n*

(949)

lim((Det Y) sw-ll'sl+

T

C (1,Y,s))

=

s 0+1 - sO+ ~$(10+ 1 0+1 )

(Det Y) sw*-{&'s}+

-

n C. (z,Y,s)

(0=l,...,w*).

o

THEOREM 85 : (95o)

1<_~<~<_w

Let

1,p

8(l,s)

~=o

~=o

be defined by (222), (430), (431). Set =

146 Then o

k(1,Y,p) = const.~(l,s)(Det Y)-II'SIA(I,Y,s)

(951) PROOF:

.

Apply theorem 84.

By theorem 70 the left-hand-side of (951) is an integral function in s. The factors of ~o(1,s) are contained among the factors of ~(1,s). Hence by (933) there is an ~(1,s) consisting of factors like in (950) such that o

k(l,Y,p) = const =(l,s) . k(l,Y,s)

(952)

.

Now let (953)

11

=

...

=

Iw

=

m

.

Then we set [m,w] = 1 = (m,...,m)

(954)

,

w times. Especially (955)

1

=

[1,n]

.

From (953) we deduce (956)

n = m w

THEOREM 86:

.

It is n*

(957)

Res .... Res ~ ([m,w],Y,s) = (Det Y ) - ~ ul=m Uw_l=m

Especially .

(958)

Res

u1=1

...

Res

Un_l=l

o

C (l,Y,s) = (Det Y)

n-1

147 PROOF:

Apply theorem 84.

Set m o

C(m,z) = T l " l ~ ( z t=1

(959)

+ {)(-z

+ ½)}

.

Then (960)

{o([m,w],s) = E([m,w],s) =

"]-[- C(m,s v- s u) • l
Let

m

F(z) be defined by (I03), i. e., o

F(z) = (1-z)~(z).

(961) Then

C(1,z) = F(Z + ½) •

(962) Now let (m) (963)

s

m =

(s 1 , s I

+

~

,...,

s I

+

)

•

Then

il:)}

(964)

=Sl+

n*

T

.

Because of (959) the function C(m,z) has exactly one zero of first order for z = ~m . Hence by theorem 86:

THEOREM 87 (965)

:

It is /([m,w],Y,(~ )) = const

with a constant not depending on

Y

and

s 1.

148

THEOREM 88: permutations PROOF:

The function x([m,w],Y,s) of Sl,...,s w.

Apply (878), (936),

(937), (939),

is invariant under all

(940) .

CHAPTER IV.

SELBERG'S EISENSTEINSERIES

Selberg's Eisensteinseries

are generalizations

of Eisensteinseries

for the elliptic modular group. They depend on several complex variables.

Again we prove analytic continuation and functional

equations.

We proceed like in Diehl [11].

§ 14.

SIEGEL'S UPPER HALF-PLANE

The real symplectic group Sp(n, ~ )

Sp(n, ~)

and Siegel's modular group

operate on Siegel's upper half plane }(n) = {Z = Z' : X + iY; Y > 01.

Some elementary results on Siegel's upper half-plane are collected.

The following results may be found for instance in Christian

[7].

Siegel's upper half plane of degree n is given by (966)

>(n) = {Z = Z' = X + iY; Y ~ ~(n)}

.

It has

(967)

d(n) =

complex dimensions. (968)

Form the 2nx2n matrix i °

with nxn zero and identity matrices group (969) Set

Sp(n, ~)

0

and

E. The real symplectic

consists of all real 2nx2n matrices I[M 3 = I

M

with

150

with nxn matrices A, B, C, D. Then by (971)

M = (AZ + B)(CZ + 0) -I = X M + iY M

the group Sp(n, ]~) operates transitively on ~(n). In (971) X M and YM are the real and imaginary part of M. We set (972)

mzl

= cz + D

One has

Det M{Z} ~ 0

(973)

(M E Sp(n, B ) ) ,

~Z = (M{Z}),-Iy(m~I)

(974)

-I

,

hence Det YM = (abs M { Z } ) - ~ e t

(975) An easy computation

shows

(M I,M 2 ~ Sp(n, m))

(976)

(MIM2)

= MI(M2>

(977)

(MIM2){Z}

= MI{M2}M2{Z}

Furthermore one has

M = 2

for all

2 £ ~(n) if and only if

m

The Jacobian of the substitution

A volume

(979)

(971) is

~M(Z) = (Oet M{ZI)-n-I element in

~(n) is given by

[dXl[dYl d~ Z = ( D e t y)n+l

,

CMI,M 2 6 SpCn, I~)).

M=+E.

(978)

Y .

'

151 where [dX], [dY~ denote the Euklidean volume elements in the X- and Y-spaces respectively. One has (980)

d~M = dw Z

(M E Sp(n, I~))

THEOREM 89: Let ~ c ~ n ) be a compact domain. Then there is a constant c17 = c17(n,~) ~ 1 with (981)

YM <- c17 YM

for all Z,Z PROOF:

(M E Sp(n, ~))

E ~.

See Christian [61, page 271, Hilfssatz 15.

THEOREM 90: Let ~ c ~(n) U theorem 89. Then for

be a compact domain and

(982)

Z,Z* E ~

(983)

M E Sp(n, m)

c17

like in

,

the following inequalities hold (984)

1 * * c~7(YM) v ~ (YM)v ~ c17(YM) v

(985)

c;~((YM)-I)v ~ ((YM)-I)~ ~ c17((YM)- )v (~=l,...,n),

(986)

C1~ * v Det((YM)~)(~=1,'",n), * - Det( (YM)v) ~ Det((YM) ~) ! o17

(987)

c~Det(((Y~)-1)~) ~ Det(((YM)-I) v) ~ c~7 Det(((YM)-I)~)

(v = 1,...,n) , •

(V

1

=

1,...,n)

.

PROOF: From (981) follows y~l ~ c17Y~-1 . From this and (981) follow (984), (985) and from this (986), (987). Theorem 90 is proved.

152

DEFINITION

10:

Let

m

Sp(n, ~)

and

I < j _< n .

(988) Then let

be a subgroup of

~j

(989)

denote the subgroup of all /Aj 0 Bj * *

P~

*

Cj

0

D.

0

0

M E m

of type

.

M = \

\0

*

J

p~ljl

\

with a jxj matrix P4e and (n-j)x(n-j) matrices Aj,Bj,Cj,Dj. group 9j is called the "j-th cusp group" of m.

The

11: Siegel's modular group F(n) = Sp(n, ~) consists of all integral matrices of Sp(n, B). The Borelgroup

DEFINITION

W

(990)

~(I)

=

n

r k (n)

v=l

v

consists of all matrices (991)

M :

with (992)

12:

(993)

Let

~(Y) =

THEOREM 91: constant

SU-I 1 U-I

U 6 A(1)

and integral symmetric DEFINITION

If

'

S = S' Y 6 ~(n)

and set

min Y[g] > 0 g 6 ~n-{0}

Let ~ c ~(n)

be a compact domain. There exists a

c18 = c18(n,~ ) Z I

such that

153

(994)

Det(((YM)-I) v) ~ c~

(995)

~((YM )-I) ~ c ~

for all

Z E ~

PROOF:

By enlarging ~ we may assume iE 6 ~ . = iE we see t h a t i t s u f f i c e s t o p r o v e

with

Z

and

(v = 1,...,n) ,

M 6 I'(n) holds.

(996)

Det(((EM)-I) v) ~ 1

(997)

.((EM )-I)

(v

=

Applying (985)

1,...,n) ,

~ 1 .

But from (972), (974) we obtain (EM)-I = CC' + DD' > 0 .

(998)

This matrix is integral and hence (996), (997) follow. Theorem 91 is proved. Set

(999)

W

=

/W(n) 01) Q 0 W(n

E F(n) .

Then the group v

(1000)

~

V

U

~(i) = W~(1)W

consists of all matrices (1001)

with integral (1OO2)

M =

S = S'

UoI

and U' 6 A(1) •

154

THEOREM 92 : given by

A fundamental domain

abs

(lOO4)

Y 6 f~(n)

,

(1005)

- ~1 ~ x

< ~1 --

~

F(n)

in ~ (n)

(N E r(n))

(1003)

N{Z}

~(n) of

1

(t,x

is

,

= 1,...

,n)

for

x

(1oo6)

PROOF:

=

(x~)

.

See Christian [7], Kapitel IV.3.

DEFINITION 13: is defined by

Let

~ > O. The "elementary set"

(1007)

Y ~ r(n,u)

(lOO8)

-1 Yl >U

(lOO9)

-U<X

THEOREM93: (1010) PROOF:

PROOF:

,

(t,~


There exists a constant

= 1,...,n)

c19 = c19(n) > 1

.

such that

~ ( n ) c ~(n,c19) . See Christian [7], page 225, Satz 4.86.

THEOREM 94: (1011)

~(n,u) c ~(n)

There are only finite many M<~(n,p)> ~{(n,p) ~ ~

M E £(n)

•

Christian [7], page 225, Satz 4.87.

such that

155

be compact. Then there are only finite

THEOREM 95 : Let ~ c } (n) many M £ F(n) such that

(lO12) or

(lO13) or

(1014)

M<~ > ~ ~(n)

+ @•

PROOF: For sufficiently large U* we have ~, ~(n),~(n,#)c~(n,p*). Now theorem 95 follows from theorem 94.

§

15.

SF/~ERG'S EISENSTEINSERIES

Selberg's Eisensteinseries is defined and its convergence investigated.

DEFINITION 14:

Let

(1015)

Z 6 }(n)

(lO16)

r ~

,

u o

and form "Selberg's Eisensteinseries" (1017) ~(1,r,Z,s) =

~ Det(M{~)2rf(1,YM, S+(r + ~ ) e ( w ) )

,(1)\r(n) DEFINITION 15:

Let ~ c ~

w

be the domain 1+

(lO18)

al > - - 4 - - -

11 '

156

> Iv+ I + I ~

(1019)

°'~+1 - qv

(v = 1, .... w-l) .

4

Set #=~+i~

(1020)

w

The series E(l,r,Z,s) converges absolutely in THEOREM 96: and represents a holomorphic function there. If ~ c ~ , ~ c ~(n) are compact subsets the series converges absolutely uniformly for

s~

(1o21) PROOF:

k+i~

w, z ~ .

From (975) we see (Det y)r (abs M{Z}) 2r =

(1022)

(Det y ~ r Hence it suffices to prove that the series with positive terms (1023)

~(I,0,Z,o)

=

~f(I,YM,O ~(1)~F(n)

+ ~-! e(w))

converges uniformly for

(1o24) Let ~

o ~ c ~

(1025) (1026) From (424), (1027)

,

z ~

be a domain with M<~>

n~

= ~ j

(433), (984),

(M ~ + E, M £ F(n)),

dm Z > 0 . (1024),

(1026) we deduce

n+1 [ ~ ~ ~ n+1 f(I,YM,~ + -~-- e(w)) ~ dl ~ f(I'YM'O - T e(w) )d~ Z

Z But

@

157 (1028)

YM = (y(~[))-1

In (1023) with M also MW runs over the sum (1023) is majorized by (1029)

HI =

~

~

¢(1)\F(n).

Hence up to d I

f(~,(yM)-I ' ~~ - Tn+1 e(w))d~z

"

Set (1030)

~ =

M<S~ > ,

U M E ~(l')\r(n)

We may assume that ~ lies in a fundamental Because o f ( 1 0 2 5 ) we h a v e

(lO31)

domain of

@(i).

n+l e(w) )dmz

H1 -- I

From (994) we deduce

(1032)

Det(Y -I) %) >- - cI~

(Z 6 ~;

v = 1,...,n)

Hence if H I converges in ~ it converges uniformly for q E ~ . Therefore it is left to prove that H I converges for ~ 6 ~ . From (355),(979) (1033)

we deduce

n+l [dX~dVy • d~ z = (Det Y)- 7

Let ~ be a fundamental mental domain of ~(1)

(1o34)

domain of in ~(n)

y-1 E ~ ; X = ( x ) ;

Hence by substituting

y ~ y-1

&(1) in ~(n). is given by - ~ < x

we get

_< ½

Then a funda-

(~,~=1 .... ,n)

158 H I -<

(1o35)

; . f(l,Y,~ + T n+1 e(w))dVy -

Det Y

(u=l,...,n)

Now we proceed like in the proof of theorem 61. We apply the generalized Jacobi trnasformation (379), (380), (381) with i instead of 1. Instead of (718) we get w 1 _~ _~J (1036) dVy = -~-{(Det R )~(n k k 1)dv R }[dD] .

v=l Furthermore

~

(lO37)

w

f(l,Y,o + ~

The fundamental domain

# of

(IO38)

)

R

e ~(i

~o + ~ ( k

1+L+1)

e(w)) = -~-(Det R ) ~ v=l &(1)

may be described by

(,~ = 1,...,w)

and the condition that all elements of D are between 0 Putting this into (1035) and immediately integrating over

(1039)

H1 ~

;

T[-{~et

~ C]~ ~ - ~ - ( D e t

2n-k -k

+

w

V4 ~-1

R ) "

and 1. D gives

+1 dv R } •

(v:l,...,w)

Ru)

Obviously

(1040)

~

+

2n-k -k 1+1 v~ ~-

because from (1018), (1041)

~

> {(k

= -~w+l-~

+

k

w- v

+k +~ +1 w 1- v < 0

4

(1019) we deduce + k ~-1 + I) "

Hence we may apply (726) upon (1039) and we immediately see that

159

up to a constant

H 1 coincides with w

k

w-v

+k

.

w+1- V

+1 -

(1042)

H2 =

~

-1

-~-(r

Ow+1_

~

4

dr

~)

r

< - ~ r v=l

c 1 8 -- u= 1

U

Make the substitution (728), i. e., (1043)

t

=

-iT r tt=l

(~=l,...,w) U

From (729) we deduce dr I (1044)

dr w ...

rI

dt I =

rw

dtw

tl

"''-~w

"

Furthermore k w-~l +k w,,+I - V +1

w

(1045)

Tr" r

4

w

-

~w+l-~

v=l

= ~

-g

t

v=l

with _

(1046) g~ = ~w+l-v- °w-~,

lw-v+lw+1-V 4

(v = 1,...,w-1)

11+I (1047)

gw = °1 - - - T - -

And from (1018),

(1019) we get

(1048)

g

> o

Hence W

(lO49)

"

H2 = -[7~=1

-1-g~ t dt -1 ~ c18

This converges. Theorem 96 is proved.

(v = 1,...,w)

•

,

160

Let ~ be a row of principal characters. Then the series E(1,r,Z,s) contains Selberg's zetafunction ~*(1,$,Y,s) as a partial sum. Hence ~*(1,~,Y,s) must converge in the domain (1018) (1019). But

~*(1,@,Y,s)

does not change if one substitutes

s

by

s + ae(w) (a ~ C). Hence the condition (1018) can play no rSle in the convergence of

~*(1,~,Y,s). T h ~ l e a v e s t h e c o n d i t i o n s

(1019)

which are identical with (779). Hence we got a new proof for the convergence of Selberg's zetafunction in the domain (779). An easy computatin shows (1050)

~ (1,r,M,s) = (Det M{~}) - 2 r E ( 1 , r , Z , s )

§ 16.

REPRESENTATION WITH SIEGEL'S EISENSTEINSERIES

(M 6 F(n))

.

Like in Diehl [11] Selberg's Eisensteinseries is represented with the aid of Siegel's Eisensteinseries. This gives analytic continuation to a bigger domain and a certain functional equation.

From now on we assume (1051)

11

=

...

=

lw

=

m

.

Then (1052)

n = mw .

We define Siegel's Eisensteinseries of degree (1053)

E(m,r,Z,u) =

m

by

~ (Det MIZ})-2r(Det YM )u-r M E Fm(m~F(m)

with a complex variable u.

161

THEOREM 97:

For

(1054)

Re u

the series

>

(1053) converges

function

in u.

PROOF:

This follows

m+l

absolutely

and represents

easily from theorem 96 applied with

or from Braun [5]. From (990),

(1051) we deduce w

(1055)

~ = ~([m,wT)

=

~ F m (n) . ~=1

Set w-1 (1056)

~* = ~*([m,wq)

=

~

Fmv(n)

v=1 Then (1o57)

~ =

~([m,w~)

=

w

~ w

.

Set u

(lO58)

~* = w ~* ~ .

Obviously (1059)

~ c 4*; ~ c ~ * .

THEOREM 98 :

Let

(1060)

run over

~? = (A B) 6 V(m)

Tm(m)\F(m).. \

a holomorphic

Then

.

w = 1

162

A

O

E (n*) 0 J =

(1061)

0

D

o

o

E (n

runs over a complete set of representatives of ~\~*

E(n*)

and

0 0 01

J

(1062)

=

O

E (n*)

C

0

J

runs over a complete set of representatives of

PROOF: It suffices N E ~*

to prove the first assertion. The matrices

are of type

(1063)

V

¢\~* .

A 0 B N =

'\

P'

*

*

'i

0

D

*

i

0

0

p-1 / /'

J with (1060) and

(lO64)

P E AC[m,w-1])

.

Hence O

(1065)

N = K J

with (1066)

K E ~ .

Therefore each residue class of

$\~*

contains a representative

of type (1061). o

o

Now let J1' J2 be two matrices of type (1061) and (1066). Then one easily computes

o

J2 = KJI

with

163

o

(1067)

K =

~i

I

E(n*)

0

0

0

K11

0

0

E(

with

o

(lO68)

K~I//

6 rm(m)

•

This proves the theorem THEOREM

Let

99 :

Z1

(1069)

Z2

6 ~(n)

Z = Z2 '

Z

with (1070) Hence

Z 6 ~(m) Z

.

is the right lower m×m submatrix of Z° Set

(1071)

Z = X + iY .

Then ~ ([m,w~,r,Z,s) =

(1072)

~(Det YM ~ )2r (Det Ml~})2r£([m,w-1~,(YM1)n.,(s)*-(r + ~-T~)e(w*)) M £ ~*~(n) × PROOF:

E(m,r,M,s I + ~ !

) .

From (433), (1017) we deduce

(1073) ~([m,w~,r,Z,s) = ~

(Det M{~l)2rf([m,w~,YM, S-(r + D-~r!)e(w)) •

M ~ ~\r(n)

164 From (977), (999) we get

(1074)

= Oet(M{Z})

Det((WM){Z})

Furthermore we have (1028)o With M Writing M i~stead of WMW we get

(lO75)

6 ([m,w],r,Z,s)

also

MW

runs over

~\F(n).

=

(Det M { ~ } ) 2 r f ( [ m , w ] , ¥ M l , s M E ~\r(n)

- (r + ~ - ~ ) e ( w ) )

Hence

(lO76)

~ ([m,w],r,Z,s)

=

~ (Det(JM){~})2r f([m~w],YjM,S

_ (r + ~ ) e ( w ) )

M E ~*\r(n) J E ~\~*

Here we assume that

J is of type (1063).

Now

(1077)

Det((JM){7})

= (Det Mi21)(Det J IM(2)})

,

hence

(lO78)

Det((JM)I~ ~) = (Det MI~l)(Det YM)(Det(YM)j)

Furthermore (1079)

(y~M) -1 = ((YM)j) -1 ,

(1080)

( (YJM)-l)n*

(lO81)

Det(YjM) = (Det YM)abs(J{M(2>l

From (434) we deduce

= (Y~I1 )n* ' ~

)-2

.

(Det J{M(Z)IT I.

165 (1082)

Det YM = (Det YM)(Det((YMI)n .)

)-1

.

From (1081), (1082) we get

(lO83)

Det(YjM) = (Det((YM 1)n.))-1(Det(Y~M)~)

From (427), (1080) we deduce

(lO84)

f([m,w],(YjM)-l,~-

(Det YjM )sl+(r + ~ )

(r + ~ ) e ( w ) )

=

f ( [ m , w - 1 ] , ( Y M 1 ) n . , ( s )~ * + (s 1 - ~ ) e ( w * ) ) .

The formulas (426), (1083), (1084) give us

(1085)

f([m,w],(¥jM)-l,s

~ f([m,w-lq(YM1)n.,(s)-(r+

- (r + -~)e(w))

=

~ ~)e(w*))Det((YM)

sl+r+

m+l

~)

From (1076), (1078), (1085) we get

(1086)

l (Det

E ([m,wq,r,Z,s) = ~ )2 r(Det Ml:l)2rf([m,w-1] ,(YM1)n.,(:)*-(r+ ~ ) e ( w * ) ) YM

M E ~*\F(n)

m+l

l

(Det J{M}) -2-~(Det(YM) ~ ~) s~"

-r+ --~

JE rm(m)\~(m) From (1053), (1086) we get (1072). Theorem 99 is proved. Axiom A: There exists a function properties:

K(m,r,u) with the following

x

166

a)

K(m,r,u) is meromorphic for many poles.

b)

The function

(1087)

and possesses only finitely

T(m,r,Z,u) = K(m,r,u)E(m,r,Z,u)

is holomorphic for equation (1088) c)

u ~ ¢

u 6 C

and satisfies the functional

T(m,r,Z,u) = T(m,r,Z m+l

There exist a constant and linear functions ,~(~,,Re

(1089)

u)

c19 >_ 1, a certain natural number

U) = , j * ( l , ) R e

u + 3(~,)

h

(1, = 1,...,h)

with

(lO9O)

J*(1)

=

-1,0,1

such that h (1091)

abs T(m,r,Z,u) _< c19 ~ (Det Y P ( : ' R e ~=I

THEOREM 100:

(1092)

PROOF:

For

m =

u)

(Z ~ ~ (m)) .

Axiom A is true with

o~

F(s)

(r =

~(r,s)

(r > O)

K(1,r,u) =

See § 3.

DEFINITION 16:

A domain ~ c ~ w exists a constant d with

is called a "# -domain" if there

167

(1093)

Ov+l - ~

(1094)

~2 + ~1 > d

A domain ~ 6

Cw

,

is called a "~-domain" if w

(1095) with a ~ - d o m a i n THEOREM 101: (1096) A

(~, = 1 , . . . , w - 1 ) ,

> d

#.

A # - d o m a i n is invariant under the substitution rSl " -rSl' ~,~

q'o

(v = 2, .... w) •

~ - d o m a i n is invariant under the substitution

(1097) PROOF:

s I -- -s I ; s

-o

sv

(~ = 2 , . . . , w )

•

Clear

THEOREM 102:

Let axiom A be true. The series

(1098) S(m,w,r,Z,s) = K(m,r,s I + ~ )

i([m,w],r,Z,s)

=

~ * -(r+ ~ ) e ( w * ) ) (Oet YM)Zr(Det Ml~i)2rf([m,w-1],(YM1)n.,(s) M 6 ~*\F(n) T(m,r,M,s I + ~ ) converges absolutely in a ~ - d o m a i n ~ . In invariant under the substitution (1097).

~it

is holomorphic and

PROOF: The invariance under (1097) follows from (1088). Like in theorem 96 it suffices to prove the absolute convergence in a v~ domain. By multiplying

M

with a left-hand-factor

obtain (1099)

M
.

~ ~*

we may

x

168 Applying formula (1022) we see that it suffices to show that sums of type (1100)

~ (Det YM1)rf[m,w-1~,(YM1)n.,(~)*-(r+

~-)e(w*))

h

×

M E ~*\r(n) (Det YM )$1(~'~1) converge with

(t=l,...,h)

~1(~,~ I) = ~(~,~1 ) + 2r .

Like in the proof of theorem 96 the sum converges if the integral

( 1101 )

H1 =

.[ (Det Y-1)rf([m,w-1],(Y-1)n.,(~)*-(r+

converges. But now we may assume that domain of ~*. Hence we get

(1102)

~r)~$1( ~,~1 ) d~ 2

lies in a fundamental

Det Y ~ d~ 1

with some constant

(11o3)

~-)e(w*)XDet

d I ~ 1. Then instead of (1035) we obtain

H 1 ~_ *

dw 2 -

YE# Det Yk >-- cll (v=1 ..... w) Det ~ 1

> dl I

Now apply again the generalized Jacobi transformation. (1104)

y-1 = R w-I

Hence instead of (1042) we get

Then one gets

169

H2--

(11o5)

(

-TT r H=1

H

< @r tt=l

(~#=1 . . . , w - l )

~2 ( : '~1 )

drl drw) rl

C1~

dl ~

(w-u)m m-1 2 + T - (~w+l-p

W-1

j~

rw

•

drw-1 "" rw-1

( ~ = 1,...,h)

41...rwI_I with linear functions ~2(~,nl ) of type (I089), (1090). The inner integral may be estimated by a sum of 3 terms of type m-1

J3 (*'~)

(const)

~3(*'~1 ) - T

1 (r I ... rw_1 )

with linear functions ~ 3(x,~1) of type (I089), (1090). Hence it suffices to compute the integrals ,[ r

(1106)

(w-v)m 2

+~3(~'al ) - ~w+l-~

dr I . drw-1 -rI •. rw_ 1

(x=1,...,3h).

-1 c18 ~ ' t ~ ' r (~,=1,...,w-1) ~=1 ~t Make again the substitution

(11o?)

t %J

=TFr

(v

= I,...,w-1)

tt

tt=l Then

(11o8)

(11o9)

dr I rl

"'"

w-1

~

TFr ~=I

with

drw_ 1 rw-1

dt I = -~1

dtw_ 1 "'"

tw-1

'

w-1 _ + ~3(~'~1 ) - ~w+1-~ = -~- t g~ ~)

~=I

.

170

(111o)

g

--

v = °w+l-v

(1111)

--

m

(v=l,...,w-2)

Ow-~)

(~=1,...,3h)

gw-1 = n2 - ' ~ 3 ( ~ ' ~ 1 )

.

Then (1107) becomes (1112)

This converges for (1094) and (1113)

w-1 -]-F-

co

,)=1

Cl I

~ t-l-g~ v dry .

gl .... 'gw-1 > O

which is identical with (1093),

~2 > d

with a suitable constant d. But because of q2+nl c2 = ~ +

(1114)

n2-~ I

2

the condition (1113) is a consequence of (I093), (1094) with another d. Hence we may confine to (1093), (1094). Theorem 102 is proved.

THEOREM 103: The poles of the function C(m,u) order. They lie in the points (111

)

PROOF:

=

--

(~

-

=

are of first

0,1,...,m-1)

Apply (959) •

THEOREM 104:

Let axiom A be true. The function w

(1116)

(

P(m,w,r,Z,s) = ( - ~ - K ( m , r , s ~=1

+ ~))

x

~ (C(m,s -s~,)C(m,s,)+su ))) ~(rm'w]'r'Z's) l
is holomorphic in a q ~ - d o m a i n ~ a n d

invariant under (1098).

.

171

PROOF:

For m+l K(m,r,s1+ -~-)~ ([~w],r,Z,s)

(1117)

this was proved in theorem 102. For

2 <_ ~ < w

(1118)

)

K(m,r,s

is independent of

+ ~

the function

s 1. From (1114) and (1094) we deduce

(1119)

o"%) > d

(~ : 2,...,w).

Since K has only finitely many poles we may choose that (1118) is holomorphic in ~ .

d

so large

It is easily seen that the product (1120)

Tr1
(C(m,sv-su)C(m,s~+Su))

is invariant under (1097). From theorem 103 and formulas (1093), (1119) we see that it is holomorphio in ~ . Theorem 104 is proved.

§ 17.

REPRESENTATION WITH SELBERG'S ZETAFUNCTION

Like in Diehl [ 1 1 ] Selberg's Eisensteinseries is represented with the aid of Selberg's zetafunction. This gives analytic continuation to a bigger domain and the invariance under the permutations of all variables. THEOREM I05: (1121)

It is

~([m,wT,r,Z,s)= ~(Det Ml~})2r(Det M £ Fn(n)\F(n)

PROOF:

Apply (918), (1017).

n+l ~)r+-~ A([m,w],YM,S)

•

172

We set W

Isl = I [ m , w ] , s l

(1122)

= ~

s "W

.

v=l Then from (933) we get

(1123)

~o([m,w],s)A([m,~,Y,s)=(Det Y)IS}x([m,wT,Y,s)

.

From (1121), (1123) we deduce (1124)

~o([m,w],s)~([m,w],r,Z,s) = n+1 ~(Det M{~l)2r(Det YM) Is}+r+ ~ x([m,w],YM, S) •

M E £n(n)\P(n) DEFINITION 17:

A domain # c ~ w

is called an " ~-domain",

if the

following holds: There are finitely many (say a) linear functions ~ ( x , . . . ) (~ = 1,...,a) such that

(1t25) Here

v --> ~ ( ~ ; ~ , - o ~ v ( 1 ) ~

.... ' d v - o ~ ( v - 1 )

runs over all permutations of

'~v-~vtv+l)t

' .

~O -~

. v . ~.v ( w )

1,...,~)-1,~+1,...,w;

v = 1,...,w; x = 1,...,a. A domain~c

Cw

is called a " ~ - d o m a i n "

if

with an # -domain ~ THEOREM 106:

Each # -domain is invariant under all permutaions

of ~1,...,aw. Each ~ - d o m a i n of s 1,...,s w. PROOF:

C] ear.

is invariant under all permutations

)

173

THEOREM 107: rences s -s PROOF:

depends only on the diffe-

The function X(I,Y,s) (1 < u < v <_ w).

Apply formula (947), i. e., n*

X(1,Y,s) = ¢o(1,s)(Det Y) sw-{l's~+ T

(1127)

(l,Y,s)

Because of (927), (928), (930) we see that ~o(l,s) depends only on the differences sv-su (I ~ ~ < v ~ w). The same is true for Sw-{1,s I. And it is true for ~ (l,Y,s) = * (1,Y,u). Theorem 107 is proved. THEOREM 108: (1128)

The series

Q(m,w,r,Z,s)

= ~o([m,w],s)~([m,w~,r,Z,s)

(Det M{~})2r(Det

YM )Is}+r+ ~ x ( [ m , w ~ , Y M ,

= S)

M E ~n(n)XF(n) converges absolutely in a 7 - d o m a i n s . In ~ i t invariant under all permutations of Sl,...,s w.

is holomorphic

PROOF: The invariance under all permutations of Sl,...,s w from theorem 88. Like in theorem 96 it suffices to prove the lute convergence in a y - d o m a i n . Applying formula (1022) we that it suffices to consider the case r = O. Furthermore we assume in (1128) that

YM 6 V'/f~(n) c ) " ( n , c 1 1 )

(1129)

Now we apply the estimate (1130)

abs

~(rm,w],Y,s)

follows absosee may

.

(844), i. e., ~ c15(Det Y) ~w-~a}+ 4 n*

c16(Det

and

g(1)

Y)~W-Ie}-~ ~ Yn

g(l) ~

-~(i, ~,~). ~=1

Yl ZI,~,~)

+

174

From theorem 107 we deduce, that the linear functions ~ may be chosen in such a way that they depend only on the differences -e (1 <_ ~t < ~ <_ w). Furthermore H n+ I + n* (net Y)-- T -~and y~ y~ may be both estimated by a finite sum of suitable powers of Yl and Yn" Hence it suffices to prove the convergence of finitely many sums of type (1131)

~ (Det YM )cw+ ~n+l (yM)~(~) M E £n(n)\r(n)

(8=1,n; ~=l,...,h)

,

where ~ ( ~ ) are linear functions in ~ -q (I < u < ~ < w), u provided that e lies in a suitable ~-domain ~ . Like in the proof of theorem 96 the sum (1131) converges when the integral n+1 (1132) H 1 = I(Det Y) ~w+ -~- y~(~) d~ Z (~=1,n; ~=l,...,h) converges. But now we may assume that ~ is in a fundamental domain of Fn(n). By theorem 91 we have w

(1133) But

@(y-l) > c118

.

Y E ~/~(n) and hence (1133) is equivalent with

(1134)

Yn

with some positive constant Then we get

<- dl d I. Apply (1034) and integrate over X.

(1135) H 1 <_ ~ (Det y)~W ~y~(~) dVy Y E ~(n)

yn <_ dl

(~ = 1 , n ;

~ = 1,...,h)

175 Now apply the Jacobi transformation (357), (358), (359). Then from (385) we get n ~(n-2~+I) dr dVy = "]'[-(r V )[dI)] r "

(1136)

~=1

Applying this to (1134) and immediately integrating over D we get up to a constant factor, the following two types of integrals:

(1137)

[

r~n ~)( "]~-r ~w+ ½(n-2v+l)dr

r

v= 1

v

(~

=

1,...,h)

,

(x

=

1,...,h)

•

(~

=

1,...,n-1)

r,~

'V., _< d2 (v=l,...,n-1)

rv+l

rn ~ d 2

(1138) ~ r~l (') (-mrr ~ < d2 rv+1

~w+ {(n-2,,+1)

dr--~ ) -.r

r v=1 ~' (~--1,...,n-1)

v

rn ~ d 2

From r

(1139)

~ rv+1

(1140)

r n <_ d 2

<_ d 2

we get

(1141)

o

d21-n rl _< rn --< d2 = d2 rl

Hence the integral (1137) may be majorized by integrals of type

(1138). The integral (1138) has the upper estimate

,

176

(1142)

d3 ~ + ~(n-1) + ~ ( ~ ) (; rl w 0

dr I n d3 ~(n-2~+I) dr rl )-[[- ~ rv ~w+ r v ~;=2 0 ~; (~=1,...,h)

with a certain constant

d 3 > O. These integrals converge for

(1143)

~w ~ -1 - ~ ( ~ )

(1144.)

a w >-

n

2

(~

= 1,...,h)

,

•

On the right-hand-side of (1143), (1144) stand linear functions in ~ -~It (1 --< ~ < ~ --< w). Ajoining some more inequalities which may make the domain smaller we see that we have convergence in a y-domain. Theorem 108 is proved. THEOREM 109: Let axiom A be true. The function (1117) is holomorphic in a ~ -domain and invariant under all permutations of Sl,...,s w. Because of (960) and theorem 108 it suffices to show that the functions

PROOF:

W

(1145)

-FFK(m,r,s

+ "~

)

"~=1

(1146)

-~-

C(m,s + s )

are holomorphic in a y-domain. The invariance of (1145), (1146) under all permutations of Sl,...,s w is immediately seen. Now assume (1147)

a

> d

(~; = 1 , . . . , w )

with a proper constant d. Then the holomorphy of ( 1 1 4 6 ) follows from theorem 103 and the holomorphy of (1145) follows because K has only finite many poles. Ajoining (1147) to the conditions (1125) the assertion follows.

177

§ 18.

ANALYTIC CONTINUATION

We multiply Selberg's Eisensteinseries

with certain factors.

The

product is analytically continued to the whole complex space.

It is

invariant under a certain finite group.

THEOREM 110:

Let

w ~ 2

and let ~ c ~ w

a function is holomorphic extended to

/

(~)

See H8rmander

PROOF:

DEFINITION 18:

Let

(1148)

Here

~ + i ~w

~

is the convex hull of ~ .

[18], page 41, theorem 2.5.10. A(w)

denote the group of all substitutions

%(~)

(~

runs over the permutations

of

run independently over

~ 1.

= 1,...,w)

.

Sl,...,s w. Furthermore

is exactly the group of all integral orthogonal

substitutions.

Its order is (1149)

2nn!.

THEOREM 111: Let Axiom A be true. Then the function P(m,w,r,Z,s) may be holomorphically continued to ~w. It is invariant under ~(w). PROOF:

Let ~

be a ~ -domain and ~

(1150) The domain ~ (1151)

~

n ~

- ~

> d

an

m-domain.

We show that

~ •

is given by (I093), ~+1

If

then it may be holomorphically

, where / ( ~ )

s~ ~ ~

Cl,...,~ w A(n)

in

+ i ~w

be a connected domain.

(1094),

i. e., (~ = 1 , . . . , W - 1 ) ,

178 (1152)

~2 + ;1 > d .

The domain

#

(1153)

>Z~(X;~ - ~ ( 1 ) , . . . , ~

~

is given by (1125), i. e.,

-~(v_l),~v-o~(v+l),---,q

-q~(w))

(v) with linear functions ~ . ~ runs over ~ (w*) where ~ ( w * ) denotes the symmetric group of degree w* and (v) means that it operates on 1,...,v-l,v+1,...,w. Furthermore in (1153) we have v = 1,...,w; x = 1,...,a . Now choose

(1154)

o,j+ 1- o v

(v = 1 , . . . , w - 1 )

such that (1151) holds. Then the right-hand-side of (1153) is fixed. Now let al,...,~ w simultaneously increase such that the differences (1154) remain unchanged. Then for sufficiently large ~l,...,~w the conditions (1152), (1153) are fulfilled. This proves (1150). For abbreviation set (1155) If

~ E ~ (w)

P(s) = P(m,w,r,Z,s) then

~ = ~(~)

.

hence

n

Hence the set

(1156)

¢ * =/(~- U

V

~(~))

is connected and by theorems 104, 109, 110 the function P(s) i s holomorphic in ~* + i ~ w. The domain ~(~) is described by the inequalities (1157)

d~(1) + ° 0 ( 2 ) > d ,

(1158)

a~(v+l)

- acp(v) > d

(v = 1 , . . . , w - 1 )

.

179

But since a fundamental (1159)

domain of ~(w) in

o 1 ~ 02 ~

and since

S

...

is given by

~ ow

denotes the convex hull,

conditions ~(1158) may be dropped. %

~w

it is easily seen that the

Further:more

% ~ contains a domain

described by

(1160)

0 + a

> d

(1 ~ ~ < u ~ W)

/

w i t h suitable d. In N o w let

T

~ + i~ w

is P(s)

holomorphic.

be the substitution

(1161)

01 ~ - o I , c

Because of

< (@) = ~

(v = 2 , . . . , w )

o

•

the set

(1162)

is connected and

P(s)

is holomorphic

in

~ + i ~ w . The domain

is g i v e n by (1163)

o

+ o

> d

(2 ~ ~ < u ~ w)

and (1164)

The domain (1165)

au - o I > d

<(~)

(m = 2 , . . . , w ) .

is described by ( 1 1 6 3 ) a n d

ou - o I > d

It is g e o m e t r i c a l l y cI

.

clear that the forming of the convex hull takes

away the two conditions by (1163). Here

(t, = 2,...,w)

(1164),

(1165). Hence

$

is simply d e s c r i b e d

does no longer appear hence it m a y be a r b i t r a r y

chosen in ~ . From the latter remark it is clear that

~ ~'(n)

180 Hence P(s) is holomorphic in ~ w + i ~ w = Cw. It is invariant under the group A(W) which is generated by ~ ( w ) and ~. Theorem 111 is proved. THEOREM 112:

morphic in

(1167)

Let be m = I then the following functions are holoCn and invariant under A(n): P(1,n,O,Z,s)

=

n

(TF'F(s ~=1

(1168) n

+ ~))( "TT'IF(s +su + ~)F(s~-su + ~1)}){([1'nl'°'z's) r

I

~u<

P(1,n,r,Z,s)

~ ~ n

=

(-l~- .~(r,s + ~))( -l-I- IF(s +s,+ ~)F(s -sp+ ½)})~([1,n],r,Z,s) v=l l 0). PROOF:

Apply theorems 100 and 111.

I: In axiom A the inequality (1092) has to be true. I do not know if this inequality follows from the inequalities given in Gri~enko [12], page 596, theorem 3 and Maa~ [34], page 236 formula

(32). From theorems 85, 112 we see that all Eisensteinseries (1,r,Z,s) for arbitrary 1 may be analytically continued by forming residues of Selberg's zetafunctions.

REMARK 2:

CHAPTER V.

SIEGEL'S EISENSTEINSERIES

Siege1's Eisensteinseries are defined . It is shown that they may be derived from Selberg's Eisensteinseries (Representation with Selberg's zetafunction) by computing residues of Selberg's zetafunction as it was done in § 13. Hence the analytic continuation of Selberg's Eisensteinseries gives us analytic continuation of Siegel's Eisensteinseries. From the functional equations for Selberg's Eisensteinseries one gets a functional equation for' Siegel's Eisensteinseries.

§ 19.

SIEGEL'S EISENSTEINSERIES

Analytic continuation and the functional equation for Siegel's Eisensteinseries is obtained.

Like in (1054) define Siegel's Eisensteinseries (1169)

E(n,r,Z,u)

by

T (Det MiZ})-2r(Det YM )u-r

=

L

M E rn(n)\r(n) Obviously

(i17'0)

S(n,r,M,u)

= (Det M{ZI) 2r E(n,r,Z,u)

(M E F(n))

Set

(1171) (1172)

(m) s

(u)

m+1 _ ~(w-v)

= u - T

(m) s (u)

(m) =

( Sl,...,

(m) s w)

Then from (964), (1122) we deduce

(1173)

f(~)(u)t

An easy computation shows

= u - ..~..~ .

(~

= 1,...,w)

,

182

(1174) THEOREM 113:

(1175)

Let axiom A be true and set B(m,w,r,u) :

w-1 (-~- K(m,r,u - ~ ) ) ( •~ = 0

T~ 1 <: t! <

l+m(,,+v) 2

C(m,2u "¢ <

W-1

R(m,w,r,Z,u) = B(m,w,r,u)E(n,r,Z,u)

(1176)

)),

.

Then

(m)

(1177)

R(m,w,r,Z,u) = y(Det Y)-2rp(m,w,r,Z, s (u))

with some constant v. The function R(m,w,r,Z,u) is holomorphic for u E C and it satisfies the functional equation R(m,w,r,Z,u) = R ( m , w , r ,7~ ,n+1 -~--u)

(1178) PROOF:

Let ~ be a ?-domain.

(1179)

(m) s (u) E

.

For large enough ?

Re u

we have

•

Hence theorems 87, 108 and formulas ~116),

(1171), (1172), (1173),

(1175), (1176) give (1177). From theorem 1 1 1 it follows that R(m,w,r,Z,u) is holomorphic for u E G. By theorem 111 the function P(m,w,r,Z,s) is invariant under A(W). Hence

(118o)

P(m,w,r,Z,s) = P(m,w,r,Z,s).

From (1174), (1180) we get (1178). Theorem 113 is proved.

183

THEOREM 114:

(1181)

The function R(1,n,0,Z,u)

=

n-1 (-['[-F(u - -~))( a=o

"]']- F(2u - Y-~ ))E(n,0,Z,u)

1 <_ ,~ <: U < n - 1

is holomorphic for

u ~ ¢

and satisfies the functional

equation

(1178).

PROOF:

Apply theorem 100 and 113.

From (1181) one sees where

E(n,O,Z,u)

can have poles. More about

this may be found for arbitrary

n

[19] and for

n = 2

[20] and Maa~ [34].

THEOREM 115:

Let

(1182)

in Kaufhold

in Christian [8] and Kalinin

r > O. The function

R(1,n,r,Z,u)

=

n-1 (-IT

~(r,u

- ~))(

u=o

T[-

F(2u

- ~J~))E(n,

r,Z,u)

1 < u < v < n-1

is holomorphic for

u E C

and satisfies the functional

equation

(1178). PROOF:

§ 2o.

Apply theorems 100 and 113.

POLES AND HECKE'S SUMMATION

It is shown that for certain small weights the analytic continuation of the Eisensteinseries has Hecke summation.

is holomorphic.

From (151) we get (1183)

~(r,s) = ~-Sr(s+r)C(2s)

Hence for these weights one

184

where

~

is Riemann's zetafunction.

(1184)

~(s)

= ~(O,s)

Then the functional

=

Set

~-Sr(s)~(2s)

.

equation of the F - f u n c t i o n

gives us

r-1

(1185)

~(r,s)

= (~

(S+v))g(s)

%)=0

Set n-1

(1186)

A(n,r,u)

= (-FT- ~(r,u - ~2))(

"T]-

F(2u - 2 ~ ) )

1 < U < v < n-1

u=o Then by t h e o r e m 115

(1187)

R(1,n,r,Z,u)

is h o l o m o r p h i c

for

= A(n,r,u)E(n,r,Z,u)

u 6 C. From (1185),

(1186) we deduce

n-1 r-1

(1188)

A(n,r,u)

= (-iT

Tr"

tl=O

%)=0

(u - ~ + v ) ) x

n-1 (-IT

-

-IT

U=I

F(2u

T H E O R E M 116:

The function I

m

(1189)

-

1 <_ U <-- %) --< n-1

G(s)

= s(½

-

s)g(s)

_

S

= -2 - F ( s )

1 - s is h o l o m o r p h i c PROOF:

s E C

See Landau [22],

T H E O R E M 117:

(119o)

for

and possesses no real zeros. § 128.

It is A(n,r,r)

n-1 [~]) 4 0 (r = 1,2,[--Z- ],

185 For u=

n ~ 3

the function

A(n,l,u)

(1191)

[n-~2 ]

3 ~ r ~

the function

A(n,r,u)

f (1192)

=

S(r)

has a zero of order r - 2

(3 <_ r < ~ ) -

PROOF:

Express

g (s)

Because of theorem linear factors.

(1193)

and

F(s)

by G(s) and linear factors. 116 all real zeros and poles are given by the

For another proof see Christian

E(n,r,Z,r)

~ ~o

(r =

E(n,l,Z,1)

= 0

(n >_ 3)

Apply theorems

THEOREM 119: (1195) PROOF:

(1196)

Let

115,

1,2,[-~],[~-])

[10].

has at

,

u = r

at the most a

117.

n < 8. Then

E(n,r,2,r) Apply theorem

k E ~

[8],

counting of

It is

For (1191) the function E(n,r,Z,u) pole of order S(r). PROOF:

_
r

Now the result follows by an elementary

poles and zeros. THEOREM 118:

w

) -

Let

has a pole of first order at

I. For

~

(r = 1,293,¢)

118.

such that E(n,r,2,u)

= (u-r)-kD(n,r,Z,u)

.

186 and

(1197)

D(n,r,Z,r) ~ oo .

Then

(1198)

D(n,r,Z,r) =

lim {(u-r)kE(n,r,Z,u)} u ~ r

From (1170), (1198) we obtain

(1199)

D(n,r,M,r) = (Det M{Z})2rD(n,r,Z,r)

(M ~ r(n)).

If one can prove that D(n,r,Z,r) is holomorphic in Z then for n > 2 this functions must be a modular form of weight 2r. m

It was proved by Wei~auer [47 ! that (1200)

-

n+l

E(,,,-~- , Z

,~)

(n

1 mod 2)

is holomorphic. Theorem 118 may be also expressed that way, that the Eisensteinseries (1201)

~

(Det M{Z}) -2r

M E Fn(n)\r(n) possesses "Hecke s1~mmation" for

(1202)

rn-ll rn+1~ r = 1,2,L-~-j,t--~-- j

For the explanation of "Hecke s~mmation" see Hecke [16], [ 1 7 ] . It is an important open question wether it is possible to take the poles away in the interval (1191). Of course for the function K(m,r,u) in axiom A one may take A(m,r,u) defined by (1186). Assuming that then (1091) holds one can make the whole computation with this K(m,r,u). This leads again exactly to theorem 118. But

if it is possible to find for a certain m a "better" K(m,r,u) than our A(m,r,u) then it may follow from theorem 113, that theorem 118 can be sharpend for all multiples [12]

finds an

K(2,r,u),

n = wm. For

n = 2

but this is identical with our

Gri~enko A(2,r,u)

and cannot be used for sharpening theorem 118. If it should be possible to find a

K(m,r,u)

which helps to sharpen

theorem 118 this will be only for the multiples of

m. Heace the

question arises if it is neccessary to assume (1051). with an arbitrary

1

each

such that the assertion of axiom A holds for

v = 1,...,w

we may assume that there is a

If we start K(lv,r,u)

for

each v. I did not check what comes out if one gives up (1051). But this case becomes more difficult in so far as the functions ~(1,Y,s), vv(l,s ) (v = 1,2~ (see (889), but only meromorphic.

(890)) are not holomorphic

Let

mod q whith

q E ~

and

X

a Dirichletcharacter

(1203)

x(-1)

Then Gri~enko

=

(-I) r

[12] considers for

•

n = 2

the Eisensteinseries s

(1204)

~ x(Det D)(Det(CZ + D))r(Det YM )~ • "'~ c ~ mod q

He proves analytic continuation and it should be remarked (Gri~enko does not mention this explicitely) at s = 0 summation.

for

that the function is holomorphic

r = 1,2, i. e., this Eisensteinseries

has Hecke

LITERATURE ANDRI~NOV, A. N.: On zeta functions of systems of quadrat$c forms. Math. Notes Acad. Sci USSR 20, 571-674 (1974).

[2]

ANDRIANOV, A. N.: Symmetric squares of zeta functions of Siegel modular forms of genus 2. Proc. Steklov Inst. Math. 142, 21-45 (1979).

188

[3]

ANDRIANOV, A. N. and MALOLETKIN, G. H.: Behaviour of theta series of degree N under modular substitutions. Math. of the USSR Izvestija 9, 227-242 (1975). ARAKAWA, T.: Dirichlet series corresponding to Siegel's modular forms. Math. Ann. 238, 157-173 (1978).

[5]

BRAUN, H.: Konvergenz verallgemeinerter Eisensteinscher Reihen. Math. Z. 44, 387-397 (1939).

[6]

CHRISTIAN, U.: Uber Hilbert-Siegelsche Modulformen und Poincar&sche Reihen. Math. Ann. 148, 257-307 (1962).

[7]

CHRISTIAN, U°: Siegelsche Modulfunktionen. 2. Edition, GSttingen 1980/81 (Lecture Notes).

[8]

CHRISTIAN, U.: Bemerkungen zu einer Arbeit von B. Diehl. Abh. Math. Sem. Univ. Hamburg 52, 160-169,(1982).

[9]

CHRISTIAN, U.: Eisenstein series for congruence subgroups of GL(n, ~). Amer. J. Math.

[lo]

CHRISTIAN, U.: On the analytic continuation of Eisenstein series for Siegel's modular group of degree n. Monatshefte f. Math.

[11]

DIEHL, B.: Die analytische Fortsetzung der Eisensteinreihe zur Siegelschen Modulgruppe. J. reine angew. Math. 317, 40-73 (1980).

[12]

GRICENKO, V. A.: Analytic continuation of symmetric squares. Math. of the USSR Sbornik 35, 593-614 (1979).

[133

GUNDLACH, K.-B.: Dirichletsche Reihen zur Hilbert'schen Modulgruppe. Math. Ann. 135, 294-314 (1958).

[141

HARISH-CHANDRA: Automorphic forms on semisimple Lie groups. Springer Lecture Notes in Mathematics 62 (1968).

[151

HASSE, H.: Vorlesungen Gber Zahlentheorie. Springer Verlag, Berlin, GSttingen, Heidelberg 1950.

[161

HECKE, E.: Analytische Funktionen und algebraische Zahlen. Zweiter Tell. Ges. Abh. 381-404 and Abh. Math. Sem. Univ. Hamburg 3, 213-236 (1924).

[17]

HECKE, E.: Theorie der Eisensteinschen Reihen hSherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Ges. Abh. 461-486 and Abh. Math. Sem. Univ. Hamburg 5, 199-224 (1927).

[181

H~RMANDER, L.: An introduction to complex analysis in several variables. D. van Nostrand Company. Princeton, N.J., Toronto, London.

189

[19]

KALININ, V. L.: Eisenstein series on the symplectic group. Math. of the USSR Sbornik 32, 449-476 (1977).

[2O]

KAUFHOLD, G.: Dirichlet'sche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades. Math. Ann. 137, 454-476 (1959).

[21]

KUBOTA, T.: Elementary theory of Eisenstein series. Kodansha, Tokyo and John Wiley, New York, London, Sydney, Toronto 1973.

[22]

LANDAU, E.: Handbuch der Lehre v o n d e r Verteilung der Primzahlen. Chelsea Publ. Comp., New York 1974.

[23]

LANGLANDS, R. P.: Eisenstein series, Proc. Sympos. math. 9, 235-252 (1966).

[24]

LANGLANDS, R. P.: On the functional equations satisfied by Eisenstein series. Springer Lecture Notes in Mathematics 544.

[25]

MAASS, H.: Automorphe Funktionen und indefinite quadratische Formen. Sitzungsber. Heidelberger Akad. Wiss. 1949, 1-42.

[26]

MAASS, H.: Ober eine neue Art yon nichtanalytischen automorphen Funktionen und die Bestimmung Dirichlet'scher Reihen durch Funktionalgleichungen. Math. Ann. 121, 141-183 (1949).

[27]

MAASS, H.: Modulformen zweiten Grades und Dirichletreihen. Math. Ann. 122, 90-108 (1950).

[28]

MAASS, H.: Die Differentialgleichungen in der Theorie der elliptischen Modulfunktionen. Math. Ann. 125, 235-263 (1953).

[29]

MAASS, H.: Die Differentialgleiehungen in der Theorie der Siegelschen Modulfunktionen. Math. Ann. 126, 44-68 (1953).

[3o]

MAASS, H.: Lectures on Siegel's modular functions. 1954 - 55.

[31]

MAASS, H.: Lectures on modular functions of one complex variable. Bombay 1964.

[32]

MAASS, H.: Some remarks on Selberg's zeta functions. Proc. intern, conf. on several complex variables. U. of Maryland, College Park, Md. 1970, 122-131.

[33]

MAASS, H.: Siegel's modular forms and Dirichlet series. Springer Lecture Notes in Mathematics 216.

in pure

Bombay

190

[34]

MAASS, H.: Dirichlet'sche Reihen und Modulformen zweiten Grades. Acta Arith. 24, 225-238 (1973).

E353

NEUNHOFFER, H.: Uber die analytische Fortsetzung von Poincar@reihen. Sitzungsberichte Heidelberger Akad. Wiss. 1973, 33-62.

[36]

RAGHAVAN, S.: On Eisenstein series of degree 3. J. Indian math. Soc. 39, 103-120 (1975).

[373

ROELCKE, W.: Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene I, and II. Math. Ann. 167, 292-337 (1966) and 168, 261-324 (1967).

[383

SATO, F.: Zeta functions in several variables assosiated with prehomogeneous vector spaces. Proc. Japan Acad. 57, part I: 74-79, part II: 126-127, part III: 191-193 (1981) and . , I: TShoku Math. J. 34, 437-483 (1962) II: TShoku Math. J. 35, 77-99 (1983) III: Annals of Math. 116, 117-212 (1982)-

[39]

SELBERG, A.: Harmonic analysis 2. part. Lecture Notes G~ttingen 1954.

[40]

SELBERG, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47-87 (1956).

[413

SELBERG, A.: A new type of zeta functions connected with quadratic forms. Report of the Institute in the theory of numbers, Boulder, Colorado 1959, 207-210.

[42]

SELBERG, A.: Discontinuous groups and harmonic analysis. Proc. Intern. Congr. Math. 1967, 177-189.

[43]

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[44]

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[45]

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[4-6]

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[47]

WEISSAUER, R. : Eine Verallgemeinerung eines Satzes von Raghavan. To appear.

LIST OF S Y M B O L S

Capital latin letters

Lower case latin letters v

d(n) 2

1 4, 36

e

1

58

U.~

s

36

e(w) 62

~ 37

f(l,Y,s) 14, 62

~

f(1,Y,u,a)

1 38

s 61 61

s* 61

1 A

39

1 A ~ H 39 A~ 89

h

56

66

37 A(n,r,u) 184

V

v

1

s

61

s 144

38

B~y 90 B(m,w,r,u) 182

ha(l) 56

1 37

h~y 90

1 38

h~y(~) 90

£

h 91 ^~

i* 38

h (~) 91

1 74

s

145

(m) s 147 (m) s(u) 181

38

(m) s (u) 181

C

91 Y C(m,z) 147

O(q,t) 5 D(q,T) 79

o*(t) 5 O*(k) 60

vY

~

h 91 vV h (~) 91 V h 91 Y h * (~) 91 Y j(Y) 7, 84

1 144

p 63

j*(~) !67

p~ 62

F(w) 18

k 36

q 3, 36

F*(q,x,~ ) 18

k

q~ 29

~(i) 5~

36

145

66

v 103

n*

w

r 26 r(1)

D*(k,Y)60

v n 56

38

37

k 38

u

n 36

q* 84

v

k

9

37

37 v

1

u 102

36

w* 38

D(n,r, Z,u) 185 E(n,r,Z,u) 160, 181

F

56

V(n,Y,s) 119 51

s 61

o(x) 3 G(x,a) 3

38

~ 14, 61

G(m, X) 69

k

38

Y

G(m,x, C) 69

k

58

A

61

192

I 9, 149

S(m,w,r,Z,s) 167

I(m,p) 19

T(m,r,Z,u) 166

J 162

15, 48

Lower

4r 167

aM(z) 15o

W 14, 45

K

W(n) 45

4# 172

K(1) 52

W 153

V~ 172

K(m,r,u) 165

X 163

/(~)

59

XM 150

L 59

14,58 V

L

59

Y 4, 58

letters

Zr 156

U#

1

Gothic

89

J(l,X,s) 64

89

case

167

177

179

~ * 178

. (Y)

152

A

YM 15o

L 59 L(X,m ) 12, 13

163

L(l,s) 139

Z 163

~(1,u) 14o

L(1,¢,s)

179

~ ( 1 ) 43 38,61,99,137 ~ * 38, 61

113

A

L(I,x,u) 110

Capital Gothic letters

M n 59

~1 (l, ~,,~) ¢o

Pk 60

~2(z, ~,,~) 40 ~ 1 ( 1 , :,~) 40

P(1) 23, 50 P(s) 178 P(m,w,r,Z,s)

170

Q(1) 4, 50

~2(1,~,~) 40 £ 1 ( 1 , ~ , ~ ) 4o

R(x) 8t

£ 2 ( 1 , ,,,~) 40 lo5 (n,~) 154

R(m,w,r,u) 182

~(1) 8

S 122

~c~(1) 56 ~ I (1) 56

Q(m,w,r,Z,s) 175

S(r) 185

193 ~(n) 154

1o3

~(1,~,Y,s) 128

157

(n) 2

~ (q,Z,x,Y,s) 19

~ I(i,

~,~)

40

# 2(1, ~,~) 40

#>(z, ~,~)

~(n) 8,149

~(q,1,~,Y,~) 17

L.c. Greek letters

40

xI(q,I,~,Y,s ) 21 X~I(1,~,Y,s) 128 *( 1,r,z,s) 26

# > ( ~ , ~,~) 40

,F(s)

#*(l, :,~) 41

yl(Z,¢,s) 137

~(y,s) 18,116

#1(z,~,,) ¢o

¥2(l,¢,s) 138

142

# 2 ( l , ~,~) 40

~ (l,s) 139

~(l,s) 139

6(q,~) 17

~(l,~,s)

,,)

40

81

~>(i, ~,~) 40

6(1,s) 117

~

162

51(l,s) 127 62(1,s ) 128

161

C(I,Y,s) 139

(~,~,~) ¢I

40

117 117

~ (Y,Y*) 60 ~(1,~,u) iOl

61

{(l,~,Y,s) 113

13, 61 X) *

~ 2(I, ~,~) 40

~(q,l,x,Y,s) 13

~>-(z, ~,~) ~>(z,~,~)

40

~*(1,Y, s) 139

40

~*(1,~,Y,s)

113

"rl(1 ,•) 74 "r2(1 ,X) 74 mj 152

A * ( I , $,*) 41 ~(~,c~) 21,26,166

~*(q,l,x,Y, s) 13

m(l,x,~) 100

~(I,Y,u) 140

x3

Z(1, ~,c) 129

.C(I,x,Y,u) 103

Xv 29, 98 A

(n) 55

X 86

~7~l(n) 56

#*(1,Y,u) 140

~(1) 93

~*(1,~,Y,u) 103

~(1) 93

~*(q,l,x,Y,m) 12

~

~ (n,u) 55

a~(1,~)

Xu 75

(C,B) 67

~(z)

lO¢

~(I) 103

~(u)

lOO

7

X(l,Y,s) 139

Xv 86 74

74

{vw 29, 98

X(~) 29 98

194 Other symbols 9v 32, 99 99

A(z,~,Y,s) t14

' I

A(q,l,x,Y,s) 16

I < )2,

118

$~ 99

39

~(!,s) 143

M 8, 150

~v 99

~(l,~,s) 127

M{Z} 25, 150

99

~(1,,,s) 127

K[L] z

¢ 99 V

A

[kl,-..,k ] ]

~v 99

161

$* 99

}* 161

{l,sl 62

161

Is} 172

m(~) 37

~(~)

~* 161

37

¢(i) 152 u

~(z) 15 3

r(n) 8, 152

~(Z,s) 143

ri(I) 25

~o(l,s) 143

A 11

~(l,~,s)

A(w) 177

¢o(l,~,s) 128

128

v

~(1,¢,s) 128 v

~o(l,¢,s) 127

~2(l) 39

~(1)

39

S(q,l,x,Y,t) 6

~2(1)

39

I ~(z)

48

e(q,i,x,Y,t) 4

Q(n) 11, 39

%*(r,z,t) 9 79

G v

e (1,x,Y,T) 74 A(1,Y,s) 139

Det

I I

exp 2 Re 12 Rk

I

Tr

I

Sp(n, [~:{) 8, 149 Sp(n, 7z) 8, 152 ds 58

6(l,r,Z,s) 25,155

(1,X Y,T)

I

Dg

Capital Greek letters

At(l) 39

abs

O(n,q) 39

dvy 54 d~ Z 150 [dY] 54

-~~Y 58

INDEX adjoint operator 59 ~-domain 166 chain, descending 89 character 3 character, primitive 3 characterrow 98 characterrow, primitive 98 complex matrix 2 convex hull 177 ~-domain 167 descending chain 89 differential 58 d±fferential operator 59 Dirichlet's L-series 13 distance 60 domain, fundamental 8,56,154 ~domain 172 eigenfunction 62 Eisensteinseries, elementary 24 Eisensteinseries, Selberg's 155 Eisensteinseries, Siegel's 160 elementary Eisensteinseries 24 elementary set 154 Epstein's L-series 12 Epstein's zetafunction 12 equations, functional 137 equivalent 89 functional equations 137 function, theta 74 fundamental domain 8,56,154 Gaussian sum 3, 69 ~-domain 172 group, modular 8 group, symplectic 8, 149 group, unimodular 11

halfplane, Siegel's upper 149 Hecke summation 186 hull, convex 177 integral matrix 2 invariant operator 58 isotropic vector 2 Jacobi's transformation 55 L-series, Dirichlet's 13 L-series, Epstein's 12 L-series, Selberg's 103 matrix, complex 2 matrix, integral 2 matrix, positive 2 matrix, primitive 39 matrix, rational 2 matrix, real 2 matrix, semipositive 2 Minkowski's pyramid 55 modular group 8 operator, adjoint 59 operator, differential 59 operator, invariant 58 positive matrix 2 primitive chmracter 3 primitive characterrow 98 primitive matrix 39 pyramid, Minkowski's 55 rational matrix 2 real matrix 2 reduced 93 residue 139 Selbergts Eisensteinseries 155 Selberg's L-series 103 Selberg's zetafunction 103 self-adjoint 60 semipositive matrix 2

196

set, elementary 154 Siegeldomain 55 Siegel's Eisensteinseries 160 Siegel's upper halfplane 149 space, weakly symmetric Riemannian 58 special 93 sum, Gaussian 69 summation, Hecke 186 symplectic group 8, 149 thetafunction 4, 74 transformation, Jacobi's 55 unimodular group 11 volume element 54, 150 weakly symmetric Riemannian space 58 zetafunction, Epstein's 12 zetafunction, Selberg's 103