Hilbert Modular Forms

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Eberhard Freitag

Hilbert Modular Forms

Springer-VerlagBerlin Heidelberg New York London Paris Tokyo Hong Kong

Eberhard Freitag MathematischesInstitut Universitit Heidelberg Im Neuenheimer Feld 288 D-6900 Heidelberg Fed. Rep. of Germany

Mathematics Subject Classification (1980): IO-XX, 32-XX

ISBN 3-540-50586-5 Springer-Verlag Berlin Heidelberg NewYork ISBN o-387-50586-5 Springer-Verlag NewYork Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. Freitag, E. (Eberhard) Hilbert modular forms / Eberhard Freitag. p. cm. Includes bibliographical references. ISBN O-387-50586-5 1. Hilbert modular surfaces. I. Title. QA573.F73 1990 516.3,52--dcZO 89-26258

CIP

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication ofthispublication orparts thereofis only permittedunder the provisions oftheGermanCopyrightLawofSeptember9,1965,initsversionofJune24,1985,andacopyright fee must always be paid. Violations fall under the prosecution act ofthe German Copyright Law. 0 Springer-Verlag Berlin Heidelberg 1990 Printed in the United States of America 2141/3140-5 4 3 2 10 - Printed on acid-free

paper

To Fred and Ursel Dieterle

Contents

Introduction

1

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

Chapter I. Hilbert Modular Forms. ................ 51 32 53 34 55 56

Discrete Subgroups of SL(2, R) ................. ................ Discrete Subgroups of SL(2, R)n .................. The Hilbert Modular Group. Automorphic Forms ....................... ............ Construction of Hilbert Modular Forms. The Finiteness of Dimension of a Space of Automorphic

Chapter II. $1 $2 33 $4 55

5

Forms

73

Dimension Formulae ..................

73 81 89 112 122

................... The Selberg Trace Formula The Dimension Formula in the Cocompact Case ........ ..... The Contribution of the Cusps to the Trace Formula. ................ An Algebraic Geometric Method Numerical Examples in Special Cases ..............

Chapter III.

5 20 32 43 55 66

The Cohomology of the Hilbert Modular Group ...

5 1 The Hodge Numbers of a Discrete Subgroup r c SL(2, R)” in the Cocompact Case ..................... ...... f 2 The Cohomology Group of the Stabilizer of a Cusp 5 3 Eisenstein Cohomology ..................... ........... 5 4 Analytic Continuation of Eisenstein Series ................. 5 5 Square Integrable Cohomology ......... 5 6 The Cohomology of Hilbert’s Modular Groups 5 7 The Hodge Numbers of Hilbert Modular Varieties Ibv \ * C. Ziegler) ~, ”

I33 133 142 148 158 174 182 185

Contents

VIII

Appendices

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

203

....................... I. Algebraic Numbers ........................... II. Integration ................. llI. Alternating Differential Forms

203 214 221

Bibliography

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

245

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

249

Index

Introduction

The Hilbert

modular

group l?K = SL(2,o)

is the group of all 2 x 2 matrices of determinant 1 with coefficients in the ring o of integers of a totally real number field K > &. This group and the corresponding spaces and functions - the Hilbert modular varieties and Hilbert modular forms - have been subject of many investigations starting with the Blumenthal papers [6]. In this book we seek to develop the theory to the extent necessary for us to understand the Eilenberg - Mac Lane cohomology groups HVK,

C)

( rK

ads trivially

These cohomology groups are isomorphic of the Hilbert modular variety

On 63).

to the singular

cohomology

group

XK = H”/l-‘,. Here H” denotes the product of n upper half-planes equipped with the natural action of rK. This action being properly discontinuous, we have H’( rK, c ) =

&(H”/rK, (singular

c ) cohomology)

Since the Hilbert modular variety carries a natural structure as a quasiprojective variety, the cohomology groups inherit a Hodge structure, which will also be determined in the course of the book. From the point of view of the cohomology theory of arbitrary arithmetic groups, the Hilbert modular group is nothing but a simplified example. It is, however, the only special case in which the cohomology can be determined explicitly; this even includes the computation of the Hodge-numbers. In contrast to the very deep and involved methods of the general theory, the case of the Hilbert modular group can be treated in an absolutely elementary manner. For these reasons the study of the Hilbert modular group is strongly justified although it should merely be considered an introduction to more

Introduction

2

general theories. Everything necessary to determine the cohomology developed in this book. The principal topics discussed in this book are

will

be

1) The reduction theory (compactification of H”/I’, by h “cusps”, h = class number of K). 2) The elementary theory of (holomorphic) Hilbert modular forms. 3) The evaluation of the Selberg trace formula to determine the dimensions of spaces of Hilbert modular forms of weight r > 2. (This has been done in a very important paper by Shimizu [57], whose lines we will follow closely.) 4) We use an algebraic geometric method to come down to the border case r = 2 in the dimension formula. (This case has to be treated if one is not only interested in Betti but also in Hodge numbers.) 5) We need the definition of an Eisenstein series in the border case r = 2 where convergence is not absolute. We will achieve this in the usual way, namely by Hecke summation and analytic continuation of Eisenstein series. Applying the methods of Hecke and Kloosterman the analytic continuation will be obtained in an elementary way. The idea is really quite simple: Compute the Fourier coefficients and continue them! With this preparatory work finished, the determination of the Hilbert modular group will then be based on two papers: 6) Matsushima and Shimura [49] determined H*(l?, C) in the case of an irreducible discrete subgroup r c SL(2, R)” with compact quotient H”/I’ instead of the Hilbert modular group. 7) It was Harder [26] who transferred the theory of Matsushima and Shimura to the case of the Hilbert modular group and its congruence subgroups. He showed that the cohomology splits into two parts: a) The square integrable cohomology, which can be treated like the cocompact case. b) The Eisenstein cohomology, which is due to the cusps. It is a part of the cohomology that maps injectively if one restricts the cohomology to the boundary. We will also determine the mixed Hodge structure in the sense of Deligne [9]. Th is was the subject of Mr. C. Ziegler’s Diplomarbeit and has been revised by him to be included as the last paragraph of this book. Altogether the book is somewhere in between a graduate text and a research report. It can be used as an introduction to the theory of Hilbert modular forms, the Selberg trace formula, etc. There is in fact only little intersection with van de Geer’s book on Hilbert modular surfaces and as both books have a different line of approach they fit together well. Several parts of the book cm also be used for seminars. Therefore I have included some appendices in which the basic facts about algebraic numbers, integra-

Introduction

3

tion, alternating differential forms and Hodge theory are described, mostly without proofs. Finally, I would like to express my gratitude towards Mr. Holzwarth and Mr. von Schwerin who produced the ‘l&X-manuscript and especially to Mr. Ballweg who corrected many mistakes in the original manuscript.

Chapter

I.

Ql Discrete

Hilbert

Subgroups

Modular

Forms

of SL(2, R)

A discrete subgroup f’ c SL(2, R) acts discontinuously on the upper half-plane H. The parabolic elements of I? give rise to a natural extension of H/l? by the so-called cusp classes. We are mainly interested in the case where this extension is compact. Our basic example is l? = SL(2, Z). The method of construction is such that it can easily be generalized to the case of several variables, i.e. discrete subgroups of SL(2, R)” acting on the product of n upper half-planes. This will be done in the next section ($2).

It is well known that any biholomorphic mapping of the upper half-plane

is given by z H Mz := $$, where M = (z i) is a matrix with real coefficients and determinant 1. The set of all these matrices is the group SL(2, R). The matrix M is uniquely determined up to its sign Mz=Nz

forall

ZEH

M

We shall frequently make use of the formulae

W’W = (MN)(z),

The mapping SL(2,R)xH-+H (iv, z) H Mz

M=fN.

Chapter

6

is continuous. Here SL(2,R) Euclidean metric of R4.

carries

the natural

I.

Hilbert

topology

Modular

induced

Forms

by the

Description of H as a Coset Space. The point i E H is a fixed point of the M E SL(2, R) I‘f an d only if a = d and b = -c or equivalently

transformation

M’M=E=

;

;

(

of i is the special orthogonal group

So the stabilizer

SO(2,R) = {M E SL(2,R) which

is obviously

and surjective.

Mi = Ni e we obtain

a bijective

1M’M

= E}

a compact subgroup of SL(2, R). The mapping

SL(2, R) --t is continuous

. >

mapping

H,

MHMi

Since

M - SO(2, R) = IV. SO(2;R) from

the coset space to the upper

SL(2, R)/SO(2, R)-t M-S0(2,R)-

half-plane:

H Mi.

If we provide the coset space with the quotient topology (a set in the coset space is open iff its inverse image under the natural projection SL(2, R) + is continuous. But we SL(2, R)/S0(2, R) is open in SL(2, R)), th’ is mapping can show even more:

1.1 Remark. The mapping SL(2, R)/S0(2,

R) + H

Me SO(2, R) H Mi is topological.

PTOO~.A bijective mapping is topological we have to show that the mapping SL(2, R) +

H,

iff it is continuous

and open.

So

M++Mi

is open. It is sufficient to show that the image of a neighbourhood U of the unit matrix 23 is a neighbourhood of i. This is easy to be seen (it is sufficient z) E V). 0 to look at the upper triangular matrices (i

$1

Discrete

Subgroups

of SL(2,

7

R)

The description 1.1 of the upper half-plane with the action of SL(2, R) (the group SL(2, by multiplication from the left). An important

1.11 Corollary.

The mapping p : SL(2, R) t

is

as a coset space is compatible R) acts on SL(2, R)/S0(2, R) application of 1.1 is

H,

M H Mi

proper, i.e. the inverse image of a compact set is compact.

Proof. Let K c H be a compact compact subset

subset.

it c SL(2, R),

We first

p(g)

prove

the existence

of a

> K .

For this we choose a compact neighbourhood U(X) for every point x E of p(z). We need only SL(2, R). The image p(U(x)) is a neighbourhood finitely many of those neighbourho_ods to cover K. The union of the corresponding neighbourhoods U(x) is K. We obviously have

p-‘(K)

c i7. SO(2, R) .

So p-‘(K) is compact (because which is the image of the compact mapping (multiplication)).

it is a closed subset of .k! . SO(2, R), set k x SO(2, R) under a continuous 0

1.2 Proposition. A subgroup r c SL(2,R) discontinuously on H.

is

discrete if and only if it acts

Let us recall: a) A subset I’ c SL(2,R) is discrete if the intersection of I’ with any compact subset K c SL(2, R) is a finite set. b) A subgroup I? c SL(2, R) acts discontinuously if for any two compact subsets K1, K2 c H the set

is finite. It is sufficient to consider may replace K1, K2 by K1 U K2.

the case K = Kl = K2 because

Proof. 1) Assume bourhood H under

we

that I’ acts discontinuously. We choose a compact neighU of the unit matrix E in SL(2, R). We denote by V its image in the projection p. We obviously have

MEU=kM(v)nv#P). By assumption

there

exist

only

finitely

many

M.

Chapter I. Hilbert

8

Modular

Forms

2) Assume that I’ is discrete. Let K c H be a compact subset. Its inverse image k = p-l(K) in SL(2, R) is compact (1.11). We obviously have M(K)nK#0+M&.k-? The latter set is compact because it is the image of g map ((2, Y) ++ x9-l ).

x

k under a continuous El

Points. We want to investigate the conditions under which a matrix M E SL(2,R) h as a fixed point in the upper half-plane. The solution of the fixed point equation az + b -=z cz + d Fixed

gives us .Z=

a-d+&+d)2-2c

ifcZO

From this simple calculation we see immediately that a transformation M different from the identity (M # fE) has a fixed point in H if and only if Ia + dl < 2, and in this case M has a single fixed point in H. In general, a matrix M E SL(2, C) with lo(

< 2

(o(M)

= a + d)

is called elliptic. We summarize: A matrix M E SL(2, R), M # fE, has a fixed point in the upper half-plane if and only if it is elliptic. In this case it has a single fixed point in H. 1.3 Remark.

A point a E H is called 5111elliptic if the stabilizer I’,={MeI’,

fixed point

of a subgroup I’

c

SL(2, R)

Mu=u}

contains an element different from the identity (M # fE). 1.4 Remark. The set of elliptic fixed points of a discrete subgroup r c SL(2,R) is a discrete subset of H. Each point a E H has a compact neighbourhood U C H. There are only finitely many M E r with the property M(U) n U # 0, and so we have only finitely many elliptic fixed points in U. I3 Proof.

9

$1 Discrete Subgroups of X(2, R) 1.5 Remark. We assume that M is contained in a discrete SL(2, R). Then the following three conditions are equivalent: a) M is elliptic OT M = fE. b) M is of finite order, i.e. Mh = E GOT some h E N. c) M has a fizzed point in H. PTOOf.

We already know a&)

and so it is sticient

subgroup

r

of

to show c)+b)+a).

c)+b): The stabilizer ra of the fixed point a of M is a finite subgroup of the (discontinuous) group I? and therefore each element of rc is of finite order. b)=+a): Each matrix M E SL(2, C) of finite order is diagonalizable, i.e. there exists a matrix A E SL(2, C) with the property

This follows, for example, from the theory of the Jordan canonical form. The number C is necessarily a root of unity. If C # fl we obviously have la(M)1 = IC + [-‘I

< 2.

q

Transformation of a Fixed Point into the Zero Point. The upper half-plane H is biholomorphically equivalent to the unit disc E = {w : [WI < 1). The biholomorphic mapping a:H+E 2 H

(2

-

U)(Z

-

Ti)-l

transforms a given point a E H into zero. If 7 : H + mapping with fixed point a, then 70 = a-w -’ : E -

H is any biholomorphic

E

is a biholomorphic mapping with fixed point 0. From the Schwartz lemma we know that each such ^/o is of the form 7s~ = C.zwhere C is a complex number of absolute value 1. If 70 is of finite order, C is a root of unity. We know that each finite group & of roots of unity is cyclic (because 2 = {u E R I ezriO E E} is a discrete, hence cyclic subgroup of R). The image of the 1.6 Remark. Let r c SL(2,R) b e a discrete subgroup. stabilizer ra of any point a E H in the group SL(2,R)/{fE} is a finite cyclic gTOUp.

10

Chapter

I.

Hilbert

Modular

Forms

The Quotient Space H/l?. Two points z, w E H are called equivalent with respect to our discrete subgroup r c SL(2, R) if there exists a M E l? with Mz = w. If we identify equivalent points we obtain the quotient space H/I’ with a natural projection p:H--+H/I’. We provide H/I’ with the quotient topology: A set in H/I’ is open iff its inverse image in H is open. To investigate the local structure of H/I’ we prove the following

1.7 Lemma. a) Each point a E H has an open neighbourhood U with the following property: Two points of U are equivalent with respect to I? iff they are equivalent with respect to ra. b) Let (a, b) b e a p air of I’-inequivalent points of H. There exist neighbourhoods U(a), U(b) such that no point of U(a) is r-equivalent with any point of U(b). Notice. We may stabilizer

assume

in both

cases that

U(a)

is invariant

under

the

I’,: M(U(a))

because

we may replace

= U(a)

U(a)

for all

M E ra ,

. t ersection by the (fi m ‘t e ) m

of all M(U(a)),

M

E

r II' PTOOf

1.7.

Of

a) If the statement

is false we can find

sequences

a,+a, such that

b,-ta

a, and b, are equivalent

M,,(a,)

mod I’, but inequivalent

= b, ,

mod

I?,:

M,, E r .

As I? acts discontinuously, the sequence M, belongs to a finite set. Taking subsequences we may assume that M, is constant, M,, = M. Taking limits we obtain Ma = a, which contradicts our assumption M # ra. b) The

proof

1.71 Corollary. (A surface

is similar The

to a) and therefore quotient

is a HausdorfI

we leave it to the reader.

space H/l? is a surface.

space which

is locally

homeomorphic

Proof. 1) Two different points of H/l? can be separated which follows easily from 1.7,b). 2) From 1.7,a) we conclude: The natural projection

H/l?,

-H/l?

to R2)

by neighbourhoods,

El

$1

Discrete

induces

Subgroups

a topological

of SL(2,

11

R)

mapping

of some neighbourhood U of the image of a in H/I’, onto some open neighbourhood of the image of a in H/I’. For this reason the local structure of H/I’ at a is determined by ra. During the considerations which led to the proof of 1.6 we constructed a finite group & of roots of unity with the following property: The mapping 2 H induces

(z - u)(z

- q-l

a homeomorphism

H/I’,Let

zaiv/n

C=e

E/E. . 9

(v)=l

be a generator of the cyclic group E. Obviously & consists of all roots of unity of order n. From this it is clear that two points 20, w’ of E are equivalent mod & iff W” = wfn. We obtain a unique bijective mapping o such that the following diagram commutes

E i

1 E/E

qmodE

=Eq

A

1 E

1 qn.

a! is a homeomorphism because the two other arrows open mappings. The proof of 1.71 is now complete. We have shown

H/l? r-.,H/r, t locally

Cusps.

The

We consider

the closure

= E/E = E

ZH-

and

.

at a

cl

of H in the Riemann

formula

are continuous

sphere:

uz + b cz + d

defines an action of SL(2, R) on the larger ventions~=ooifK:~Randcrc+d=O(note:c~+d=O+u~++#O)

space Hi. We use the usual

con-

Chapter

12

I.

Hilbert

Modular

Forms

e are interested in the structure of the ands=% (=ooifc=O).W stabilizer rK of a discrete subgroup l? c SL(2, R) in a boundary point K. For this purpose we choose any matrix AESL(~,R), (for example A = (

-4

;

>

AK==

)*W e now consider the conjugate group AI’A-1

which is again a discrete subgroup of SL(2, R) instead of l?. The conjugation M I-+ AMA-’ obviously defines isomorphisms l?t

AI’A-1

rKA

(AI’A-l),

.

A matrix M E SL(2,R) fi xes 00 iff it is an upper triangular matrix. corresponding transformation is then of the form Mz=~z+b;

The

e>0,b~R.

We are highly interested in the special case of translations Mz=z+b

(Ed)

i.e.

In this case we call M a translation matrix or simply a translation. 1.8 Definition. (# fE)

The group l? is said to have cusp 00 if it contains a nontrivial translation.

1.9 Lemma. If the discrete subgroup l? c SL(2,R) has cusp 00, each element of the stabdizer PO0 is a translation. Moreover the image of FM in SL(2,

R)/{~E)

is an infinite

cyclic

group.

Proof. We consider the set t:={aERIzt+z+aiscontainedinI’} of all real numbers a E R such that

is contained in l?. Obviously t is a discrete subgroup of R, hence cyclic: ‘t = Z . as. We have to show that each matrix of the form

Er

is

$1 Discrete

Subgroups

a translation,

of SL(2,

13

R)

i.e. .s2 = 1. This follows (8

This calculation

Ebl)(;

from the simple calculation

‘;> (;

E”l)-l=(;

shows that multiplication t +

t,

&:,).

by e2 defines an automorphism

a++E2a,

which obviously implies E2 = 1. The second part of Lemma 1.9 is also clear, because the image of I’= in SL(2, R)/{fE} is isomorphic to t. 0 Before we give the definition of an arbitrary cusp, we notice that an upper triangular matrix M E SL(2, R) is a translation matrix if and only if a(M)

= f2.

A matrix M E SL(2, R) with the property a(M)

= f2,

M#fE

is called a parabolic matrix. Notice. A parabolic matrix M E SL(2, R) has exactly one fixed point on the extended real axis R U {oo}. 1.10 Lemma. Let I? C SL(2, R) be a discrete subgroup. FOT a boundary point K E R U {oo} the foIlowing three conditions are equivalent: 1) There exists a matrix A E SL(2, R), AK = 00, such that AI’A-1 has cusp infinity. 2) The latter condition is satisfied 3) TheTe exists a parabolic element

GOT

each A E SL(2,

in the stabilizer

R) with

AK = 00.

rK.

The proof of lemma 1.10 is an immediate consequence of the preceding remarks and the fact that trace is invariant under conjugation. A boundary point K E R U { 00} is called a cusp of I? if the conditions formulated in 1.10 are satisfied. We denote by H* the union of H with the set ~~CUSPS 0f I?, H* = H u {cusps of I’} . This set of course depends on our given discrete subgroup I?. Let 00 be a cusp of I’. The stabilizer Lyononly contains translations (1.9) and so it acts on the open set lJc={z~HIIrnz>C}. The relevance of the cusps to the structure of the quotient space H/I’ obvious from the following

is

14

Chapter

1.11 Proposition.

If 00 is a cusp of r, then Uc/roo

is an open imbedding

fOT

-

suficiently

I.

the natura2

Hilbert

Modular

Forms

projection

H/r

large C > 0.

(“Open imbedding” means a topological mapping onto an open subset.)

Proof.

The projection is obviously continuous and open. Therefore it is sufficient to show that it is injective for large C. This means: From Imz>C, ImMz>C, MEI’ we have to deduce

M E rm

(i.e. M = (i

z)).

We first prove 1.111

Lemma.

Let A C SL(2,R)

be a discrete

subset

with

the following

PTOpeTty:

There

are two Teal numbers

ii&A+ Then

there

exists a number

a, b # 0, such that for

n;)iVl(;

ali n, m E Z

Tb)~A.

6 > 0 such that

GA

and

c#O

implies

PTOOf

. Let

be a sequence in A such that c, converges to 0. If we multiply suitable translation matrices from the left and from the right

we get a new sequence in

M,

by

A with the same c, and with the further property

la, - 11I Clc,l,

I& - 115 Clc,l.

15

$1 Discrete Subgroups of SL(2, R) Here C is a suitable

constant

depending

lCnbnl = I(% - l>(&

on a and 6. We obtain

- 1) + (%I - 1) + (Al - 111

5 c2c2, + 2Clc,l. Now it is obvious that the sequences a,, b,, c,, d, are bounded. By assumpq tion A is a discrete subset and hence the set of all M,, is finite. We now deduce from 1.111 a lemma which obviously implies 1.11. subgroup

with

A E SL(2,

R) .

fIJTtheTmOTe that a constant C > 0 is given. C’ > 0 such that

Then

1.112 Lemma.Let 00 Assume constant

M(A-lUc)

I? C SL(2,R) and

b e a discrete

K=A-loo,

n UC, #

0,

+

M E I?

the cusps

there

exists

a

MK.=00.

The set A = I’A-’ has the properties formulated in 1.111, because 00 and R are cusps of I?. If Lemma 1.112 were false, we could find sequences

Proof.

z, EUC

and

M,

Er

such that ImN,z,+oo,

where N,, = M,,A-1

and M,K. # 00

(i.e. N,oo # oo) .

But then we have Im Nnz, = and hence which is a contradiction

to 1.111.

0

The structure of the quotient Uc/I’oo is very easy. Let z I+ z+u , a > 0, be a generating translation of the stabilizer. The mapping UC -

u,‘(o) = (‘2 10 < 141< 1”)~

T = e--27w~,

Chapter

16

induces a topological

Roughly quotient

I.

Hilbert

Modular

Forms

mapping

speaking we may express this as follows: H/I’ looks like a pointed disc U,‘(O).

Close to the cusp 00 the

It looks natural to add the centre 0 of the disc to the quotient. This is done for all cusps simultaneously by means of the following construction: We introduce a topology on H* = H U {cusps of I’} (which is very different from the topology sphere). If 00 is a cusp of l?, the sets

induced from the Riemann

UCU{~l, UC= {z 1Imz > C} will form a basis for the neighbourhoods of 00 (and not the complements of discs as in the usual topology of the Riemann sphere). 1.12 Lemma. The set H* carries a unique topology erties: a) The topology induced on H is the usual one. b) H is an open and dense aubspace of H*. c) If K: is a cusp of I’ and A E SL(2, R) a matrix

with

the following

prop-

with

AK = cm, then

the

sets A-l(Uc)u{~},

C>O,

form a basis foT the neighbourhoods of K. The proof of 1.12 is very easy. Of course one must know that the system of the sets A-l(Uc) U {K} (th e so-called horocycles, open discs which touch the real axis at K together with the point K)

0

K

does not depend on the choice of A. The topology of H* has some strange properties. We summarize some of them, the simple proofs are left to the reader. 1) H* is a Hausdorff space. 2) The set of cusps is a discrete subset of H*.

$1 Discrete

Subgroups

R)

of SL(2,

17

3) H* has a countable topology. (Notice: The discreteness of r implies that I’ and therefore the set of cusps is at most countable.) 4) A cusp never has a compact neighbourhood. Otherwise the set {zEC~Imz2C}U{oo} would be a compact set for large C. But the sequence n + Ci, contains no convergent subsequence! 1.13 Proposition. mappings.

The

The (discrete) quotient

group

I? acts on H* as a group

n E N,

of topological

Xr = H*/r (equipped with th e quotient topology) is a (connected) surface, especially a The set of classes of cusps is a discrete locally compact Hausdorff apace. subset of Xr. The canonical mapping

H/r-kH*/r is an open imbedding.

Remark. For homeomorphism

any matrix (=topological

A E SL(2, R) the mapping mapping) XI’-

z H

A.z induces

a

XArA-1

Proof. If ICis a cusp of l? then AK is a cusp of AI’A-I. This follows immediately from the definition of a cusp and the above remark is clear from the cl definition of the topology of H*. We now want to investigate the structure of Xr close to a class of cusps. Because of the foregoing remark we restrict ourselves to the cusp co. From 1.11 it follows immediately that the natural projection ucu

{0+r,---+H*p

is an open imbedding for large C. Moreover the mapping 2 H

induces a homeomorphism

e2rrirla

(=

0

for 2 = oo)

Chapter

18

So, analogous a neighbourhood

to the case of inner points (1.71), of a class of cusps. It remains

I.

Hilbert

Modular

Forms

Xr looks like a disc in to prove the Hausdoti

property: 1) We separate the image zs E H from the image of a cusp, for example co. A simple consequence of 1.111 is that in each class of r-equivalent points in H there exists one with maximal imaginary part. Obviously

depends continuously on z. The inequalities y > C (including oo) and ]z zs] < C-l define open sets in H* whose images in H*/l? separate the two points [zs] and [co], if C is large enough. 2) We want to separate two We choose a transformation A the images of A-l(Uc U {co}) neighbourhoods of the two cusp

different cusp classes, for example [co], [K]. E SL(2,R), AK = co. By Lemma 1.112 and UC U {oo} in (H)*/l? define disjoint classes if C is large enough. cl

We are interested in the case where number h of cusp classes is finite

H*/I’ is compact.

In this

case the

h = #(H*/r-H/r). Notice. Let rs c I’ be a subgroup of finite index. Each cusp of rs and conversely. We therefore obtain a natural

cusp of I’ is also a mapping

which is obviously continuous. It is easy to see that this mapping is proper (the inverse image of an arbitrary compact set is compact). We therefore obtain: Xr

is compact

if and only

if Xr,

is compact.

But in general the number of cusp classes h(I’o) number of cusp classes h(r) of I’.

Fundamental Sets. A subset

F

c

H=

H is called u

of l?s is larger

than

the

a fundamental set of r if

M(F).

Ma(Of course H itself is a fundamental set, but we are interested in smaller fundamental sets which reflect some of the global structure of H/I’.) If H/I’ is compact, we can always find a compact fundamental set by means of the following

$1 Discrete

Subgroups

1.14 Lemma.

of SL(2,

19

R)

Let f:X-Y

be a surjective continuous and open mapping between locally compact paces. If K c Y is a compact subset of Y we can find a compact subset K c X with the property f(k)>K.

Corollary. The discrete subgroup I? c SL(2, set if and only if H/I’ is compact. The

proof

R) has a compact

of 1.14 is easy and can be omitted

Cusp Sectors. For two positive V(s,t)

numbers

(compare

s and t we define

fundamental

proof

of 1.11).

the domain

= {z E I-i I 1x1 I s , Y 2 t} .

Let K be a cusp of r and A E SL(2, K to infinity AIE=CO.

R) a transformation

which

carries

We call the domain A-1(V(s, a cusp sector choice of A.

with

1.15 Proposition. is compact.

respect

to K. This

t>> notion

is obviously

F?) b e a discrete

Let I’ c SL(2,

independent

subgroup

such

of the

that H*/l?

Let ~l,***,~h

be a set of representatives fundamental set

of the r-classes

of cusps.

Then

there

exists

a

F = K u VI u . . . u Vj where K is a compact (1 5 j 5 h). ‘cj

subset

of H and Vj is a cusp sector

Fundamental Domains. Fundamental are minimal in a certain erties. For our purposes

with

respect

domains are fundamental sets which sense and which have reasonable geometric propthe following definition is sufficient.

to

Chapter I. Hilbert Modular

20

Forms

1.16 Definition. A fundamental set F C H of I? is called a fundamental domain if the following properties aTe satisfied: a) F is measurable. b) There exists a set S C F of measure 0 such that two different points of F - S are inequivalent mod I’. Of course one has to clarify what “measurable” means. In our context it is sufficient to use the usual Lebesgue measure. Example: The famous fundamental domain of I’ = SL(2, Z) is defined by the inequalities 1

For S one can take the boundary of F. The existence of reasonable fundamental domains follows from 1.15 and the remark below, which is an immediate consequence of general facts about measurable equivalence relations. (A 11.10). 1.17 Remark. Each

measwable fandamental

set contains a fundamental

do-

main. Final Remark. For discrete subgroups of SL(2, R) there is quite a simple construction for a very nice fundamental domain (so-called “normal polygons” in non-Euclidean geometry). But the method described above carries more easily to the case of several variables.

32

Discrete

Subgroups

of SL(2, R)n

We will generalize the constructions of $1 to the case of discrete subgroups I? C SL(2, DZ)n and describe the extension of H*/r by cusps. This construction will be justified by the fact that in the case of the Hilbert modular group the extended space is compact.

We want to generalize the basic constructions of 51 to the case of several variables. The group SL(2, R)” acts on the product of n upper half-planes: Mz := (Al&, where

. . . ) M&)

)

$2 Discrete

Subgroups

of SL(2,

21

R)n

M=(M1,...,

Mn),

z=(a

,... ,GJ.

We shall also use the notation

if

and Mz

= (az + b)(cz

+ d)-’

.

We shall occasionally use the notation sz = Zl + . . . + zn Nz = Zl - . . . - z, for z E C”. Many proofs of the one-variable case ($1) carry over immediately to the case of several variables. In these cases we omit the proof. A good example for this is 2.1 Proposition. A subgroup discontinuously on H”.

r c SL(2,

R)”

is discrete if and only if it acts

Recalk “To act discontinuously” means that for each compact set K c Hn the set

{MEI’IM(K)~K#~} is finite. Proof.

cl

Compare 1.2.

We want to introduce the notion of a cusp. For this we extend the action of SL(2, R)” to H” where H=Hui’?,

R=RU{w}.

The Cusp oo = (CO,... ,oo). We assume that a discrete subgroup l? c SL(2, R)n is given and we are going to define what it means that l? has cusp infinity. The justification of the following definition is the fact that our main example, the Hilbert modular group, has cusp 00 and moreover that the quotient H”P > l? Hilbert modular group,

Chapter I. Hilbert Modular

22 can be compactified by adding a finite cusp 00 = (00,. . .) oo)). The condition sense that the stabilizer

is as large

as possible.

We first look

number of cusps (one of them is the that co is a cusp of P means in some

at the translations

Put

Here “z H z + a lies in I’“means of course E P with Mz = z + a. There are 2” possibilities

in I’,:

r) .

t = {a E FP 12 H z + a lies in

M

Forms

that there for M,

is an element

iM=(*(; “;>,...,*(:, a;)). group There

t is a discrete subgroup t of Rn is isomorphic exist

k R-linearly

of R”. It is well-known that each discrete subto Z” for some Ic, 1 5 Ic 5 n. More precisely: independent vectors al,...,ar,

E R”

such that

t = zal

+ . . . + ZUk .

In the case k = TZ,t is called a lattice. So t is a lattice basis al,... ,a, of the vector space R” such that

t = ZUl + . . . + Za, Of course

the basis al,.

. . , a, is not uniquely

P={zEA”,z=~tjaj

iff there

exists

. determined.

We call the set

o
a fundamental parallelotope of t. It is a compact set with its translates a + P , a E t, cover the whole R”.

The

definition

of “cusp

00” involves

2.2 First Condition. The translation

two conditions module

t is a lattice.

the property

that

a

$2 Discrete

A vector

Subgroups

of SL(2,

E E R” is called

R)”

23

totally positive (E > 0) iff

El >o ,...,& > 0. We call a totally positive element E E R” a multiplier of I’ if there exists a vector b E R” such that the transformation z H EZ + b lies in I’. The set A of all multipliers is a subgroup of the (multiplicative) group of all totally positive vectors. We denote the multiplicative group of all positive real numbers by R+ = {t E R 1 > 0).

t

2.3 Remark. If the first A is a discrete subgroup

condition of (R+)“.

(Z.2) is satisfied, Each multiplier

the group satisfies

of mdtipiiers

El * . . . . &n = 1 .

Proof. If E is a multiplier, a transformation of the form z H EZ + b is contained in I?. We may replace b by b + a, a E t, and therefore assume that b is contained in a certain bounded set. The discreteness of I? now implies the discreteness of A. To prove the second statement of 2.3 we note that A acts on t: Axt---+t

(&,+-+E~a=(Elal)...) Multiplication

with

E defines

&an).

a linear

mapping

me : R” --t R” a c--t EU whose determinant is ~1 . . . . . cn. The matrix of the lattice is integral. Therefore we have

same applies

The morphic by

to E-I

multiplicative to the additive

instead

of e, and we obtain

By means a discrete

to a basis

det m, = fl.

0

group of positive real numbers is topologically isogroup of all real numbers. An isomorphism is given log : R+

into

respect

E Z.

detm, The

of m, with

of this isomorphism subgroup of IV: 1ogA loge

N + R . we transform

the group

c R” := (log&r,.

. . ,logc,)

.

of multipliers

Chapter I. Hilbert

24

Modular

Forms

If our first condition 2.2 is satisfied, log A cannot be a lattice because the condition el . . . . - Ed = 1 implies that log A is contained in the hyperplane (=subspace of dimension n - 1) v = {u E Rn 1 a1 + . . . + a, = 0). We obtain

that A E Zk ,

ksn-1.

2.4 Second Condition.

2.5 Definition. The discrete subgroup I? c SL(2,R)” has cusp 00 = . , oo) if the fir& and the second condition (2.2 and 2.4) am satis(0%. fied, i.e. A z Z”-1 . tczn,

We now give the definition

of an arbitrary

cusp

K = (/cl,. . . ) Kc,) E R” . We can always find a transformation infinity, AK = 00.

A E SL(2, R)n which

transforms

K to

2.6 Remark. Let I? c SL(2, R)n be a di3CTek subgroup and A E SL(2, R)” be an element such that AI’A-I has cusp infinity. Then for each B E SL(2, R)” with A-l(m) = B--‘(m) the group BI’B-’

Proof.

has also cusp infinity.

It is sufficient

to treat the case A=(E

i.e. B(z) = EZ + b. Using an obvious

which

,...,

notation

E),

B(oo)=cm,

we have

t(Bl?B-I)

= ,z2 t(r),

A(BI’B-‘)

= A(r),

gives our assertion.

The preceding

remark

2.6 justifies

the following

$2

Discrete

Subgroups

2.7 Definition. Some (every)

of SL(2,

25

R)n

A discrete &group

r

C

SL(2, R)” has map K E En #for

AESL(~,R)~, the group

Al?A-l

AK==,

has cusp 00.

We again use the notation (H”)* = H” U set of cusps of I?. 2.8 Lemma. The with the following

set (H”)* (which properties:

a) The topology

depends

on I’)

on H” is the usual

induced

carries

a unique

topology

one.

b) H” is an open and dense subset of(H”)*. c) If tc is a cusp of I? and A E

SL(2,R)"

a transformation

with AK. = 00,

then the sets A-l(U+J{~},

C>O,

with

Uc={r~H~,fiImz~>C) j=l

form a basisfor the neighbourhoods

of K.

The proof is the same as that of 1.12, so we omit it. We only mention two important facts: a) The system of sets

A-1(&) u {K} , c > o, does not depend on the choice of A. b) The stabilizer I’m acts on UC, because each transformation of the form Mz=ez+a

M E I?= is

with Ne

:=

El - . . . - En =

1 .

We also see immediately: If K is a cusp of I?, then AK. is a cusp of Al?A-’ for every A E sL(2, R)“. We obtain especially that I’ acts on (Hn)* and therefore we can consider the quotient space

xr =(Hy*/r, equipped with the quotient topology.

Chapter I. Hilbert

26

A

If induces

is any element a homeomorphism

Therefore arbitrary

of SL(2,

R)“,

then

I? C SL(2,

subgroup

Forms

z -

AZ

of Xr

(for

the transformation

we may reduce the investigation of the local l’) at a cusp IE to the case K = 00.

2.9 Proposition. For any discrete

Modular

structure

R)”

the quotient

space

Xr = (H”)*/r is a locally compact HMLS~OT~~ space. If 00 is a cusp of I?, then the canonical mapping uc u +4/r, e’s an open imbeddkg for the neighbourhoods For the proof 2.91

A

w-7*/r

suficiently ZaTge C. of the class of CO.

GOT

of 2.9 one needs

Lemma. Let

-

C SL(2,

a suitable R)”

This

generalization

be a discrete

subset

system

defines

of 1.111. and let

r,r’ c s~(2,~)n be discrete

subgroups

with

cusp 00. we

assume

rk. Ad?b,cA. Then

there

exists

a number

6 > 0 with

EA

the following and

property:

c#O

implies

As a special

case we obtain Nc=Odc=O.

Proof.

We first

prove

the last statement.

EA,

Assume

c#O,

the existence

q=o.

of a

a basis

$2 Discrete

Subgroups

of SL(2,

R)n

We choose a sequence of 1y E t(I”,) a#O,

such that

(Y2-+0 )...)

a,+O.

Such a sequence exists by AI.19,2). We notice that by AI.19,1) nents of (Y are different from 0. We now consider

After of

that we may choose a sequence of p E t(I’,)

N(i),

N=(i

!)

(:

i)

(i

all compo-

such that the real part

7)=(:

d&J

remains bounded. We obtain a contradiction to the discreteness of A (since the sequence N(i), hence N, because of 1.11 , is contained in a compact subset ) . We now come to the proof of the first part of 2.91: Assume that there exists a sequence

After multiplication assume

where lattice

with

GA,

c#O,

suitable

matrices

of I’m from

the left we may

6 is a suitable constant. This follows from the fact that log A is a in the trace-zero hyperplane of R”. But now we have Cj+O

for

Now the proof can be completed n=l (1.111). An immediate 2.92 Lemma. There

Nc+O.

exist

in exactly the same manner

consequence

of 2.91 is (compare

Let IC, IC’ be cusps neighbourhoods

qu)nu’#0,

U,U’

l<j
1.112)

of the discrete subgroup in (H”)* such that

kfEr

+

as in the case 0

I’ c SL(2,R)“.

Mlc = tc’ .

PTOO~of 2.9. From 2.92 we may deduce as in the case n = 1: a) Xr is a Hausdorff space. b) The mapping uc u b4/r, (w)*/r

Chapter

28

I.

Hilbert

Modular

Forma

’

is an open imbedding for sufficiently large C (if 00 is a cusp of I’). What remains to be shown is, that Xr is locally compact. Of course Hn/I’ is locally compact because Hn is so and the projection H” ----t H”/l? is open and surjective. So we only have to show that each class of cusps has a compact neighbourhood. We actually show 2.10 Lemma.

The space

with UC= is compact

GOT

C

>

{z E H” 1 Ny 2 C}

0.

The group roe of course acts on the space rc U {oo} topology induced from (H”)*. The quotient is provided topology.

which

with

carries the the quotient

PTOO~. We first construct a fundamental set for roe. A subset P of an n-dimensional real vector space is called a psrallelotope there exists a basis al,. . . , a, with the property P = {Q 1 U = 2tjCXj

if

, 0 2 tj < 1 for 12 j 5 ?2}.

j=l

A parallelotope

is of course compact.

2.101 Lemma. Let r c SL(2,R)" There exist parallelotopes

be a discrete subgroup with cusp infinity.

a>

P c R”

b)

Q c {v E R” 1Sv = 0)

such that the domain

is a fundamental

set of roe acting on 77~.

Proof. Take for P a fundamental parahelotope of t in R” fundamental parallelotope of log A in the vector space

{v ER” 14%= 0). Notice

that u = log &

and for Q a

\

29

$2 DiscreteSubgroupsof SL(2, R)n

is contained in this vector space. A transformation of the type z I+ EZ + p has the effect u H u + log E. A transformation z H z + Q leaves u unchanged and has the effect z H z + cr. Now the proof of Lemma 2.101 is clear. q We now consider the closed, hence compact interval [e, oo] in the extended real axis. The image of the following, obviously continuous mapping

(t4ogY)

-

{

5 + ity if t < oo 00 ift=oo

is a compact fundamental set of roe acting on vc U {oo}. Therefore UC U {00)/l?, is compact.

q

Cusp Sectors. Consider any parallelotopes PCfP, Q c {v E R” 1Sv = 0) and a positive C > 0. The domain v = {z E UC 1x E P , log &cQl is called a cusp sector at co. If K is an arbitrary Ati=oo,

element of w” ,

A E SL(2, R)n ,

we call a domain of the type A-l(V) independent of the choice of A.

a cusp sector at K. This notion is

An obvious generalization of 1.15 is 2.11 Proposition. space

Let I? C SJC(~,R)~ be a discrete subgroup such that the Xr := (H”)*/l?

is compact. Let be IE~,. . . ,~h be a set of representatives of the (finite !) number of cusp classes of I?. There exists a fundamental set F=Kuvl

u...uvh,

where K is a compact subset of H” and Vj is a cusp sector at Kj h)*

(1 5 j 5

Chapter I. Hilbert Modular Forms

30

Singularities. In contrast to the one-variable case the cusp classes are never smooth points of Xr if 7~ > 1. (A point is called smooth if it has an open neighbourhood homeomorphic to an open ball in R”). We want to indicate the reason for readers who are familiar with homology groups. The local homology group of a topological space X at a point a E X is defined as H(X, {a}) := l@l Hr(U - {a}, Z) ) u30. where U runs over the system of neighbourhoods of a. It is enough to take a fundamental system. If n > 2 the space R” - (0) is simply connected. It follows that H(X,

M)

= cl

if a is a smooth point of X with dimension X > 2. On the other hand the fundamental group of UC/I’~ is I’m itself because I?= acts freely on the contractible space UC. The first homology group is the abelianization

where

[I’,,

I’,]

d enotes the commutator

subgroup

of IYoo. We obtain

qxr, [q E rEb for each cusp K. The commutator subgroup of I’= contains only translations! We obtain that the abelianization I’$ is an infinite group in the case n > 1. It is not hard to show that raL % Z”-l $ finite group.

Elliptic

Fixed Points.

2.12 Lemma. We SL(2, R)“. Then a) All components b) M is of finite c) M has a jixed

An immediate

consequence

of 1.3 and 1.5 is

assume that M is contained in a discrete subgroup the following thTee conditions are equivalent: Ml,. . . , M, of M are elliptic. OTdeT. point in H”.

I? of

The fixed point of such an M is not uniquely determined even if M is nontrivial. Some of the components of M could be trivial, others not. But in the case of Hilbert’s modular group the following condition will be satisfied:

$2

Discrete

Subgroups

2.13 First jections

of SL(2,

condition

31

R)n

of irreducibility.

The restriction

: SL(2,R)" --+ SL(2,R)

pj

(1 <j

of each of the n pro-

<

n)

to l? is injective. Let r c LITL(~,R)~ b e a discrete subgroup which satisfies the of irreducibility. The image of the stabilizer ra of any point a E H” in the grozlp (SL(2,R)/{fE})” is a finite cyclic group. The set of elliptic fixed points of r (=aet of points where this image is non-trivial) is a discrete subset of (H”)*.

2.14 Remark. first

condition

Proof. Compare 1.4 and 1.6 and make no non-trivial elements of finite order). 2.141

Corollary.

use of 2.8 (notice

f’-

contains Cl

to the irreducibility

If ( in addition

that

condition)

the quotient

apace

xr =(H")*/r is compact, there exists only (as well as of cusps).

a finite

now determine 1.71). Let a be any point

structure

We

the local

number

of H”. By means z -

of classes of elliptic

of H”/l? (compare

with

fixed points

the proof

of the transformation

(z - u)(z

- q-l

we may transform H” to E” and a to 0. The group of transformations corresponds to a finite cyclic group G of transformations of E” onto A generator of G has the form (w,...

, %>

-

((1

w,

because have

we can replace (j

The

group

= e 29&j

the generator /cZ

,

G is completely

(el,...,

e2ri/e

by a power

determined

e,).

, en)

that

the

,

(ej,e) = 1 , 1 5

(el,e2,...

We call G oftype

=

r(l itself.

* * * , Cn%),

where
of

coprime

5 e

ej

by the system (el

=

to e, and we then

(1 5 j 5 n) . of numbers

1).

As in the case n = 1 we now obtain:

Chapter

32

I.

Hilbert

Modular

Forms

2.15 Lemma. Let I? C SL(2,R)n be a discrete subgroup which satisfies the irreducibility condition 2.13. Let a be any point of H”. The natzlral projection H”/I’,

+

H”/I’

is a local homeomorphism at a. The transformation induces a homeomorphism Hn/l?,

z I+ (z - a)(z - 7i)-l

N PEn/G,

where G is a finite cyclic group of order e and a certain type (el = 1 , (ej,e) =

(el,...,e,)

f?j 2

1 , 12

e for

1 2 j 2 n) .

But in contrast to the one-variable case the image of 0 in En/G is never a smooth point if n > 1 and of course e > 1. (A similar consideration as in the case of cusps shows that in the case n > 1 (where E” - (0) is simply connected) the local homology group of En/G at the image of 0 is G itself.) We may summarize: Let r c SL(2,R)* b e a discrete subgroup satisfying the irreducibility condition 2.13 and such that Xr = (H”)*/I’ is compact. The space Xr contains a fmite number of singularities (non-smooth points), namely the cusp classes and the classesof the elliptic fixed points. The complement Xr - classes of cusps and of elliptic

fixed points

is a manifold of (real) dimension 2n (i.e. is locally homeomorphic to R2”). The

results

of this section,

especially

the central

Lemma

2.91 are essentially

due to H. Maad

[471.

33 We

The Hilbert have

to use some

Modular basic

facts

about

Group algebraic

numbers.

For

the

convenience

of the

reader we have summarized them (without proofs) in appendix I. The Hilbert modular group IK = sL(2,oK) (K totahy real number field) has precisely h (=class number) cusp classes. The extended quotient (H”)*/r is compact. This result is also due to II. Maa% [4f31*

Let K be a field of complex numbers QcKcC.

$3 The

Hilbert

Modular

33

Group

We assume that K is an algebraic number field which means that the dimension of K as a vector space over & is finite. This dimension is called the degree of K, n := [K:Q] := dimQK. Such a number

field admits n different

into the field of complex numbers of K onto a subfield of C).

imbeddings

(An imbedding

of K is an isomorphism

We assume that K is totally real, i.e. the image K(i) imbeddings is contained in the field R of real numbers K--r, 0, H

K(j) .(j)

of each of the n

c R (1

5

j

5

?2)

.

We put the n imbeddings together into a single Q-linear, obviously mapping K-PR”, 0, H (a(‘), . . . , .(“)) . For the sake of simplicity (a(l), . . . ) &)).

it is sometimes

useful to identify

injective

a and the vector

If we attach to the matrix E GL(2, K) the tuple (M(l), we obtain

. . . , MC”))

an imbedding

,

M(j)+;;

s),

jzl,...,

n,

of groups GL(2, K) L) GL(2, R)” .

Also we occasionally 3.1 Definition. is

identify

The Hilbert

GL(2, K) with

its image.

modular group of the totally l?K = SL(2,o)

where o denotes the ring of (algebraic)

)

integers

of K.

real number field K

Chapter

34

We recall with field

that the set of all algebraic integers of fractions K. Moreover the image K +

Hilbert

contained of o under

Modular

Forms

in K forms a ring the imbedding

R”

a H is a

I.

(a(l),

. . . ) CP))

lattice of R”. We have especially

as additive groups. The discreteness of o in R” implies SL(2, o) in SL(2, R)“. From 2.1 we obtain:

the discreteness

of

3.2 Remark. The Hilbert modular group l?~ = SL(2,o) of a totally real number field K acts discontinuously on the product of n upper half-planes. We want to determine the cusps of the Hilbert modular of 00 consists of all transformations of the form

%i+E22+a, (o* is the multiplicative this we see: a) The morphic

translation to Z” :

group

module

UEO,

of units,

The stabilizer

EEo*

i.e. invertible

of the Hilbert

group.

modular

elements

group

of 0). From

is o , hence

iso-

t=oGizn. b) The

group

of multipliers

A is the group

of squares

of units

A = {E2 1E E o*} . In the case of a totally

real field

the famous

o* s z-l From

this

it follows

immediately

Dirichlet unit theorem states

x z/22.

that

What we have shown is that 00 is cusp of the Hilbert modular group. Before we determine the other cusps, we mention a simple fact: Let I’s c I be a subgroup of finite index. If l? has cusp 00, then Is also has cusp 00 (and conversely). (Th is of course implies that l?e and l? have the same cusps.) The mentioned fact follows from two easy group theoretical lemmas.

$3 The Hilbert

Modular

Group

35

Lemma 1. Let G be a group and Go c G a subgroup of finite index. Let H C G be any subgroup of G. Then Ho = H fl Go has finite index in H (because the natural map of cosets H/Ho

-

G/Go

hHo H

hGo

is injective).

Lemma 2. Let t C Z” be a subgroup

of finite

index.

Then

t E Z” .

Important examples of normal subgroups of finite index of the Hilbert modular group are the principal congruence subgroups: Let a c o be an ideal (different from 0) in o. The principal congruence subgroup of level a is the kernel of the natural homomorphism

SL(2,o)

-

SL(2, o/a) .

The group SL(2,o/a) is finite because o/a is a finite kernel is a normal subgroup of finite index. We denote

ring. Therefore it by

the

rK[a]={MErKIM-Emoda}. 3.3 Remark. Let from 0 such that

be A E GL(2,

K).

I’,[a]

Proof. There have integral

There

exists

c AI?KA-~

an ideal a C o different

.

exists a natural number I such that the matrices IA and lA-’ coefficients. The ideal a = (Z2) has the desired property. 0

Two subgroups Ii, I’s of a given group are called commensurable if I’i nrs has finite index in both the groups Ii, I2 . An immediate consequence of 3.3 is

3.31

corollary.

modular

group,

If

r c sq2, R) n is any group then the same is true of AI’A-1

,

A E GL(2,

commensurable

K)

.

with

Hilbert’s

Chapter

36

3.4 Proposition. KU {co}.

The cusps of the Hilbert

(We recall that an element E Fr.) (a(l), . . . ,&))

I.

Hilbert

Modular

Forms

modular group are the elements of

a E K has to be identified

with

the vector

PTOO~. 1) Let be a E K. The matrix A= transforms of rK. 2) Let

a to 00. By 3.2 00 is a cusp of AI’KA-’

IE=(/q,...

and therefore

a is a cusp

,Ic,) E-iin

be a cusp of rK. The stabilizer of K contains translation and therefore an element

#fE,

a conjugate

of a non-trivial

a+d=f2.

The fixed point equation

has only one solution,

namely

Cusp Classes. We have to clarify when two cusps are equivalent action of the Hilbert modular group. For this purpose byaandl

under the

we consider for each element a E K the ideal generated 0, H

(U,

1)

.

We extend this by sending cc to the principal 00 H We are not really interested class. If we denote by C(K)

ideal (1)

(1).

in the constructed ideal but only in its ideal the (finite) set of ideal classes, the above con-

$3 The Hilbert

Modular

37

Group

struction gives a mapping Ku {CO} +

C(K)

a-

class of (a, 1) if a E K 1 trivial class if a = 00 .

3.5 Lemma. The mapping KU {oo}

----t C(K)

is surjective. Two cusps are equivalent with respect to the Hilbert modular group if and only if the corresponding ideal classes coincide. This means that we have constructed a bijective mapping Ku

{oo}/lk

N > C(K).

3.51 Corohry. The Hilbert modular group has only finitely classes. Their number equals the class number of K. Proof.

many

cusp

The proof of 3.5 will be an easy consequence of the following

3.52 LemmL Let a =

(cl,&)

=

(~2,d2)

be an ideal of K. There exists a matrix M E rK = SL(2,o) with the property (cl,

Proof. The (fractional) Therefore we have

4)

. M

=

(~2,

d2)

.

ideals of a number field form a multiplicative lEo=a-a-l

and can find elements aj, bi E as1 ;

j = 1,2

such that the matrices Mj =

;

j=1,2

group!

Chapter I. Hilbert Modular

38 have determinant

has integral

Forms

1. Obviously

coefficients

and therefore (cdl)M

=

is contained

(O,l)Mz

=

in SL(2,o).

We have El

(~242).

Proof of 3.5. 1) Let K, K’ be equivalent

cusps

IC’=MK

7 M E SL(2,o).

We obtain (K, 1) = ((K, 1). M’) = (UK + b,c/c + d) - (K’, 1)

M’ denotes

where

2) Assume alent :

that

the transposed

matrix

IE, IE’ are cusps such that

of M. the corresponding

ideals

are equiv-

(K’, 1) = (Y * (Kc, 1) . By 3.52 we find

M E SL(2, o) such that

a matrix

= (CKK, (Y) . M’ ,

(d,l) ie. . d=Mtc. proof

0

The finiteness for

of the number

of cusp classes is a hint,

but of course

3.6 Theorem. The space xr,

= H” u K u {cm}/I’~

is compact.

PTOO~. It is sufficient

(because

of 2.10)

/ci =A;‘(m); and a number

to construct

finitely

many

cusps

j = l,...,h

6 > 0 such that

lj A;‘(&) j=l

is a fundamental

3.63).

set of I?K. We arrange

the proof

in several

lemmata

(36-

no

$3 The

Hilbert

Modular

39

Group

3.61 Lemma. There ezists a constant C = C(K) such that for each 2 E R” and for each E > 0 we can find integers c, d E o, c # 0, which satisfy the inequalities

Here

we use the notation

and of course cx = (C(l)Xl,

. . . ) Jn)x,)

.

Proof. Let P C Rn be a fundamental parallelotope of o it Rn. We can find a finite number of E-balls (with respect to the norm II.II) which cover P

P c ul u . . . u UN . The number achieve

where C(K)

N of course

depends

N’ = N’(K) depends such that the number

on E. But

only on K. We now of all CEO,

IlcllI

is greater than 5. This is possible because exists a constant 6 > 0 such that the number

is greater

or equal

SP.

it is easy to see that

We now

choose

CEO,

choose

one can

the constant

C =

c 2e for any lattice of points

t C R” there

for each c

C llcll 5 -2E

a d E o such that cx + d E ul u . . . u UN . At least one of the balls Vi has to contain two points cx + d with c. The difference of such two cz + d is a solution of our inequalities.

different Ill

Chapter

40

3.62 Lemma. There exists a pair

is a constant

6 = 6(K)

(v4-x0,

of integer3

which

Hilbert

for

Modular

Forms

each z E H” there

(4)#w)

satisfies the inequality Ny 2 Spv(cz

(FhLll:

such that

I.

+ d)12 .

A?%= .%l* . . . * .zn for z E C”)

Proof. We may replace z by z - e2,c E o*, and therefore assume that y is contained in a fundamental set with respect to the action Y-

YE2? & E o* .

We therefore may assume that the inequalities A-i

< -

Yj m’A

(1Ljln)

with a certain constant A = A(K) as in Lemma 3.61 such that

We obtain

are satisfied. We now take two integers

pv(cz + d)12 = N[(cx + q2 + C2Y21

5 (1 + C2A2)‘Wy,

and we can choose 6 = (1 + C2A2)-“.

El

We want to deduce Theorem 3.6 from Lemma 3.62. It may be helpful for the reader first to treat the simple case of class number h = 1. In this case one may achieve in 3.62 that (c, d) is the second line of a modular matrix E SL(2,o). Then the inequality 3.62 is equivalent with Ny/lcz

+ d12 = N(Im

Mz)

2 6, i.e.Mz

E Us .

$3 The

Hilbert

Modular

41

Group

Therefore US is a fundamental set. To extend this proof to the case of arbitrary prove a further lemma. Take a set of representatives of the ideal classes

the pairs (cj, dj) to matrices Mj

= (

ai 9

2

i

of SL(2, K)

E SL(2,K). >

We may assume that all the aj are integral 3.68 Lemma.

we have to

j=l,...,h

aj=(Cj,dj);

and complete

class numbers

There exists a constant

0).

(C

E = a(K)

cc, 4 E 0 x 0 3

such that for each pair

cc, 4 # (0, 0)

we can fina a) a matrix M E SL(2,0), b) an index j E (1,. . . , h} such that the following

condition

is satisfied

where

Let aj be the ideal which lies in the same ideal class as (c, d): a! (c, d) = aj for some (Y E K. We have

Proof.

JNaI

L

N(aC,

ad)

=

N

(aj)

.

We may replace cx by CYE,e E o*, and therefore assume that llorll is bounded from above by a certain constant which depends only on K. We now choose a matrix M E SL(2,o) such that (cj,dj)

* M = ( 0x2, ad)

(see 3.52) .

This means E=cYc, where

(Z, Ci) is the second line of MjM.

d”=ad El

Proof of Theorem 3.6. From 3.62 and 3.63 we obtain that for a given point z E H” there exists a matrix M E SL(2, o) and an index j E { 1,. . . , h} such that Ny 2 G”(N(cz + d)12,

Chapter I. Hilbert Modular Forms

42 where

Here s” denotes some constant depending is equivalent with N(Im MiMz)

only on K. The above inequality 1 d.

The point Mz, which is I’K-equivalent with z, is therefore contained MJyl (Us). The union of these domains is therefore a fundamental set.

in 0

of 3.6. The Hilbert modular group l?K satisfies the condition of irreducibility 2.13. The same is true of each subgroup l? C SL(2, K) commensurable with rK. From 3.6 we may therefore conclude An Application

A subgroup r ular well

c SL(2,K)

group rK has only a finite as of cusps).

wh ic h is commensurable with the Hilbert modnumber of classes of e&tic fixed points (aa

As an immediate consequence of this and of 2.141 we obtain 3.7 Remark. The Hilbert modular group (more generally, any commensurable group) has only a finite number of conjugacy classes of elements of finite OTdeT.

(When G is a group, then the conjugacy class of an element a E G is the set {ZXKC-~ ] x E G}.) In this context we mention 3.8 Remark.

The principal

rK[i] contains

no element

congruence subgroup = ker(rK

-

SW,

o/(l))

of finite order different from E if 1 2 3

(1 E N).

Each finite group G # { e} contains an element of prime order. If the statement of 3.8 is false, we can find a matrix Proof.

hf E rK[i],

hf#

E,

kfp = E,

where p is a prime number. We write M = E + B (B # 0) and denote by b the ideal generated by the coefficients of B. We have b C (1)

(because A4 E I&]).

From MP = E we obtain P

(*)

’ c(> j=l

j

Bi=O.

43

$4 Automorphic

Forms

The

coefficients

binomial

being

integral

we obtain

pB f Omod

b2,

hence

pb c b2 and therefore

p E b c (1). The binomial coefficients (p), 1 5 j 5 p - 1 are divisible cients of (T)Bj are therefore contained in

pb2 (C b3)

ifj

by p. The

coeffi-

22. of BP are also

From our assumption we obtain p 2 3. Hence the coefficients contained in b3. From the equation (*) we now obtain pBrOmodb3. This

implies

b cp-lb3 or equivalently

p E b2 c (12). But

this

is impossible

3.9 Corollary. points.

I’[l]

since Z2 cannot acts freely

$4 Automorphic

divide

p.

cl

on Hn for 2 2 3, i.e. there

are no ehptic

Forms

Let r = (rl ,...,Tn) be a vector

of rational

integers.

E zn

For

hf E SL(2,R)”

and

z E l-l”

we put Z(M,z)

=

N(cz

+ cq2,

:=

fi(CjZj j=l

+ q2q

.

fized

Chapter

44

This is a so-called factor qiwv, This condition

with

of automorphy, 2) = Z(M,

makes it reasonable

transformation

Hilbert

Modular

Form

i.e. Nz) * Z(N,

to consider

z) .

functions

law = wf,

f w4 for all A4 which

I.

are contained

zlw

in some given subgroup

I? c SL(2, R)“.

The Fourier Expansion of f at a Cusp. We first assume that 00 is a cusp of our (discrete) group I? and denote by t = {aeRn the translation

lattice

1 zHz+aliesinl?}

of I’, introduced fW-4

in $2. From the translation

formula

= w%M4

we obtain f(z + u) = f(z)

for

We recall some basic facts about the Fourier tions: Let t be any lattice to = {a E R”

aEt . expansion

in R”. The dual lattice 1 S(az)

of holomorphic

is defined as

E Z for all z E t} .

(S = trace)

Notice. Each lattice can be written in the form t = AZn , A E GL(n, An easy calculation shows to = A’-lZn. Let V

c

R” be any open (connected) domain.Then D := {z EC

1 y EV}

is the so-called tube domain corresponding to V. 4.1 Lemma. Let f:D be any holomorphic

-

func-

4:

function on the tube domain D over V. Assume

f(z + a>= f(z) , 0,Et >

R).

45

$4 Automorphic Forms where t C R” is some lattice.

Then f has a unique Fourier

f(z)

=

C agezniS(gs) . get0

(The summation is taken over the dual lattice). lutely and uniformly on compact subsets. If y V, the foTmula

J

ag = WA(P)-

p f (z)e-

(z=z+iy, holds. (vol(P) P oft.)

expansion

The series converges absoV is an arbitrary point of

E

2fmg4

&$

dz=dq-..:dx,)

d enotes the Euclidean

volume

a fundamental

of

We now return to the special case where t is the translation discrete subgroup of SL(2, R)” with cusp 00. 4.2 Remark. If t is the translation lattice of a discrete SL(2, R)n with cusp co, then each of the projections

is injective.

PToof.

t--t

R,

to +

a-

aj,

l<jln

a#O,

a>0

parallelotope

lattice

subgroup

of a

I’ C

R

Especially *

a>O.

(a 2 0

means :

Uj

2

0

fOT

1 5

(u

mean3

Uj

>

0

fOT

1 5 j 5 n)

>

0

:

j

5

?I)

q

See AL19,l).

4.3 Definition. Let r c SL(2,R)” be a discrete subgroup with cusp 00 and f : H” -+ C be a holomorphic function which is periodic with respect to the translation lattice of r. we call f regular at the cusp 00 if the Fourier coefficients satisfy the condition a,#0 We say that equivalently:

f

vanishes

at cusp 00 a,#0

*

g20. if

it is regular

==F- g>o.

and moreover

a0 = 0,

Chapter I. Hilbert

46 We want

to generalize

this

to an arbitrary

f:H” is a holomorphic

function

f(Mz)

with

= N(cz

-

Assume

I’. Let

A/c=oo,

that

law

(r = (?-I). . . ) r,)

subgroup

Forms

43

the transformation

+ d)2’f(z);

for all M in our discrete transform it to co:

cusp.

Modular

K E R”

E Z”)

be a cusp

of I?. We

A E SL(2, R)” .

Notation: rA = AI’A-l, fA(Z) = Z(A-‘,

z)-‘f(A-‘z),

Z(A, z) = N(cz + cZ)~~. A simple

calculation

shows

fA(MZ)

=

T(h!f,Z)fA(Z)

The conjugate group rA has cusp 00 and we regular at K (vanishes at K) if the same is true to verify that this condition does not depend two A’s differ by a transformation which fixes

AK=OO, we only

have to consider

BK=OO

+

the special

M E rA.

for

are tempted to say that f is of fA at 00. But first we have on the choice of A. Because 00,

AK1(co)=co,

case

i.e.

(The equivalence sign means that both sides define the same transformation, i.e. they have the same image in (SL(2, R)/{fE})“.) Let

f(z)

= c ugP-) ilEt0

)

fA(z)

= c

u-yww)

)

sEti

where tA = t(Al?A-l) be the Fourier

expansions fA(z)

=

= a2t

(=s-

t;

of f and fA. A simple

jq-q-2’

C gao

= a-V) calculation

uge-2aiS(a-2bg)e2AiS(ga-2r)

) shows .

$4 Automorphic

47

Forms

We obtain uf

= N(a)-2’e-2”is(*g)agaZ

(g E t”,> .

be a discrete subgroup 4.4 Definition. Let r c SL(2,R)” f : H” + C be a holomorphic function with transformation

f(Mz) Z(hf,Z)

= Z(M, z)f(z)

for M E r,

=

(T

N(CZ

+

d)2r

E

z”)

We say that f is regular at the cusp K (vanishes (and then for each matrix) A E SL(2, the transformed function of 4.3. Notice.

fA

R)”

,

with cusp K and law

.

at K)

if GOT

some matrix

AK = 00,

is regular at co (vanishes at co) in the sense

If f is regular at the cusp 0;) , we denote f(o0)

:=

a0

the value of f at 00. It is not possible to define the value at an arbitrary cusp K.= A-+) b ecause fA(oo) depends on A. But if one changes A, the change in fA(oo) is only a non-zero factor. 4.41 Remark. (Same notation as in 4.4.) If f is regular at a certain cusp K (vanishes at n), then the same applies to each equivalent cusp.

This follows immediately from the independence of the choice of A in 4.4. If for example K N 0;) we may choose the transformation A, AK: = 00 in I?, which implies f = fA. 4.5 Definition. Let r c SL(2,R)” be a discrete subgroup such that the quotient (H”)*/r is compact. An automorphic form of weight 2r , r = function (1”l , . . . , T,,) E Z” with respect to I’ is a holomorphic

f:H” with

the properties

a)

f(M2)

b)

f is regular

If f vanishes

=

N(c.2

+ d)2rf(2)

(M

---+ C

E

r),

at the cusps. at all the cusps,

we call f a cusp form.

If r is commensurable with the Hilbert ber field, such an automorphic form form.

modular group of a totally called a Hilbert

is usually

real nummodular

Chapter I. Hilbert

48

It is of course sufficient of all the cusps.

to verify condition

Modular

Forms

b) only for a set of representatives

Notation: [I’,2r] = linear E Z”. (r1 ,...,r,) [I?, 2r]o

= subspace

space

of all

automorphic

forms

of weight

2r , r =

of all cusp forms.

Remark. If ITo c r is a subgroup offinite index, then rO satisfies the same assumption3 as I? (formulation 4.5). The cusps of I?,-, and I’ are the same and we have [r, 27-Ic [ro, 24

4.51

[r, 2rlo = [r, 2rl n [ro, 2ylo with r E Z”. In the definition of an automorphic form we only considered even weights. This is sufficient for our purposes, but seems to be unnatural. What is really necessary for the definition of an automorphic form? Of course one first needs an automorphy factor, i.e. a mapping Z:lTxH” which is holomorphic

--*

C-(O)

in the first variable and satisfies the cocycle condition Z(MN,

%) = Z(M, iv%) . Z(N, 2) .

But in the definition of regularity at a cusp we need the definition of Z(A, z) for a larger set of A’s, because we have to transform an arbitrary cusp to infinity, which is usually not possible in T. We usually need a larger group G r c G c SL(2, R)n to transform

arbitrary

cusps to 00.

Examples: In the case of Hilbert’s modular group rK we can take G = SL(2, K). We need an extension of the automorphy factor Z to a mapping

Z:GxH”

-*

C-(O)

which is holomorphic in the second variable. But it does not need to satisfy the precise cocycle condition. It is sufficient to have Z(AB, z) = const(A, B) .Z(A, Bz) .Z(B, z) with some non-zero constant (A, B E G). In order to get a Fourier expansion at each cusp the following satisfied: For each A E G there exists a sublattice of finite index

1c

tA

such that Z(M, %) = 1

condition

should be

$4 Automorphic

49

Forms

if

Mz=z+l, Example:Letr=(ri,...,

r,)

be a vector

~~~~

by the main

branch

1~1,

+ d)v

of the logarithm.

=

group

I such

of rationals

e2*iCrj

(not

-----*

necessarily

integers).

We define

l"g(cjzj+dj)

It is sometimes W: r

for a given

MEG.

possible

to define

a mapping

C-(O)

that

Z(M, z) = N(cz + cl)’ . v(M) is an automorphic factor (if r is integral one can take any character of F for v , for example w E 1. For arbitrary rational T the conditions are more involved. Such a system v(M) is called a multiplier system of weight r with respect to I’. In the case of certain subgroups of Hilbert’s modular group multiplier systems of weight (l/2,. . . , l/2) are known. A general theory of these multiplier systems seems to be difficult. In the paper “Automorphy Factors of Hilbert’s Modular Group” (Proc. of the International Colloquium on Discrete Subgroups of Lie Groups and applications to Moduli, Bombay, January 1973) it is proved that each automorphy factor of a group commensurable with a Hilbert modular group is of the form

Z(M, 2) = v(M) ho h(r) N(C%+ d)’ , r E &” , I w(M) I= 1, where

h is a holomorphic

function

without

zeroes.

The

denominators

of f are bounded.

Modular Forms of Weight 0. Let I? C SL(2, R)” be a discrete subgroup with cusp 00. Let f(z) = c age*“-) gw be a Fourier series which converges in some domain UC = {z E H” , Ny > C}. We assume V to be a cusp sector at 00. In V we have an estimate 6-l m

2

Yj

2 6m

(6 some constant)

and therefore obtain

If, moreover, f is invariant under the whole stabilizer I’m, we obtain

without

any further restriction on z. As a special case we obtain

Chapter

50

I.

Hilbert

Modular

Forms

Let f be an automorphic form of weight 0 = (0,. . . ,O) with to r. Then f is r-invariant and therefore defines a function on which we denote again by f. This function extends continuously to

4.6 Remark. respect

H”/I’ (H”)*/I’.

Proof. The preceding remark shows that f extends continuously to 00. For Cl the other cusps one uses the technique of “transformation to 00”. For the rest of this section we assume that is compact. An important corollary of remark 4.6 is

4.7 Proposition. Each

automorphic

Proof. It follows from 4.6 a cusp form, it has to be follows that f has to be a case we denote the values that

form

the extended

f of weight

0 is constant.

that f attains its maximum in (li”)*/r. If f is attained in Hn. From the maximum principle it constant (which is equal to zero). In the general of f at the cusp classes by bl, . . . , bh . We notice

is again an automorphic form of weight 0. It vanishes, form. We obtain that f is one of the constants bi.

because

Now we want to investigate the effect of the multipliers coefficients of an automorphic form f(z)

=

C

age2niS(g+)

gEtO of weight

(2~1,

, . .

,21”,).

Let & be a multiplier, z -

is contained

in r for some b. From f(ez

i.e.

Ez+b the equation

+ b) = ET’.’ - . . . .E;‘”

f(z)

=

N(P)f(z)

we obtain as

=

age2’%7b)NEr

especially

We give two applications.

(H”)*/l?

quotient

The

first one is obvious:

7

it is a cusp I7

on the Fourier

§4 Automorphic

Forma

51

4.8 Remark. If f is an automorphic Tl

form, =

. . .

=

but not a cusp form,

then

T,.

The second application is the so-called GMzky-Koecher states that in the case n 2 2 the regularity condition be omitted in the definition of an automorphic form:

principle which at the cusps can

Let be I? c SL(2, R)n b e any discrete subgroup with cusp co. As usually we denote by t the translation lattice and by A the group of multipliers.

4.9 Proposition.

Let 12 2 2 and f(z)

=

C

age2niS(gr)

gw be holomorphic and periodic (with respect to t) on some domain Ny > C}. We assume that there is an estimation

1ag 15 Il$A with

some constant

)

1age1 for E E A

A. Then

Corollary. In the case n > 2 the regularity an automorphic

UC = {Z

condition

b) in the definition

of

(4.5) can be omitted.

form

Proof. Let g E t be a translation such that E such that &l > 1,&Z
gr < 0. We choose

a multiplier

< 1

and obtain -S(gP)

21 gl 1 -ET + C

(m E N)

where C is independent of m. From the absolute convergence series of f we obtain the convergence of the subseries

This

obviously

implies

ag = 0.

of the Fourier

El

Chapter I. Hilbert Modular Forms

52

Some important results about the spaces [l’, r] are based on the following remark which is an immediate consequence of the formula lmM% 4.10 Remark. The function

=

, I”,“,

Let f be an automorphic gk)

is r-invariant

,2 )

and therefore

it4 E SL(2, R) .

form of weight 2r with respect to r.

= I f(z)

I *NY’

defines a continuous Hn/I’

which we also denote by g. If f function on the compactification

id

+

function

R,

a cudp form,

then g extends to a continuoud

(H")*/r. The values at the cusp classes have to be defined as 0.

Corollary. Proof.

If f id a cusp form,

The only fact which

g attains

remains

a maximum

to be shown

in Hn/l?.

is that

if f is a cusp form and if 0;) is a cusp of r. As g is I’,-invariant, sufficient to take the limit in a cusp sector. We claim that the series

converges uniformly have an estimate

in a cusp sector. 2HLw)

Ny’ewith

a certain

positive

where

The reason is that in a cusp sector we <- e--2-(54

bound 6 and that c

converges the limits

1ag 1 e--2~6Sw

(consider f (i6, . . . , is)). The uniform term by term. A similar estimation

convergence shows

Ny tends to 00 and iy varies in a cusp sector.

An important

it is

application

of remark

4.10 is

allows us to take

cl

$4

Automorphic

53

Forms

4.11 Proposition. Let I’ c SJ’L(~,R)~ be a discrete subgroup such that (H”)*/I’ is compact. We assume that r has cusps (‘i.e. H”/I’ is not compact). Let f be any automorphic form of weight 2r r-1 - . . . - rn = 0 ,

r # 0,

Then f vanishes identically.

Corollary. Each holomorphic diflerential fovm on H” which is invariant with respect to I? and whose degreep satisfies O
g(z) = IfMI

f

is a cusp

WY)’

attains its maximum in a certain point z(O) E Hn. We may assume rr = 0. Then the function f(z)(iVy)’ is holomorphic in zr and we obtain from the maximum principle that z1 t---+ f(Z1,Z$O),...,Z~)) is constant. Let a be an element of the translation lattice t. (We may assume that 00 is a cusp of I’.) We obtain f (z(O) f(Z1,Z~),...,Z~)) 1 ,*--> z(O))= n = f(z1

+a,22

(0)+u2,...,Zp+a,)

= f(z1 >z$O)+ o&,***,z, co) + a,) . We now have to make use of the fact that the image of the projection t -----t frl 0, -

(U2,...,%)

is dense in Ff”-l (AI.19). We may conclude that f only depends on the imaginary part of z. But f is a holomorphic function and hence constant. The constant has to be 0 since f is a cusp form. 0 The corollary of 4.11 will be basic in the determination of the cuspidal part of the cohomology of H”/I’. This will be done in Chap. III where an introduction to differential forms is also given. A holomorphic differential

Chapter I. Hilbert Modular

54

Forms

form w is an expression w =

C

.fi, ,...,i,dzi, A . - * AdZi,

l
where the components of w means

fil,...,ip

fit,...,&

(Mz)

are holomorphic

fi

dMi,

/dz

=

functions.

fi,

,...I i,(z)

The r-invariance

.

v=l

Because of dM(z)/dz

=

(cz + d)-2

this means that f is an automorphic form of weight (~1,. . . , rn) where 0 forj Tj

@{ii,...,iP}

= { 2

elsewhere . El

In the cases 1 5 p 5 n - 1 we can apply 4.11.

It is natural to ask whether 4.11 is also true in the (usually simpler) case where H”/I’ is compact. To clarify this, we define 4.12 Second Condition of Irreducibility. n projections Tj

(cancelling

: SL(2, R)” +

off one component)

The

image of I’ under each of the

SL(2, R)n-l is dense in SL(2,

, 1 5 i 5 72, R),-l.

Of course the Hilbert modular group satisfies this condition (because of AI.19,2)). But there are also groups I’ with compact quotient satisfying this condition. An argument similar to the one in the proof of 4.11 shows: 4.13 Remark. The statement crete subgroup with compact of irreducibility (4.12).

of 4.11 is also true if I? c SL(2,R)” quotient H”/I’ if it satisfies the second

is a discondition

In the next section we shall construct for each I’ a non-vanishing cusp form h of a certain weight (r, . . . , T), T > 0. Assuming this for a moment we obtain 4.14 Lemma. Let f be an automorphic form of a certain weight (~1, . . Assume that the irreducibility condition 4.12 is satisfied if I? has no If one of the components Tj is negative, f vanishes identically.

. , T,). cusps.

$5 Construction Proof.

of Hilbert

We apply

Modular

55

Forms

4.7, 4.11 or 4.13 to the automorphic

$5 Construction

of Hilbert

form

Modular

f’

- h-5.

0

Forms

I Construction of Cusp Forms. Let D c Cn be any (connected and open) domain and l? c Bihol(D) a subgroup of the group of all biholomorphic mappings from D onto itself. We assume that l? acts discontinuously (in the sense of 1.2). The group l? is countable, because D can be written as the union of a countable set of compact subsets. Basic for the construction of cusp forms is the following

5.1 Lemma. Let f:D be a holomorphic

converges measure).

function

(dv = dxl The series

converges

. . . . . dx,,dy,

uniformly

In the following

the Jacobian of an element of automorphy, i.e.

. We first = 2. We want

Proof

If(rzYIj(r7

is satisfied.

for

the usual

Euclidean

zl’

subsets

-72,z)

notice that to compare

2) =

GOT

T 2

2.

det(ayi/dzj)

y E Bihol(D).

=

each holomorphic

By the chain

j(n,72z)

rule

this

is a factor

*.i(rz,z).

it is sufficient to prove the lemma in the case the series with an integral. For this we need

5.11 Lemma. Let K c D be a compact such that

denotes

by j(r,

T

. . . . . dy,

on compact

we denote

j(-/l

c

such that

c -lathen

-

function

subset. There exists a constant : D + 43 the inequality

f

C

56

Proof.

Chapter

We choose a number Ur(u)

I.

Hilbert

Modular

Forms

r > 0 such that :=

{z E C 1 112- ull < r}

is contained in D for every a E K. It is obviously sufficient to prove 5.11 for Up(o) instead of D and {u} instead of K. We further may assume a = 0. We develop f into a power series

f(z) = c uj, ,...(j”Z:‘1 f . . . * zk . If one uses J za+pa<,a

z?

dxdy

= 0

for p # v ,

one obtains J, (o) lf(z)j’dv ,

=

c jlail...in~:‘l - . . . . z$ 12dv ,...&I U,(O) 2 (7rr2)n. laoI = (7rr2)R. If( . .

I1

To prove Lemma 5.1 it is obviously sufficient for each a E D an open neighbourhood U(u)

(because

El

of 5.11) to construct

C D such that

l.fh4121~h 412dv CJU(a)

7Er

converges. We choose a neighbourhood U(u) with the following two properties (compare 1.7)

b)

7 Era -

r(wJ)>= U(a)-

We now notice that Ij(r, z)I 2 is the real functional determinant of the transformation y. (If A : C” + Cn ’is a C-linear mapping with determinant det A, then I det Ai2 is ’ the determinant of the underlying R-linear mapping (C” E R2”)). By means of the transformation formula for integrals we obtain

I (#L)

J

* D lf(z)l”dv

< 00.

0

We want to apply Lemma 5.1 to the Hilbert modular group and therefore need examples of square integrable holomorphic functions on H”.

$5 Construction 5.2

Remark.

of Hilbert

Modular

Forms

57

The integral dv JH -IZ+a14

(dv=

dXdY)

converges GOTeach a E H. The proof can be done by direct calculation. Another way to seeit is to make use of the fact that the area of the unit disc E is finite and to transform this area to anintegralon H bymeansofthe transformationz H (z-u)(z-ti)-‘. An immediate consequence of 5.1, 5.2 is 5.3 Proposition.

Let I? C sL(2,

R)”

be a discrete

cp:H” a bounded

holomorphic

function

--t

subgroup,

C

and

w E H” ,

T

2 2 (T E z) .

The series

F(z)= q(z) = MEr c

cpGw N(Mz

- Fi?)WV(cz

+ d)2r

converges absolutely and uniformly on compact subsets of H”. It therefore represents a holomorphic function on H”. This function has the transfoTmation law F(Mz) = N(cz + d)2’F(z) fOT M E r .

Remark: There exist many bounded holomorphic functions on Hn, namely: Consider a biholomorphic mapping MO : H” N

E”

and a polynomial P(zr , . . . , zn). Then P(Z) = P(Moz) is a bounded holomorphic function on H”. For the proof of 5.3 it remains to show the transformation easiest way to prove it is to use the formula

fl(M - W = (fIWIN > where (flM&)

= MC

~)-~.Wfz)

.

law. The

Chapter I. Hilbert Modular Forms

58

Here j(M, z) denotes any factor of automorphy. The series considered here are of the type F = C.wf and we obtain El Series of the type

especially the series considered in 5.3, are called Poincard series. They give us non-trivial examples of Hilbert modular forms. We are now going to show that the Poincare series (5.3) vanishes at the cusps. It is sufficient to treat the case of cusp co: $rnm

c

-

MEI-

IN(Mz

-E)

aN(cz + d)(-”

= 0.

The seriesunder the limit has a remarkable symmetry. It remains unchanged if one interchanges z and w. This follows from the trivial 5.4 Remark. Let be ESL(2,R),

XL

We have -(Ah

- ?i?)(cz + d) = (%w - z)(Ew + 2).

It is therefore sufficient to take the limit N(Im 20) +

00 for fixed z

(instead of Ny + co). If w varies in a certain cusp sector V (which is sufficient for our purpose), each term of the series tends to 0: Imw li,mwEV N(Mz - I

- E7)-’ = 0.

We have to verify that formation of limit and summation can be interchanged. This is an immediate consequence of 5.5 Lemma. For each C > 0 there exists a S > 0 such that Iz+w12

2 61z+i12

$5 Construction

of Hilbert

Modular

59

Forms

GOT all z E H and

Imw>C,

WEH,

(Rew[ < C-r.

The proof is an elementary exercise and can be left to the reader. The lemma shows that our seriesconverges uniformly if w varies in a cusp sector, because it can be majorized up to a constant by its value at w = i. 5.6 Proposition. Let I? c SL(2, R)n be a discrete subgroup such that (H”)*/I’ is compact. The Poincare’ series F(z) defined in 5.3 represents a cusp form of weight 2(r,. . . , T). If p # 0 and w E H” are given, there es& infinitely many T GOT which F$) does not vanish. The only thing that remains to be proved is the statement about the nonvanishing of F. This is a consequence of an elementary lemma whose proof is left to the reader. 5.61

Lemma. Assume

converges absolutely finitely

many

that (a,) is a sequence of complex numbers such that

GOT all T E N and such that

the value

is 0 foT all but

T. Then 0

a n=

fog alln

.

It should be mentioned that this type of argument can be refined to prove the following Existence Theorems. I) Let a, b E Hn be points which are inequivalent with respect to I?. There exists a Poincare’ series F (hence a cusp foTm) of suitable weight

such that F(a)

II) There

= 0 ,

F(b)

= 1.

exist n + 1 Poincate’ series Fo,...,Fn

of a suitable common

weight,

which

are algebraically independent,

This means that each polynomial P(x0,. . . , x,) with the property P(Fo,...

vanishes identically.

,Fn)

=

0

Chapter

60

I.

Hilbert

Modular

Forms

II Construction of Non-cusp Forms. We assume that 00 is a cusp of the discrete subgroup I’ c SL(2, R)n. W e want to consider a series of the type c

N(cz

+ d)-2r,

r E N.

This series cannot converge if we extend the summation over the whole group I?, because for infinitely many M E I?, namely all M E roe, we have N(cz + d)--2r = 1 . From the chain rule we even may conclude: The expression Z(M, z) := N(cz + d)-2r depends only on the coset l?,M. J?.&,(z) =

We therefore define c

iv(cz + d)-2r,

MEI’m\I’

where the summation is taken over an arbitrary set of representatives of the cosets r,it4. We call this the Eisenstein series of weight 2r with respect to the cusp

mofr.

5.7 Lemma. Assume

that CCJis a cusp of r. The series

c (iI+ MErc., \r

+ q-

converges in H” for all u > 2. The convergence is uniform on each cusp of 00, especially on compact subsets of H”.

sector

We first notice: Assume that some point zo E H and a certain constant C > 0 are given. There exists a constant e > 0 such that Proof.

Icz + dl > EICZO+ dl for all real c, d and all z with 151I C)

y 2 c-’

.

The proof is easy (compare 5.5) and can be left to the reader. Assume that the series in 5.7 converges at some point zo E H”. The preceding remark then shows that the series will converge uniformly on each cusp sector (52) at co.

95 Construction We now such that

of Hilbert Modular choose

Forms

any open

61

U c H” with

subset M(U)W

for M E I?, unequal

to the identity.

N&M(z)) We now

choose

5 C

exists

a constant

C > 0 such that

of the cosets

I’,M.

apply

M we may assume

is contained

where

converges.

in a domain

B of the following

positive

for our proof

This

NEFCO,

that

A is a suitable Basic

NM,

H

follows

type

number.

of convergence

from

is: The integral

the fact that

the integral

1

J

y”dy

,

a > -1,

0

converges.

We may replace

B by the smaller

domain

u M(U)

MEM

and obtain

converges.

that

the series

We now

change

the variables

in each of the integrals

z = Mw. The

formulae

dMfdw

in H”

MEI?.

forzEU,

a set A4 of representatives

closure

0

= There

compact

= (cw + d)-2

As we may

Chapter

62

and Imz

=

I.

Hilbert

Modular

Forms

Imw 1cw + d I2

show that the volume element dv/(Ny)2 is invariant under SL(2,R)n also 11,l.l) and that the above series equals

(see

(ivy)“/2 dv

I wcz

+ 4 In 02

*

are bounded on the compact set n

The functions Ny, Ny-l therefore obtain that c

/

MEM

1 N(cz

+d)

I-”

C

Hn.We

dv

u

converges and this implies that

c 1N(cz+d)I-” (a>2) MEM

Cl

converges on U. 5.8 Proposition. Let r c SL(2, R)” be a discrete Eisenstein

13~4~)

&,.(Mz) value

Proof.

a holomorphic

= N(cz + d)2’Ezt(z)

cusp 00. The

function

on

H”

.

at 00 is 1

= 1,

E2r(m)

but Ezr

with

= c N(cz+d>-2r MErc.,\r

converges for r 2 2 (r E Z) and represents with the transformation property

The

subgroup

series

vanishes at all cusps which The first

are not I’-equivalent to co.

follows from 5.7.

part

The value at 00: We have to show N15E24z)

=

1.

The limit can be taken within a cusp sector of 00. Because of 5.7 the limit can be computed term by term. We have lim

Ny+m

N(cz + d)-2r

=

ifc#O 1 ifc=O.

0

$5 Construction

of Hilbert Modular

The second assertion have M E I’,).

is true

since d2 is a multiplier

The behaviour at a cusp K. = A-loo We have to show that

(&.]A-l)(z)

vanishes

63

Forms

which

if c = 0 (in this

is not r-equivalent

:= N(cz + d)-2’E2,(A-1z)

with

case we 00:

,

at co. We may write

and therefore

E2,1A-1

=

c 1 1 MA-l MEroo\r

=

(~72&4-l)(z)

c

N(cz + d)-2’.

ME(ArA-‘),\(rA-‘)

The same argument as at the beginning of the proof of 5.7 shows that this series converges uniformly in a cusp sector at 00. It is therefore sufficient to show $mmN(cz + d)-2r = 0

-

(in a cusp sector)

or equivalently c#O

If there

with

were

for

ME I’A-‘.

some

c = 0, we would

M = NA-1

E ITA-’

hence

Nn = NA-loo

have

Mco=oo, i.e. IE N 00 mod

= 0;) ,

r.

0

How can we define an Eisenstein series with respect to an arbitrary cusp K: of I’? Such an Eisenstein series should vanish at all cusps which are not equivalent with K, but it should not vanish at IC. A natural procedure is 1) to transform

K to 00 , AK = 00,

2) to consider the Eisenstein conjugate group AI’A-l , 3) to transform

back this

series with

Eisenstein

respect

to the cusp co of the

series by means

of A.

Chapter I. Hilbert

64

Modular

Forms

The result is

EA= 2r

c

l(M-4,

c

or, explicitly

N(cz +d)--2r. ME(AI’A-‘),\Ar

acts on the set AI’ by multiplicaNotice. The group (AI’A-I), = Al?,A-1 tion from the left. Therefore we find that the set Al? is a disjoint union of is taken over a cosets (AI’A-I), . M , M E AI’. The summation in E&(z) set of representatives M. 5.9 Remark. Let K be a cusp of r.

Et-(4=

The Eisenstein

series

N(cz +d)--2f

c

ME(AIYA-l),\Ar

AK. = co) depends only (up to a constant factor) class of 6.

on the r-equivalence

(with

5.91 Corollary. Assume that there is the case if (Hn)*/r is compact) /cj =

exist only finitely and denote by

many

cusp classes (this

l<j
A+&

a set of representatives. The Eisenstein series E$$(z) are &nearly independent.

:=

Ez(z)

They generate

E = E(r) =

,

1 5 j 5 h,

a vector

space

&Ek) j=l

which

depends

on I? only.

We call & the space of Eisenstein series of l7. That E& depends (up to a constant factor) only on the r-equivalence class of K.is a consequence of the following two observations: Proof.

$5 Construction

of Hilbert

Modular

1)

EA2r

(because

65

Forms

= Ezt-

in this case AI’ = AI’A-1

if

A E I’

= I’).

2) Let 00 be a cusp of l? and assume ACCI = co. In this case we have (AI’A-l),

= AI&A-l

and therefore E2A, =

1IM

c MeAr,A-l\Ar

=

c

l/AM

MErm\F

= const - E2,., because 1 1 A = const 5.10

if

A(m)

cl

= 00.

Let r c SL(2, R)" be a discrete subgroup such that is compact. FOT each automorphic fomn

Proposition.

(H”)*/l?

there exists a unique element E in the space of Eisenstein series such that f - E is a cusp form, in other words [r,z(T

5.101

,...,

T)]

=

[r,z(T

,...,

T)]O

6%

corollary.

dim[r, 2(r, * * * , r)] = dim[l?, 2(T,.

. . ) T)]o

+

h

(h = number8 of cusp classes). Final J&mark. What happens in the case T = l? It can be shown that Eisenstein series of weight (2,. . . ,2) can be defined by means of the limit E&z)

= ,.I:+ li

qcz c MEI’m \r

+ d)-2* 1N(cz + d) y .

It is clear that this limit transforms like an automorphic form. But there is a great difference between the cases n = 1 and n 2 2.

Chapter

66

1) n 2 2: In this properties as formulated dim[I’,

I.

Hilbert

Modular

case &J(Z) is an automorphic form with above for &r(z) , r > 1. We especially (2,. . . ,2)]

We shall prove this later group (Chap. III, $4).

=

dim[I’,(2

for congruence

,...,

Forms

the same have

2)]e+h.

subgroups

of the Hilbert

modular

2) n = 1: In this case Es(z) is usually not a holomorphic function (but a non-analytic automorphic form in the sense of Ma&). If the set of cusps is not empty in the one-variable case, the equation dim[I’,2]

=

dim[I’,2]s

+ h - 1

holds.

$6 The Finiteness

of Dimension of a Space of Automorphic Forms The aim of this lowing

section

is to give a short

and elementary

proof

of the fol-

such that the ex6.1 Theorem. Let I’ c SL(2, R)” b e a discrete subgroup is compact. The dimension of the space [I’,2r] of tended quotient (Hn)* /I’ automoTphic forms of a given weight r , r E Z, is finite. In the following

proof

Assumption. Hn/I’

we make

is

the further

not compact,

i.e. there

exist

cusps.

The case of a compact quotient is easier and we make some comments at the end of this section how to modify the proof in this case. The proof will result from the comparison of two different norms on the space of cusp forms [r, 2do.

Norm 1: We have shown

that

the function

g(z) = I f(z) I NY’ is I’-invariant

and attains

Norm 2: We choose

its maximum

a set of representatives

in Hn. We hence

may define

a norm

$6 The

Finiteness

of Dimension

of a Space

of Automorphic

Forms

of the cusp classesand transform them to infinity AjKj

= 00,

l<j
AjESL(2,R)n.

We consider the lattice tj

C R”

of translations of the conjugate group AjI’Ar’ and denote by Pj C R” a fundamental parallelotope of tj. Let 6 be a positive number.

Notation: Vj(S) = {A;‘(Z)

Remark.

1 2 E Pj , ~1 > 6,. . . , yn 2 6).

1) If 6 > 0 is small enough, the set V(6)

= I@)

u . . . u v-h(6)

is a fundamental set of I?. 2) The integral J vjt6) (dv = dz1 * . . . - &

h,

= -($2

*

= Euclidean measure) converges.

Proof. 1) follows from 2.11 . 2) The volume element d&J = dv/(ivy)2 is invariant with respect to transformations z I---+

Mz,

M E SL(2, R)n.

We hence have to prove the existence of

J

rE

Yj>6

compact set for l-Cj
dw,

and this is a consequence of the convergence of

Jrn 6

y-2dy

(6 > 0).

67

Chapter I. Hilbert Modular

68

Forms

Let

f,g E KM0 be two cusp forms of the same weight. The function dz)

= f(M4

* NY2’

is r-invariant and bounded. We hence may define for each 6 > 0 the Hermitean inner product

a=

J v(6) (P(z)* *

We obtain a family of norms llfll2,6

=

+Am=.

(If S is small enough in the sense that V(6) is a fundamental set, all the norms IIf 116axe equivalent. They are in fact equivalent with the norm deduced from the so-called Petersson inner product:

dz)* JH”,r

=

which will play a basic role in Chap. II (see 1.1). The equivalence of all these norms is a consequence of the finiteness property of V(6), i.e. the set of all M Er )

M(V(6))

l-l V(6) #

0

is finite. We do not need this and omit a proof.) We now come to the announced comparison of norms. 6.2 Lemma. that

If 6 is small enough there exists a constant A = A(&I’,r)

llfll-

I Allf II%6

for al2 f

E

[r,

2~1~ (T- E Nn fixed).

We assume for a moment that the lemma has been proved and show Lemma 6.2 _

Theorem 6.1: Let fl,...

,fm

E

w40

be a system of orthonormal vectors with respect to < ., . >6, i.e. <

fi,

fk

>6

=

bik

-

such

§6 The

Finiteness

of Dimension

of a Space

For an arbitrary

of Automorphic

Forms

69

m f

=

CCjfj,

CjEC,

j=l

we obtain from the lemma

If we specialize we obtain

Ol-

C I fj(z) I2Ny2r

5 A2 -

Integrating along V(6) with respect to the measure dw we obtain m 5 A2-

J V(6)

old < co.

Proof of Lemma 6.2. We choose 6 > 0 small enough such that V(2S) is still a fundamental set. The function

h(z) = I f(z) I (NY)’ is l?-invariant. It is hence sufficient to prove

for all 2 E V(25). W e p rove a little more, namely that for each j E { 1,. . . , h}

h(z) 5 A

h(C)“& JJb(6)

for all z E Q(26). It is of course sufficient to consider the case of cusp infinity, i.e. we may replace Q(6) by

where P is a fundamental parallelotope of the translation lattice of I’-. We now compute the integral under the root sign by means of the Fourier

70

Chapter

expansion

I.

Hilbert

Modular

Forms

of f: f(Z)

=

Cuge2*iS(gr)

.

LJ>o A simple

By means

calculation

gives

of the inequality co 2 T!. ,-a6,-(‘+1)

e -‘“yyr(jy J6 (integration

by parts!) h(z)2du

(*)

we obtain

2 vol(P).(2r-2)!.C

1 ag I2 e-4K6S(g).N(4?Fg)-2r+1.

L(6)

On the other

hand

we obtain h(z)

The

the estimation

Cauchy-Schwartz

w We now

5

C

inequality

1 ag 1 e-21rS(gy)(Ny)T. gives us

5 [ccl % I e- 3/2 *S(d)2] 1’2. [x(,-4

S(m+vy)‘)2] 1’2 .

assume z E V&26)

E Vm(S))

(not only

and obtain

with

5 B *&

1a, 12e--6x6&J)

B. If we apply

2.91 to the set

h(z)

(**) a certain

constant

(a, b, c, d E to> , we obtain

that

INgl

h as a positive e--6aWd

(C a suitable inequality.

constant).

<-

lower

bound.

This

implies

ce-4”6S(g)N(4ng)-2’+1

Comparing

(*)

and

(**)

we obtain

the

desired 111

$6 The Final

Finiteness

of Dimension

of a Space

Remark. (case of compact

of Automorphic

fundamental of all discs

with a E K is still above

subset

and the Fourier

expansion

in a compact

off

by

K

V(6)

by

z

by the Taylor

(even

analogous

to 6.2.

easier)

calculation

will

give

set K c

E of H”.

V(26)

w=A similar

71

quotient)

If HR/r is compact, we can find a compact find a positive number such that the union

contained

Forms

expansion z-a Z-ii the inequality

with

H” . We furthermore

We now

respect

replace

can

in the proof

to the variable

Chapter

II.

31 The Selberg

Dimension

Formulae

Trace Formula

We are going to express the dimension of the space of cusp forms of weight 2(r,. . . , r), T > 1, by a certain integral along a fundamental domain of the given group I’. The function to be integrated is an infinite series derived from the Poincar6 series considered in Chap. I, $5.

Basic for the trace formula

is the so-called kernel

( > =qzYo.)

1.1 Remark.

The kernel function

-2

--.I

9

2i

j=l

on Hn:

-2

Z-F 7

k(z,w)=N

function

*

has the transformation

property

where j(M,

z) = N(cz

+ d)-2 .

The measure dw,=k(z,z)dz=* is invariant

(NYJ2

under the transformations ZHMZ,

(dw, := dz1 . . . dz,dy1

M E SL(2, R)” .

. . . dy, denotes the usual Euclidean

The proof of 1.1 is trivial.

measure.)

Chapter

74

In the following

we denote

Dimension

Formulae

by

Lc, = LC,(H”) , the linear

II.

space of all holomorphic

r EN,

functions

f:Hn+C such that If(zwYY is bounded. erty.

We notice

that

1.2 Proposition. Assume

= If(z cusp forms

ZP of weight

2(r,.

f

r 2 2. Each function

E L,

. . , r) have this

satisfies

prop-

the integral

equation

(The

integTa1

is absolutely

convergent.)

Proof. If f E C,(H”), then f is contained in &(H) as a function of each of its variables. It is obviously sufficient to prove the integral equation in the one-variable case. This will also be true of the proof of convergence. We therefore assume n = 1. disc)

We transform the integral equation by means of the transformation

into

the bounded

H-E w I-+ 7 = (w - z)(w -q-l (z E H is fixed). The

inverse

transformation

is y H w = (z - Zq)(l

Because

- 7$’

.

of dw/dq

the Euclidean

On the other

volume

hand

= 2iy(l

elements

- q)-’

transform

(y = Im z) like

we have Im w = y(l

- l~j2)11

-

rllm2

model

E (unit

75

$1 The Selberg Trace Formula and hence

the invariant

volume

element

introduce

like

4dv,

d&&HdL+:= We now

transforms

(1 - lr112)2.

the function s(7)

= Cl-

(rl E E) *

d-2’f(+?))

We have 969 and a straightforward

calculation

= f(z)

>

gives us the transformed

9(O) = y

JE(1

- Isl”>“‘9(s>

integral

equation:

d%

or 9(O) = F

(dv, = Euclidean The

function

volume

JE(1

in L, if the function s(rl)(l

The

integral

d%

element).

f is contained

is bounded.

- ld2>2’-29(~)

converges

- ld2>’ if

J

(1 - lqj2)P-2dvq < 00.

E

If we introduce

polar

coordinates

q = pe'Q , dv, = Pdpdv, this

turns

out to be equivalent

with 1

J0

(1 - p2)r-2pdp

< 00

This integral can be evaluated after the transformation u = p2 (dn = 2p dp), and we obtain that it converges for r > 1. For the proof of the integral equation we make use of the power series of the holomorphic function g:

9(v)

= 2 amrIm m=O

Chapter

76

II.

Dimension

Formulae

Once again we make use of polar coordinates and first integrate along the angle ‘p (0 5 ‘p 5 27r) for fixed p < 1. We may integrate term by term, because the power series converges uniformly on compact subsets of E. But obviously

For this reason functions g. We have !!$

the integral

&-

equation

~v~2)2r-2dv9

has only to be proved

= 2(2r - 1) I’(1

- p2)2r-2pd~

= (2r - I) I’(1

We now consider a discrete subgroup of the kernel of the natural projection r + (SL(2, (Two

I?

c

elements of l? define the same transformation

From the integral

equation

- a)“-‘da

SL(2, R)“.

R)/{fE})”

for constant

= 1.

0

Let 1 be the order

. if their images coincide.)

1.2 we may deduce

We can replace Hn/I’ under the integral sign by a fundamental domain of I’. The convergence of the inner series outside a neglectible set is a consequence of general facts about integration theory (AII.7). We shall obtain better information about the convergence without using this below. We now assume that the extended space Xr = (H”)*/I’ (I, ‘$2) is compact and that f is a cusp form of weight 2(r,. . . , r) with respect to r. It follows from 1.4.10 that f is contained in C,, and we may apply our integral equation to f. If we use the formulae

fpfz) (j(M,

= j(M, r-f(z) z) = N(cz

+ d)-‘)

and k(Mw,

Mw)

= )j(M,

w)l-2+,

w> ,

$1 The

Selberg

Trace

77

Formula

where K(z,w)

= Ky&,w)

= 2-l c k(Mw, z)‘j(M,w)’ MEr 1 = @>2rn c 1 MEr N(Mw - f)wv(CW + d)2r

This type of series - a so-called Poincard series - has been introduced Chap.1, $5. Prom 1.5.3, 1.5.4 and 1.5.6 we know 1.3 Proposition.

in

The function

K(z,w)= Kr,,(z,w)= 1-l c lc(Mw, z)‘j(M,w)’ MEr -2

(k(W,Z) = Iv y (

) j(M, to) = N(cw + d)-2)

>

is (for fixed z) a cusp form of weight 2(r,. . . , r) (r 2 2) as a function It has the property K(z, w) = Iqw, z) . If f is a cusp form

of weight 2(r,. . . , r) we have

f(z) = (y)’

We now introduce We first notice

of w.

JHnIr f(g+jrW) 7

a Hermitean

inner product

1.4 Remark. The quotient space H”/I’ the invariant measure dw. (We assume that (H”)*/I’ is compact!) hoof. It is sufficient to construct volume. (AlI.8, AlI.10)

in the space of cusp forms.

has a finite

a measurable

dw, .

volume

fundamental

with

respect to

set with

finite

We use the fundamental set constructed in Chap. I, 52 (2.10) and have to show that each cusp sector has finite volume. We may restrict ourselves to the cusp cc and have therefore to show dv J v WY12

coo

where V is a cusp sector at 00. Prom the definition that there exists a constant 6 > 0 such that

of a cusp sector it follows

Chapter II. Dimension

78

b)

forj

Yj L 6

Formulae

= l,...,n

for z = 2 + iy E V. The convergence

of the integral

now follows from the convergence of

O”dy T
(6 > 0) .

cl

We now consider two cusp forms f, g of a certain weight 2r, r = (1’1,. . . , rn). The function fmw(~Y)21‘ is I’-invariant

and bounded (14.10). We may conclude (1.4) that the integral

(f,9):=in,,

fws(4PY)2’~~

exists.

1.5 Remark. of cusp forms

is a Hermitean inner product on the space [I’, 27-10 of an aditrury weight, i.e.

The pairing (f,g)

u)

(f,g)

is C-linear

in f ,

b)

(f, 9) = (9, f) ,

c)

(f, f) > 0 for f # 0.

this inner product we may rewrite the integral equation for cusp forms (1.3) in the form

Using

( f(z)= ($g.

(f,wvN (

We now choose an orthonormal basis fi, . . . , f,,, of the space [I’, 2(r,. . . , r)]s. (We have already proved that this space is of finite dimension, 1.6.1). (fi,fj)

=

{

1 for i = j 0 for i # j .

We may express the kernel function w I-+ K(z, w) by means of this basis:

$1 The Selberg Trace Formula The

integral

equation

79

gives us

h(z)=(y)‘(h,K(z,4) =

(

2r--1 4T

n-

>

uitz>

7

i.e.

We specialize along H”/l?.

this equation (z = w), multiply The result is the “trace formula”

it with

(NY)~’

1.6 Theorem. Let I? c SL(2, W)n b e a discrete is compact.

Let

T

dim[I’,

2 2 be a natural

2(r,. . . , T)]O =

such that (H”)*/lY

subgroup we have

numbeT.

and integrate

(%yJ,.,,~~,

where qz,

to) = 1-l

c

k(Mw,

MU

z)‘j(M,

lo)’

)

-

k(w,z)=N

y -2 ) j(M, 20) = N(ct0 + q-2 ) ( > du = k(z,z)dw = -+-

(dv = Euclidean

I is the order

of the kernel

PYj2 volume

of the natural

element)

.

projection

I +

(SL(2,

R)/{fE})“.

Notice: Two elements M, N E l? with the same image define the same term in the series of K(z, w). The trace formula concerns rather the image of I? (i.e. the underlying group of transformations) than I itself.

The Main Term of the Trace Formula. In the series defining tract all terms We obtain

M E l? which K(z,

belong

to the kernel

z) = qz,

z)’

+ K’(z,

of I? --f (SL(2,

K(z, z) we exR)/{fE})“.

z) )

where K’(z,z)

= z-l

c MET Mf

identity

transPormation

k(Mz,

z)‘j(M,

z)’

Chapter

80

The trace formula

can be written

dim[I’,

II.

Dimension

Formulae

as

2(r,. . . , r)],, = vol(Hn/l?)(2r

- 1)” + A(r)

,

where

4 denotes the (47p oh.

volume

vol(Hn/I’)

= (47r)+

of H”/I’

with

J,-,,

respect

clw to the invariant

We shall see later that A(r) plays the role of an error dim[r, 2(r,. . . , r)lo N vol(Hn/r)(2r

term,

measure

i.e.

- 1)” .

In this connection we would like to make some general remarks. The Selberg trace formula 1.6 can be generalized to an arbitrary arithmetic group T (instead of Hilbert’s modular group) acting on a bounded symmetric domain D c C” (instead of H”). One has to replace k(r, w) by the Bergman kernel function and j(7, Z) (7 E I’) by the Jacobian. Instead of (2~ - 1)” there occurs a certain polynomial a(r) which is characteristic for the domain D. A cusp form of weight P E Z is a holomorphic function f : D + C with the transformation property f(7z)

= j(7,

v-m

which vanishes at the cusps. There are two different the space of cusp forms [T, r]s, namely a)

the Selberg

trace

b)

generalized

Riemann-Roth

Both methods was Langlands,

have who

methods

to calculate

the dimension

of

formula, theorems.

been applied successfully to the case of a compact quotient did that in the first case and Hirzebruch in the second.

The case of a non-compact quotient D/r “Riemann-Roth method” Mumford proved dim[T,

r]s = a(r). S(r)

is much

more

vol(D/I’)

+ S(r)

= qm-‘)

involved.

By

means

D/I’.

It

of the

,

.

One may expect that this result admits the following improvement. Let Xr be the BailyBore1 compactification of D/l? [4] (which g eneralizes our compactification in case of Hilbert’s modular group). Let S be the locus of all elliptic fixed points and all boundary points. We expect S(r) where

d is the maximal

dimension

= Cl(+)

of an irreducible

component

of S.

$2 The

Dimension

Formula

in the Cocompact

$2 The Dimension

81

Case

Formula

in the Cocompact

Case

In this section we assume that l? c SL(2,R)” is a discrete subgroup with compact quotient H”/l?. We also assume that I’ satisfies the irreducibility condition 1.2.13: Each of the 12 projections r + SL(2, R) is injective.

The trace formula

1.6 has the form

WC z> :=[k(Mz,z) qz, z)1rG42)’

where

-2

k(z,w)=N 9 ) ( > j(M,z) = qcz + d)-2) 1=

order of the kernel of the image of I’ in (SL(2,

R)/{fE})”

.

The series

converges uniformly on compact of conjugacy classes

subsets.

[M,,] := {M Mu M-l,

We now write

M E

l? as a disjoint

union

r) .

We obtain

where MO runs through a complete system of representatives of all conjugacy classes. We replace the domain of integration Hn/I’ by a precise (measurable) fundamental domain F. We may assume that F is contained in a compact subset of Hn. The uniform convergence on compact sets allows us to interchange mation and integration in the following way:

sum-

Chapter II. Dimension Formulae

82 We call the occuring to the trace formula.

integral the contribution of the conjugacy class [MO] We now simplify this contribution, We have MMoM-l=NMoN-’

if N-i

M is contained

in the centralizer

I’MO={M~rI or equivalently

MMo=MoM},

if MI’M~

Whence

= NI’M~.

we obtain c MEli%

where

of MO:

k(M,z)

M runs through

=

c MEFIrM,

a complete

set of representatives

MrMo, 2.1 Lemma.

The function

k(M,z)

of the cosets

MEr. satisfies

lc(M MO M--l+) We especially

k(M M,, M-$2),

the functional

= lc(Mo, M-‘2)

have that z I-+ k(Mo, z) is invariant

equation

. under

the centralizer

rMoe

For the contribution

of the conjugacy

class [MO] we obtain

J

.

k(Mo,z)dw

the expression

Hn/r‘Ug

The advantage now is that a fundamental domain of FM,, can be determined. For this purpose we determine the centralizer of an arbitrary element MO E SW R), MO # fE, Z(M,,)

:= {M E SL(2, R) 1 MM,,

= M,, M} .

Because of Z(M MO M-l)

= M Z(Mo) M-l

it is sufficient to restrict to a suitable system of representatives of the conjugacy classes. 2.2 Lemma. An arbitrary

element MO E SL(2,R),

MO # fE

$2 The

Dimension

is conjugate 1)

Formula

in the Cocompact

an SL(2,R)

with

a translation matrix (if

2)

la(Mo)l

= 2))

MO is hyperbolic,i.e.

lu(kfo)l

> 2))

an orthogonal matrix cos $0 ( -sincp

The

MO is parabolic,i.e.

a transuection matrix (if

3)

83

Case

centralizers

sin ‘p cos $0>

(if

MO is elliptic,

i.e. 10(MO)l

< 2).

are

z(;

z

;)={k(;

(

;)I

bE,>,

cos$0 sin ‘p = SO(2, R) . - sincp cos p >

PTOOf.

1) A parabolic transformation has precisely one fixed point in R = R U {m} and this can be transformed to 00. 2) A hyperbolic transformation has two fixed points in R which can be transformed simultaneously to 0 and co. 3) An elliptic transformation has a fixed point in H. We can transform this fixed point to i. Cl

The computation of the centralizers is trivial.

We now investigate a fundamental domain of l?~,, where I’ c SL(2, R)” is our discrete subgroup. For our purposes it is always sufficient to replace FM0 by a subgroup of finite index rlMo c rMo because of

I

H” PLO

kpo,

z) dw = [FM,

: rhOi

Ic(M,,, z) dw . J H”/rMO

Chapter

84

II.

Dimension

Formulae

We also notice: If we replace Ms by MMs M-l, M E SL(2, R)“, M I’ M-l, then the integral will not change. After assume

this preparation

n/i0 = (Mf), ---

we describe

. . . , MiL), Mi”+l), hyperbolic

and (after

and I’ by the group

the fundamental

domain.

We may

. . . , M$‘), M;‘+1),

. . . , M;“)) elliptic

parabolic

conjugation)

M,(j) E SO(2 3R) ,*

I < i <- n .

We now define a certain subgroup

of fmite index. Case 1: k + I = 0 (all the components In this case I’M0 is a discrete hence finite. We may take

are elliptic).

subgroup I&

of the compact

= {id} .

Case 2: k + 1> 0 . We denote by I’lMO the group of all M =(Ml,...,M,)

M~ESO(~,R); This is a subgroup We may identify

E I?

l
of finite index of l?MO. rtMO with

a discrete R”f’

subgroup

x SO(2, Fy-“-1

of

group

S0(2,R)“,

$2 The

Dimension

Formula

in the Cocompact

85

Case

by means of the imbedding M I+ (logm,.

. . ,logw,h+l,.

. . ,br,Ml+l,.

. . ,Mn).

The projection Rk+’ x SO(2, R)“-‘-’ is a proper in II’+”

mapping,

is a discrete

---f R””

because SO(2, R) is compact.

Hence the image of I”,o

subgroup L c Rk+’ .

We denote by P c R”+’

a fundamental domain of L. From our irreducibility that the projection defines a bijection

assumption we obtain

We now obtain that a fundamental domain of I”MO is defined by: F(Mo)={z=z+iy~Hn,

Q%Yl,...

,logYk,zk+1,...,~1)

EP)

We notice that we have no restriction for a) the coordinates $1,. . . , xk (hyperbolic components) b) the coordinates yk+r , . . . , yl (parabolic components) c> z1+1, . . . , zn (elliptic components). Hence we are led to the computation of the following integrals in the onevariable case: 2.3 Lemma. Assume n = 1. We have k(MO,

1)

z) dx = 0

a # fl (hyperbolic 2)

case).

J”

k(Mo,

z) $

= 2&P-’

0

MO=( ; ; > , P#O

Chapter

86

(parabolic

II.

Dimension

Formulae

case) .

3)

j@,/&z)y

JH

= L5’ 2r--11-c

if MO is elliptic with rotation factor of MO, then MO has the form

C (If a is the unique

elliptic

fixed point

w + cw in the coordinates

w = (z - a)(z

E E).

- a)-’

PTOOf.

1) We have -2r

cY2r . Therefore

the integral

is of the type

--oo(xFa,,r ’ a 4 I3’ Jrn which,

by the residue

2) The

integral

theorem,

equals

2)~Y/Y’ Jrn~(Mo,

=

3) We express

by means of partial of the second factor the integral

y2r-2(y + p/2iy

Jrn0

0

can be computed and integration

0.

integration (differentiation of the integrand).

by means

of polar

w = (2 - a)(2 - a)-l of the unit like

disc.

We recall

from

51 that

dxdy Y2 A simple

computation

J

k(Mo,

H

z)?

dy of the first

coordinates

= pe+

the invariant

measure

transforms

; , 4P dP & (1 - p2)2 .

now yields = 4

2n’(l-P2)2’-2 ~‘Pdpd~ JJ J 0 (1 - 5P2)2’ l (1- t)2r-2 dt =

0

= 47rcy

0

(1 - stjzr

47r

Gig’

5’

52 The

Dimension

because,

Formula

in the Cocompact

87

Case

if 2r-1 G@)

(2y

:=

-

I;(,

-

1)

(

::;t

’

>

then G’(t)

= (l - t)2r-2 (1 - ct)“r

0

*

2.4 Lemma. Assume that MO E I? has no fixed point in H”. Then tribution of the conjugacy class [MO] in the trace formula is 0:

I

f(Mo,z)dw

the con-

= 0.

I-I” /I’M,,

Proof. We may assume tion

of the lemma

that MO has the form is k + I > 0.

Case 1: k > 0 (hyperbolic

components

described

do occur:

The

above. assertion

The

assump-

follows

from

2.2,1).

Case 2: k = 0 (no hyperbolic, might

occur.

We are going

Case 2a: I < n (elliptic

but parabolic as well to show that this cannot

components

as elliptic happen).

components

do exist):

We first notice that the kernel function k(Mo, z) does not depend at the parabolic components. We hence obtain that the integral

on z

J

dxl . . . dxl

P

converges. especially

This means that have isomorpbisms

the discrete

r’MoN Remark. Two

different

elements

subgroup

L C R’ is a lattice.

We

Lrz’.

M, N

of rtM,

are not conjugate

in I’.

Proof. The last components of M, N are contained in SO(2, R). But two elements of SO(2,R) are conjugate in SL(2,R) if and only if they are equal (look at the fixed point i). From our irreducibility assumption we obtain M,=N,,

=G.

M=N.

cl

We also have

r’M=r’MofOrMErMo,

M#E.

Chapter II. Dimension Formulae

88 From the computation E:

of the integrals

2.3 we now obtain for M E l?‘Mo, M #

where 6 > 0 is independent of M. (The contributions of the elliptic components can be packed into the constant 6). We now obtain that the series

converges, which is contradiction, i.e. the case 2a cannot occur. Case 2b: (all the components of MO are parabolic): As in the case 2a, we see that the discrete subgroup L c R n is a lattice, now of rank n. The difference to the case 2a now is, that two different elements

M=(; f) , N=(; f) may be conjugate: PMP-l=N. The matrix

P is necessarily

upper triangular.

We obtain

p = ep where

E is a multiplier

of the translation

lattice

.sL=L,

L ,

e>O,

especially NE = 1. We recall that the group of all such multipliers

is a discrete

subgroup

A c CR+)” of rank

5 n - 1. As in the second case, we may conclude

c’ CZEL

mod

that the sum

INal-] A

converges. The summation is now taken over a complete system of elements a E L, a # 0, which are not associate mod A. But it is easy to show by comparison with an integral that this series cannot converge. 0 We have proved that only the elements of finite order in l? give a nonzero contribution. The identity gives the main term vol(Hn/I’)(2r

- 1)“.

$3 The

Contribution

of the Cusps

to the Trace

89

Formula

2.5 Lemma.

Let a run over a system of representatives of I’-equivalence classes of elliptic fixed points. Denote by Z c I? the 1 elements which define the identity transformation. Then

of r-conjugacy

is a set of representatives not contained in Z.

classes of elements

of finite

order

The proof is easy and can be left to the reader. We finally obtain b e a discrete subgroup that I? is irreducible (12.13). Then for

2.6 Theorem. Let r c ZJ’L(~,F?)~ compact.

Assume dh[r,

2(T,.

. . , T)]

=

VOl(Hn/r)(%

-

1)”

+

c

such that Hn/r 1 we have

is

T >

&(r,

(2)

,

a

where

a runs over a set of representatives

~43) = -& M

where

< denotes

the rotation

53 The Contribution to the Trace Formula

factors

not

of r-classes

of elliptic

fixed points

c MEro the identity

of M.

of the Cusps

In the following r c SL(2, R)n denotes a discrete subgroup such that the extended quotient ( Hn)* /I’ is compact. We assume that I’ has cusps, which means that H”/I’ is not compact. We also assume the first condition of irreducibility, i.e. the n projections

are injective. We choose a precise fundamental domain of the form described in Chap. I, 52: Let Kl,...,K/&

be a system of representatives of the cusp classes. We may choose cusp sectors K,...,Vh

Chapter

90

and a set K c H” which the union

is contained

is disjoint

F is a (precise)

and such that

in a compact

of all translations,

fundamental

by V,

1 ()

ikfj=*

cv = ((~1,. . . , o,)

of H” such that

domain

of I’.

a cusp sector

O!j

1

.

(

vector

Formulae

i.e. the set of all elements

M=(Ml,...,Mn); The

Dimension

subset

We now assume that 00 is a cusp of I? and denote infinity. We also denote by l?g’

the subgroup

II.

is an element

>

of the translation

lattice

t = t(r) . The

mapping

is a homomorphism phic to the group

with finite kernel. The factor group I’,/l$,’ A of multipliers. The isomorphism is given by Ml-+&,

3.1 Proposition.

is bounded

where

Mz

= cz + (Y .

The series

on V,.

(Recall: ‘t”,

3.11 Corollary.

ia termwise The

corollary

For the proof

‘> =

[

k(Mz,z) +, z)

1 )‘

j(M,z)’

>

The integral

integrable. follows

from

of 3.1 we need

the Lebesgue the following

limit

theorem

Lemmata

(AU). 3.12 - 3.1s.

is isomor-

at

93 The

Contribution

3.12 Lemma.

of the Cusps

to the Trace

Formula

91

The series

c N(1+(a+x)2)-” c&t converges

uniformly

in x E I?’

if s >

l/2

(Here

1 denotes

the vector

(1, - * * > 1)).

PTOO~.The series depends only on x mod t. We may hence assume that the components of x are bounded by a certain constant C. An elementary inequality states: There exists an E = s(C) > 0 such that 1+

(a + x)” L E(1 + a”)

for all

The inequality constant factor

shows, that the series by the integral

J

R” N(l

Before

we state

+ x2)--8 dXi..

the next

lemma

* dx,

in 3.12 can

=

we recall

be estimated

1

m(1+x2)-8dx V -co that

n .

the three

up to a

cl

conditions

c=o, Nc=O are equivalent. We also notice that the expression INcj does not change if one multiplies it4 from the left by an element of roe or from the right by an element of I$‘,‘:

3.13 Lemma. The series c MEI',\(r-I'cxz)/rc) converges Proof.

ifs

INcl-a

> 2.

We make

use of the fact that c

ikfmm\r

N(c2

the series (1.5.7)

+ d2)-“i2

(s ’

2)

92

Chapter

is convergent.

II.

Dimension

Formulae

We now divide I’ into double cosets

and obtain for the sum the expression

+(ca+cq2)--8/2 . c 1 N(C2 MEr,\r/I$) I$’ Here a! denotes the translation vector of the corresponding trix in r2). If A4 q! rm we may write

translation

ma-

2 c

N(C2

+ (m

s’2 = INcl-S

+ tq2)-

rc)

CN(1 r(l)

+ (

But the series on the right hand side has a positive 3.14 Lemma.

exists

)-8/z

.

>

lower bound by 3.12. 0

Choose c E R” )

There

(Y + ;

Nc#O.

a constant C depending only on A and s such that c N(cc2 + l)-+’ 5 C . INcj-’ EEA

ifs >O. Proof. We want to estimate the series by means of an integral. For this purpose we consider the function , . . . ,&ml)

ml

= N(c2t

+ l)-”

,

where t = (t1 ,...,

tn)E(R+)n,

Nt-1.

Let K c (R+)‘+l b e any compact subset. Then there exists a constant Cl depending only on s such that m

,-a*

,L-1)

I

L-l&l-l)

Gf(htl,*..,

if (b..

A-1)

E K.

This follows from the trivial estimation

f(h

,...,tn-l)
l,.‘.,

L1>-

(p&F& (l+y --

*

§3 The

Contribution

of the Cusps

to the Trace

93

Formula

We now make use of the fact that the mapping A c-) RP-’ E: H

(El ,...,‘%a-1

>

defines an imbedding of A as discrete subgroup with compact quotient. Let K be a compact fundamental domain of A. In the usual way one can interprete the sum b..,&l)

Cf( CEA

as an integral with respect to the invariant measure (dtr/tr) . . . (d&-r/t,-1) along a function which is constant on the (multiplicative) translates of K. The above consideration gives us the existence of a constant Cs - independent of c - such that c

N(C2& + 1)-”

= c

rEA

f(Q)

. . *,

En-l)

EEA I

c2 (R

J

,...)

ml

&-l)$...2.

P-1

From Nt = 1 we obtain f(h ). . . ) tn.-l) Together

with

= (Cg2

+ ty2)-"

+ t,1'2)-8

.

the inequality

c2t1’2 7zR + ty2 we obtain

* * * (C;ty2

2 21&j

(S > 0)

The transformation ti + Icil-'tl is ICil-' up to a constant factor.

shows

that the value of the i-th integral El

For the next lemma we notice that the expression NC2 N(C2 + l)a does not change if one multiplies

the matrix

left or from the right by an element of I’~‘.

A4 =

E r from the

Chapter II. Dimension

94

Formulae

3.15 Lemma. The series

c

(NC)-2N(C2

+ 1)-”

MEI'~)\(I'--I'~)/I'~) is convergent

ifs

> 0.

PTOO~. Let N be a matrix

in rrn

with

multiplier

from

the right

c, i.e.

with

N has the effect

c2 H C2& . The

series of Lemma

3.15 hence

equals

(Nc)-2 c N(c2s +1)-” c ) (s>0)> ( EEA it4Erm\(r-roo)irC where A denotes 3.14 follows from PTOOf

Of

the group of multipliers of r. the Lemmata 3.13 and 3.14.

Now

the proof

of Lemma Cl

3.1.

PTOpOSitiOn

We have

S(z):= c

IqM,z>l=

kiw-rm where (Y denotes We first estimate

(1) of the corresponding matrix in I’= . equals up to a constant factor

the translation vector the inner sum which

c pv(Mz CYEt.

jCMyz)’.-j7lk(Mz+a,z)y, I @I ZY rg

= kf&)\(r-rm)

- r + a)l-2r

= c N(q2 c&t

+ (a + p>“>-’

)

where v=

Im(Mz

- Z)

p=Re(Mz-z) For the proof of 3.1 we may complement in V, is contained Vi

The

elementary

2

*

assume yi 2 in a compact 1

1 (1 5 i 2 n), because the subset of l-l”. We then have

(1 5 i _< n).

inequality (t2 + 4---r

5 (t2 + p)--re+-z’l

,

$3 The

Contribution

of the Cusps

to the Trace

Formula

95

where t>1;

x,2’

r>O,

E Fp )

shows that we may estimate the above sum up to a constant is independent of ,B and r) by the integral

I

R”

This integral result is

IN(q2 + x2)1-‘dxl

can be computed

by means

factor

which

- - - dx, .

of the transformation

2 + qx. The

jy2r+l

times

a constant

S(z)

5

factor.

Cl

Thus

we have proved N

c MEr:)\(r-rm)

=

Cl

(IrQfz

-

q)--2r+l

‘cz;m;~-2’]

1

N [y( Icz + dl-2 + 1)-2’+1 Icz + dI-2’]

c

M&)\(r-r,) I

NY

Cl

c MEr(‘)\(r-r m

with a certain and obtain

constant

m

) N(lcz

Cr. We now

S(z) I Cl . NY

c

+

d12

+

divide

1)’

into cosets of I$,’

from

the right

NC-~‘.

Merc)\(r-r,)/rg)

c

N((x + c-‘d + a)” + ~-~(l+

c2y2))+,

r2) where a! denotes the translation Repeating the same argument S(z)

I

(72

vector of the corresponding as above we obtain

c

INclN(c2:+

matrix

(1) . in PO0

1)‘~112

Mer~)\(r-rm)/r~) with

a constant S(z)

C2. An elementary I

c2

c

inequality

finally

Nc-~N(c~

gives us + l)l-‘.

Merg)\(r-r,)/rg) This series is actually convergent proved Proposition 3.1.

for r > 1 by Lemma

3.15. We have thus 0

Chapter

96

3.2 Proposition.

c IWW MErm

3.21 Corollary.

Dimension

Formulae

there exists an estimation

On V,

where the constant

II.

< CNY ,

C only depends on I’-

and V,

(but not on z).

The function

c IWf,z)l(N~)-

>

3> 0>

MEFCO is integrable

along V,

3.22 Corollary.

(with

respect to the invatiant

volume

element dw,).

The function

c k(Jw

MEI-, is bounded on V,

JC

(which

follou~s from 3.1 and l.J),

J

k(M, z) dw = )I?+

By the fird

corollary

(NY)-’

VC.3

v- MEI'-

and we have

c k(Wz)~.

MEI-

this equals lim

s+o+

C

k(M, z)(Ny)-“dw

.

MUe., J vca

Again the corollary follows from the Lebesgue limit theorem (AlI.5).

PTOO~. We have c Ik(MYZ)l = c MEl-m M=(; !)a,

Ny2’IN(d-1(az

+ b) - ~)l-~‘.

(The quotient E = acF1 is a multiplier, hence NE = 1. From ad = 1 we obtain Na = Nd = 3~1.) We now divide I’m into cosets of I’(&’ from the right and obtain c

(NY)~’ c IN(d-‘(az rg) MfxaJ/rC) c MErm/r$i)

(NY)~’ c rC)

+ aa + b) - ~)~-2’ =

IN[(l + ~-l)~y~ + (a + x - c--l2 + u-1b)2](--r,

03 The

Contribution

of the Cusps

to the Trace

97

Formula

(1) where a! denotes the translation vector of the corresponding matrix in PO0 . The sum over (Y can be estimated by an integral like that described in the proof of 3.1. The result is c Ik(M,z)l MUao

5 CNy

c r,/r$y

N(1 + &)I-=.

cl The sum on the right hand side is convergent by Lemma 3.14. By means of Propositions 3.1 and 3.2 we are now going to express the rank formula as a sum of contributions of the conjugacy classes of P. First we choose transformations which transform the given set of representatives of the cusp classes to infinity Njnj

= 00

We may assume that there is a (large) mental domain of PKj in

(1 I j I h) * constant

C such that Vj is a funda-

Uj = N,y’Uc UC = {z E H”; Ny > C) (Recall: F = K U VI u - . - U Vh is a precise fundamental domain of I’.) We denote by S’j (1 2 j 5 h) the set of all elements of the stabilizer PSj which are different from the identity (as transformations) and by

We now split the trace formula k(M,z)dw

din-@, 2r]0 =

=A+B,

where A=

w4

JcF

MEr-S

JcF

MES

B=

2) cJ4d

lc(M, z) d.d .

In the first integral we may interchange summation and integration. In the following A& runs over a complete set of representatives of the conjugacy classesof l?. We have r - s = upfo]‘, MO

Chapter II. Dimension Formulae

98

where

[MO]’= [Nd f-l(r - s> ([MO]= {NMON-l, N Er}) * For the first integral A=x

we obtain

c /t(M,+h=x MO ME[Mol’ F

where

/ Mo

k(Mo,+%

F( MO)’

F(Mo)’ = u M-‘(F). “,“o’;lMO ZS

Here “ mod rMo” means that it4 runs over a fixed system of representatives with respect to the equivalence relation M NN

MMoM-’

e

= NMoN-’

.

The domain F(Mo) ’ is part of the fundamental domain F(Mo)

= ME[Mo],

of the centralizer

U

M mod

M-‘(F) IIitio

I?M~ of MO in r. In the special case

[Mo]cr-s we have of course F(Mo)

= F(Mo)’

.

We now treat the second integral

B$Bj,

Bj = Bi + By ,

j=l

where

In the integral Bi summation and integration can again be interchanged

=~JF(Mo~Wo~~W 7 B;=c C J k(M,z)h MO

ME[Mo]flSj

F-g

3

§3 The Contribution where

99

of the Cusps to the Trace Formula

Fj(M0):=

u

M--l(F- 4).

MMOM-'ESj, M mod l-~~ We obtain

where F(Mo)*

= F(Mo)’

u lj

Fj(M0)

j=l

M-l(Vj). = F(Mo) - 6 u j=l MMOM-lESj, M mod r&f,,

In the remaining integrals By we have to introduce factors. If we apply Proposition 3.2 to the conjugate of l?, then after a simple transformation we obtain

convergence generating group NjI’NJrl instead

MO MMoM"-lESj, M mod I-M,, 3.3 Proposition. Let MO run over a complete system of representatives of the conjugacy classes of I’. FOT each MO we select a jked set of representatives of the cosets r&\r, where rMo denotes the centralizer of MO in I?. Then the trace formula

may be written dim[l?,

2(T,.

as follows

..,T>lo =.I:?+ cMO h(Mo,z)h

where

F(Mo)* =F(Mo) - lj

u

M-‘(Q)

j=l MMOM-IESj, M mod

and F(Mo)

rMo

is a certain fundamental domain of rMo.

100

Chapter

II.

Dimension

We call the expression between the big brackets the contribution are now going to compute this contribution in several cases. Case 1. MO is either the identity fized point of MO.

(as a transformation)

Formulae

of Ms. We

or no cusp of I? is a

This is the case if, for example, MO is of finite order! In this case we have F(Mo)* = F(Mo) and the contribution to

of MO reduces

~(Mo, z)du . JF(Mo)

We may integral compact MO is of case.

apply the method of the cocompact case (32) to compute this and we will obtain the same formula for this integral as in the cocase. The integral especially vanishes if MO is not of finite order. If finite order one obtains the same contributions as in the cocompact

Case 2. MO is not the identity 1).

and fixes two diflerent

cusps (especially

n >

We will show that the contribution of MO vanishes. For this purpose we may conjugate the group I’ and hence assume that co is a cusp of l?, i.e. Moz = E,,Z + b,

Eo # (L-71)

*

It is easy to see that each component of ~0 is different from 1. Hence after a further conjugation we may assume that the second fixed point of MO is 0, i.e. b = 0. We have to determine the centralizer of MO. Denote by As the group of all multipliers E such that z~+ez

(notonlyzHez+b,)

is contained in l?. The centralizer group of transformations, consists

of MO, more precisely of all

the corresponding

z H ez ) E E no. We now have to investigate the domain F(Mo)*. Let K,, 6~ be the two cusps in our (fixed) set of representatives which are equivalent to 00, 0. (Of course a! = ,8 is possible.) tc.a=A(oo), We must determine

q=B(O);

all M E I’ such that MMoM-1

equivalently

A,BEI’.

E S,

$3 The

Contribution

of the Cusps

to the Trace

MMoM-lKj

101

Formula

= Kj

Or

MoM-lKj

= M-l&’ 3

for some j. This is the case if M-l&j

E {m,O}

e

Our result is MMoM-’

ES

e

ME~,,Ao~MEI’,,B

and therefore F(Mo)*

= F(Mo)

where

U M-1(V’)

F, =

(analogous

- (Fu u Fp) ,

for Fp).

Notation: E, =

u

M-‘(Va)

MErc,A

&

=

u MErra

From the definition

M-l&). B

of the cusp sectors

it follows

E, = {z E H” 1 Ny > Cl} .,@p= {z E H” 1 N(Im

(

+

>

>>C2)

with certain constants Cl, C2. It is ,no loss of generality to assume Cl > 1, C’s > 1. We have especially E, II Ep = 0 (which is automatically true_ if (Y # p). From the description of l?~~ we see that rMo acts on E, and Eg. This implies 1) F, (resp. Fp) is a fundamental domain of rM,, in E, (resp. &) and as a consequence 2) F(Mo)* is a fundamental

domain of rMo in H” - (E,

U Ep).

After these preparations we are able to compute the contribution 3.3 we find the expressions j(N,M,z), and

where M E lTKaA

of MO. In

Chapter

102

j(NpM,

z), where

In both cases these expressions j(N,M,

M E l?,,B

II.

Dimension

Formulae

.

do not depend on M. For example

z) = j(N,AA-%,

z) = j(N,A,

A-‘Mz)

. j(A-%I,

z) .

But j(A-lM, z) = 1 since A-lM fixes 0;) and 00 is a cusp of the conjugate group A-‘I’A. The contribution of Ms in the right hand side of the formula in 3.3 is the sum of the following three integrals 1)

k(Mo,

2) OLJ

2) OLJ 2,JF, k(Mo, A-Qfz)l” J F(Mo)*

(NY)~I~(KA,

3) see 2) but replace LY,A with In all three cases the integrand z H Mz This is obvious

for k(Ms,

&A(ccI) N$3(0)

/3, B .

is invariant

(= EZ with

under a transformation

NE = 1) for M E l?~~ .

z) and follows

for the occuring

j-factors

= 00

(+ j(N,A,

z) = const.),

= 00

(*

z) = const. . Nzm2)

j(NpB,

We may now replace Fey, Fp, F(Mo)* in E,, Ep, H” - (E, U IQ).

by any fundamental

from

.

domain

of I&

We are now able to prove that each of the three integrals vanishes. The easiest is the second one. We notice that a fundamental domain of FM0 in E, = {z E H”,Ny > C,} can be described by certain conditions on the imaginary part y and no restriction for the real part z of z. So it is sufficient to prove

J

R”

(E,,z - z)-“.

dx = 0 .

This follows immediately from the residue theorem (compare 2.3.1). The same method can be applied to the third integral after the transformation z I+ -l/z. The most involved integral is the first one. Here one has to determine a suitable fundamental domain of rM,, in the domain {z E H” 1 Ny < Cl,

Ny < C21Nz12}.

Recall that FM,, consists of transvections

z H ez ) E E no, where As is a certain discrete subgroup of the norm-one space in R:.

$3 The

Contribution

We introduce

of the Cusps

to the Trace

Formula

103

new coordinates

where O
Pj>O, The exponential function determinant is Np dp do,

(1 I j 5 n>.

has to be taken hence

component-wise.

The

functional

dp dQ = Np-f

The

new inequalities

for our domain C;lNsincp

The

group

of multipliers

N(sin

p)2

'

are

< Np < CrN(sincp)-l

.

As has the effect

and no effect on the variable Q. Hence we may determine a domain such that a fundamental domain of A0 is described by (Pl 7 . * . ,pn-1) and no condition

for pn and

(2i)2,, * We first have

integrate

along

Q.

Our

E B

integrand

Np-’ - N(sin Q)“-’ N(eo& - .&y

the p-variable.

o
k(kfs,

z) dw equals

dPdQ *

Up to factors

independent

dP 7 J WPF

where a = cFIN To compute

the integral Ul

The

integral

equals

sincp ,

we introduce

= p1,

. ..)

tin--l=

B C RF-’

b = CrN(sincp)-’

.

the variables pn-1,

u?z =

NP*

of p we

Chapter

104

II.

Dimension

Formulae

Notice. F’rom the convergence of the second integral one can conclude that ho is actually a group of maximal possible rank n - 1, hence a subgroup of finite index of the group A of all multipliers. The value of the integral

is up to a constant

factor

log b - log a = log( Cr CzN(sin To finish the second case it is obviously

cp)-“)

.

enough (use n > 1) to prove

(sini42’-2 =o7 (+P - dp Jo?r X&-“P)2’

where

1x1 # 1.

The transformation cp H r - cp shows that it is sufficient to treat the case [XI > 1. The transformation cp H cp + K shows that we may integrate from 0 to 27r (instead of x). Hence we have to consider the curve integral (z - z-y--2z-1

(jz

(2 - x2-y

f where the path of integration this integral as

’

is the border of the unit disc. We may rewrite

f

(2” - 1)2’% dz (2” - X)2r *

The integrand is an analytic function in a domain which contains unit disc (because of (Xl > 1). The integral vanishes by Cauchy’s Case 3. A& is not the identity transformation. further fixed point in H” which is not a cusp.

It fixes a cusp and it has a

We shall see that this case cannot occur. We proceed We can assume that

MO2 = eoz,

as in the second case.

co # (L.71)

and that 00 (but not 0) is a cusp of I’. Again the centralizer by a discrete subgroup ho of the norm-one subspace of R”,. The domain

F(Mo)*

now is a fundamental

H“-E,;

the closed theorem.

E, = {z E H”,

domain

of MO is given

of FM,, in

Ny > Cl).

But in this case the integral

q~o,z)~ JF(m)*

$3 The

Contribution

of the Cusps

is not absolutely

to the Trace

convergent,

105

Formula

because the integral b

J

% -’

du,

(b ’ 0)

0

does not exist (compare the computations in the preceding case). The Contribution of the Parabolic Transformations. A transformation MO E I? is called parabolic if its fixed point set in Hn U E” consists of exactly one cusp. If this cusp is 00, MO is a translation. Of course, the conjugates of a parabolic element are parabolic again. Before we determine a suitable system of representatives of the parabolic conjugacy classes,we notice some simple facts. 1) The fixed points of two conjugate parabolic transformations are equivalent (mod I’). 2) Two

parabolic transformations

conjugate QTOUP rn.

in the group

with the same fixed point K are r if and only if they are conjugate in the

Recall that we distinguished a system {ICI,. . . , oh} of representatives of the cusp classes and also transformations which transform them to infinity NjKj

= 00

(1 I j I h).

We denote by tj the translation lattice of NjI’iVJ~l and by hj its group of multipliers. Of course each r-conjugacy class of parabolic elements contains a representative which fixes one of the /Cj. A simple calculation in roe (if 00 is a cusp) now yields 3.4 Lemma. Let ‘FIj be a set of Tepresentatives set of all M forms

E rfij

a complete

,

Nj.i14NJT1(Z)=Z+U,

of tj UE3-Ij

system of non-conjugate parabolic

(0) mod hj. Then

the

(lljlh), elements

of r.

We now fix one Ktj and investigate the contribution of all iI& E rKj in our set of representatives. For sake of simplicity we assume K = 00, hence Moz=z+a. The centralizer of Me is the group of all translations

The same considerations as in the second case show that the contribution of MO is k(Mo, 2) qMo,z)~ + L, J (H”-E)/I’c) J E/l-$) (NYP

Chapter

106

II.

Dimension

Formulae

where

E={zEH~I with

a certain

constant

C. This h(Mo

Hence

the integrands

vol(t) lattice

constant z)

N(”

NY

2r--2d

NI
on a. We have

* the contribution 2r-2-s

NY dy - J Ng>C N(u + 2iy)2p

is

1 ’

denotes the volume of a fundamental parallelotope of the transt. We want to compare the expression in the big bracket with

Ny2’-2-sdy

J and have to estimate so(C) > 0 such that [NY274

bound

_

Ny2’-2-S

For every

Il.5

for the difference

transformation

+ 2i~)~r

the difference.

O<s

exists

an so =

5 SO.

is

dv

R: N(u2

yi H uiyi

e > 0 there

for

therefore

J

’ The

+ 2iy)2’

on 2. Therefore

Ry N(u

An upper

does not depend

w2nrwY)2T

=

do not depend

[J

(2;)%ol(t) where lation

>

iVy>C}

+ 4y2)r

*

(1 5 i 5 n) shows

that

this

is

~JNuI-~~‘+~ times a constant. be written as

We therefore

NY

RF N(u

the error term p,(s) that the series

tends c a:(t-{O})/A

converges

(see AI.20).

that

the contribution

of MO may

J 2r-2-sdy

(2i)2?ol(t) where known

have proved

This

+ 2i~)~’

+ IN~l-~~+‘pa(s)

to 0 uniformly

in a if s --f O+. It is well

INuI--~~+’

implies

JAY+ c IN~l-~~+~p,(s) =0.

f3

The

Contribution

of the Cusps

to the Trace

Formula

107

Hence in the final dimension formula (3.3) the term (pa is neglectible we may replace the contribution of MO by the modified contribution

We investigate

the integral

J

O”y2P-2-Bdy

0

where

and

a is a real number

(a + 2iyy

different

from 0:

1) a > 0: By Cauchy’s theorem we may deform the path of integration from the right real axis to the lower imaginary axis, where the complex power z’ = eelogr has to be defined by the main branch of the logarithm which is holomorphic outside the negative real axis including 0 and real for positive real z. The result is that the integral equals

2) In the case a < 0 upper imaginary axis. formula remains valid -i--slul--(l+s). Uniting

one has to deform the path of integration to the An analogous computation shows that the above in the case a < 0 if one replaces (-i)-‘~--(~+‘) by both expressions, this may be written as -(l+s)e$is

h34bl The transformation

J

O”Y

0

t = (y + 1)-l

sgn a .

finally gives the result

2r--2-Sd

(u+

2iy)l

-sgnu.eT

= (2$-1

7r ia sgna

Ial- (l+g)

J

’ ts(l

- t)2r-2--s

dt .

0

The integral on the right hand side is a usual Beta integral. We only need its limit for s + O+, which is (2r - 1)-l. The modified contribution of MO may now be written as g(s)(2i)%ol@) (2r - 1)” where

g(s) is a function

sgn(~u)

independent

prul-(~+l)e$i~ of a and

SKY+ g(s) = 1 .

S(wa)

108

Chapter

II.

Dimension

Formulae

Of course S(sgna):=sgnal+...+sgna,. We now sum up over a system Ms of representatives of conjugacy classes of parabolic elements fixing 00. We will see in the following that the limit g(s)(2i)%ol(t) (2r - 1)n

Ef+

C sgn(Nu)lNul-(l+S)eqiS M.

Skwa)

exists. Assuming this for a moment we may call this expression the contribution of our cusp to the right-hand side in 3.3. Recall that the system of representatives is given by it!foz=z+a, where a runs over a system of representatives of t - (0) modulo A. But each Ms occurs 2 times (I = order of the kernel of the natural projection I? --+ (SL(2, R)/{fE})“). Hence the limit will be equal to 1(2i)nvol(t) p. - 1)”

s

d$+

~.(t-(o)),a c

s~n~Nu)wP+

,$is

S(sgna)

)

.

We treat the cases n = 1 and n > 1 separately. 1)

n=l:

Wehave t=dZ,

d>O.

The sum equals d-(l+s)

Oi) n-(1+s)

1

e,is-e-$ia

.

c

n=l

Making

use of the well-known

fact that 00

lim s n -(l+s) c e+o+ n=l

= 1

one obtains d-’ for the limit.

Hence the contribution

. ?ri of our cusp is 2/T

-(2T-2)

n>l:

Weclaim

&II+ C sgn(Nu)lNul-(l+S)e~i” a

S(sgno) = )I?+ C sgn(Na)(Nal-(l+“) a

.

53 The

Contribution

of the Cusps

to the Trace

Formula

109

The existence of the limit on the right hand side is well known The limit of the difference (if it exists) is obviously $ *l.+

s C sgn(Na)S(sgn (I

a)lNal++8)

(AI.20,21).

.

We now choose a sign vector r-7

u=(m,...,GL), From AI.21 we know

Ui = *I.*

that the limit

A= ?ly+sc pu(++“) au>0 exists

and is independent

of cr. Hence the limit (*) exists

and equals

sgn(Na)S(sgn a) ,

%iA c

0 where the sum runs over all 2n sign vectors. The sum vanishes. We finally obtain

as contribution

of our cusp.

The contribution of the other cusps is obtained by transforming them to 00 and considering a conjugate group. We are now able to write down the final dimension formula:

Rt

Some Notations. Let t c R” be a lattice and A c a discrete subgroup of multipliers which has maximal rank n - 1. We define the Shimiiu L-series by qt, A) = if 12= 1; -l/2, i”

=

(2,)nvcJ(t) .:y+ .:(t-~~~,nsgn(Na)lNol-(l+‘),

if 92> 1 .

1 Here vol(t) denotes the volume of a fundamental parallelotope oft. Remark. Assume

n > 1 and that there & E R” ;

exists

N&=-l,

a vector et=t.

Then L(t,n) This

is always

the case ifn

is odd (take

= 0. E = (-1,.

. . , -1)).

Chapter

110

Remark.

Let (Y E RT be a totally

positive

vector.

II.

Dimension

Formulae

Then

qt, A) = L(&, A) .

Let now I7 c SJC(~,R)~ b e a discrete subgroup with cusp K. We choose a transformation N with NK = co and denote by tN,

AN

the group of translations, resp. multipliers of the conjugate group NI’N-l. By the above remark the value

L(r, 6) := L(tN, AN) does not depend on the choice of N. We can now write down the final formula. 3.5 Theorem. Assume (H”)*/l? is compact.

that F c SL(2,R)

n is a discrete subgroup such that the irreducibility condition is

We also assume that

satisfied.

If

T >

1 we have

dim[l?,

2(T,.

. . , r)].

=

vd(Hn/r)(%

-

1)”

-I-

c

%(r, a

a>

+

c

L(r,

K>

K

wheTe a runs over a set of representatives of r-classes of elliptic fixed points and K over a set of representatives of the cusp classes.

The contribution

E(I’, u) of an elliptic fixed point is defined as in $2

Jw7a) = &

MEr c a.

M#identlty

where C is the rotation factor of M, i.e.

The contribution L(I’, K) of a cusp is Shimizu’s L-series which we have defined just before. A Remarkable Symmetry. It follows from Shimizu’s formula 3.5 that there exist a natural number TO and polynomials in one variable pm

;

O
53 The

Contribution

of the Cusps

to the Trace

111

Formula

such that P(“)(r) The most important

= dim[I’,2(v

.

is l+(r)

sometimes called Shimizu’s the polynomial

= P(O)(r)

polynomial. &r(r)

especially

+ r-r-0,. . . , v + rre)]o

>

In the next section there will occur

:= P(l)(r)

>

its value for r = 0.

3.6 Remark.

Assume

n > 1. Then

fi(O) = (-l)“Qr(O).

The proof follows from a glance at Shimizu’s formula. This symmetry is valid for all three types of contributions. For the main term it is trivial. For the contribution of the elliptic fixed points it follows from -=-1-C

and the fact that with contribution. For the contribution

1

of the cusps it follows = 0

The values fi(O),&r(O) equip X = (Hn)*/l?

occurs in the sum defining

with

where Ux denotes the structure sheaf and X their Euler-Poincare characteristic.

this

from the fact that

if nisodd,

x(0x) x(Kx)

The

c-’ - C-1

C also its inverse 5-l

L(t,A) Final Remark section we will One has

1

n>l.

h ave algebraic the structure

geometric meaning. In the next of a projective algebraic variety.

= J%(O) + h = &r(O) + h, and Kx

the

canonical

sheaf

in the sense of Serre

equality -h

= (-lp(X(lx)

can also be proved by cohomological and the next section.

x(0x)

methods.

- h) It follows

(n > 1) from

the result

of the paper

[12]

112

Chapter II. Dimension

$4 An Algebraic

Geometric

Formulae

Method

In this section we make use of the fact that the compactification (Hn)*/r (l? commensurable with a Hilbert modular group) is an algebraic variety. Using the cohomology theory of coherent sheaves, especially the theory of the Hilbert polynomial, we will succeed in expressing dim[I’, (2,. . . , 2)] by the dimensions of dim[I’, (2r,. . . ,2r)], r > 1, which we computed in the previous sections. Another result will be a formula for the arithmetic genus of a desingularization of (H”)*/l?. W e d o not need the explicit construction of such a desingularization. We have to make use of some results about complex spaces, projective varieties and coherent sheaves, which we cannot develop in this book.

Let r c

b e a discrete

SL(2,R)”

subgroup.

We

equip

the extended

quotient

X = (Hn)*/I’ with a certain sheaf of continuous functions: Let V c (H”)*/I’

be an open subset and U c H” its inverse image in H” with respect to the natural projection. Composing an arbitrary function f:V-+C with this projection we obtain a function F:U+C

(which is I’-invariant).

We denote by WV

the set of all continuous functions f such that F is holomorphic. We obtain a sheaf Ox of continuous functions and (X, OX) is a ringed space. 4.1 Theorem. Let I? c SL(2, FQn b e a discrete

subgroup.

The

extended

quo-

tient

x =(Hy*/r, equipped Indication

with

the structure

sheaf OX, is a normal complex space.

of a PTOO~.

1) The interior of X, i.e. H”/I’, H”/I’ locally looks like

is a complex space: What we know is that C”IE,

$4 An Algebraic

where

Geometric

Method

113

& is a finite group of rotations .z I-)

((121

, - * -, m&z>

But it is well known and easily seen that C”/E (hence a normal complex space).

-

is a normal algebraic

2) X is a normal complex space at the cusps. Without we may assume that the cusp is 00. One makes use of the following

variety

loss of generality

special case of a

Criterion of Baily and Cartan. Let Y be a locally compact space with a countable basis for its topology. We assume that a E Y is a given point such that Y - {u} is equipped with the structure of a normal complex space. We extend the complex structure to a structure of a ringed space (Y, 0~). A function on an open subset V c Y is called holomorphic if it is continuous and if its restriction to V - {u} is holomorphic with respect to the given complex structure. Assumptions: 1) There is a findamental U - {a} is connected. 2) The global h o 1omorphic a is suficient).

system

of open neighbourhoods

functions

f

E Oy(Y)

separate

U of a such that points

(outside

Then (Y, 0~) is a normal complex space. We apply Cartan’s criterion to v = uc/roo

u (00)

UC = {z E H” 1 Ny > C}. We consider certain modular forms with respect to the group I?-. They are in fact invariant functions with respect to PO0and define elements of O(V)! Nevertheless the theory of Poincark series (Chap. I, §5), especially 1.5.3, can be applied to Poe. The separation properties of Poincare series (which we did not prove completely; see 1.5.6 and the concluding remark) show that these functions separate points of U~/Poo. Cl We introduce certain sheaves

defined on the complex space X. As in the definition of 0~ we denote by V c X an open subset and by U c H” its inverse image. Then Mz,.(V) is the set of all holomorphic functions f:U+C

114

Chapter II. Dimension

satisfying

the following

1) f(Mz)

= fi(cjzj

Formulae

two conditions: + dj)2rjf(~)

for all

ZEU,

Mel?.

j=l

2) f is regular

at the cusps which

What does regularity contains

are contained

at a cusp K: E U mean

UC = {z E H” 1 Ny > C} ,

in U.

? If K = 00, then

C sufficiently

large.

The set UC is invariant with respect to Lyon, especially with translation lattice t. The function f hence admits a Fourier f(z)

= c

the set U

respect to the expansion

age2”-)

go.0 in UC. Regularity

means

a,#0

3

g20.

For an arbitrary cusp K: E U one defines regularity by transforming it to 00 (compare 14.4). The GGtzky-Koecher principle (1.4.9) shows that the regularity condition is automatically satisfied if n > 1. The elements of M24V) are so-called local automorphic forms. The global sections of the sheaf Msr are the usual automorphic forms introduced in Chap. I, $4. M2r(X)

= [r,

2~1 .

The local automorphic form f is called a cusp form, cusps K. E U. In case of K: = 00 this means ag=O

3

if it vanishes

at all the

g>O,

equivalently ao=O. We denote

by M;,(V)

c

M2r(V)

the subset of local cusp forms. Obviously M!j, sheaves are Ox-modules in a natural way. 4.2 Proposition.

are coherent.

The OX -modules

is a subsheaf

of M2p.

Both

$4 An

Algebraic

Geometric

115

Method

Indication of a Proof. We show that Mzr restrict to the case of our interest q

=

... =

r,

is coherent at the cusps. We

.

If r E Z we write :=

M2r

M2(.P,...,r)

The invariance property 1) obviously means for M E l?m that f(Mz)

= f(z)

(because the norm of a multiplier is 1). In a neighbourhood UC/L

u {cQ)-

this gives us an isomorphism between Mzr line bundle close to the cusp. 4.3 Lemma.

(C >> 0)

(H”)*/r

and OX, and Mzr

is actually a q

Let ro be a natural number such that the order ro. Then the sheaves

of each elliptic

jizcd point* a E Hn divides

M are line

bundles

(i.e.

locally

29. >

r=Omodro,

isomorphic

to Ox).

If I’ is an irreducible subgroup in sense of 1.2.13 and if (H”)*/I’ is compact, there exist only finitely many r-equivalence classesof points a with non-trivial stabilizer. Hence a number ro exists in this case. (This is also true without the irreducibility assumption). Notice.

In the following the first assumption is compact, also the second assumption sake of simplicity, be considered true. 4.4 Theorem. Let r c SL(2, tended quotient X = (H”)*/I’

Mqr,...,,9,

of irreducibility of irreducibility

Recall that the order in (SL(2, R/{IIzE})“.

and, if H”/l? will, for the

R)n b e a discrete subgroup such is compact. The line bundles r = 0 mod ro

of an elliptic

fixed

that

the ex-

(rs as in 4.3),

are ample for positive r. In particular the complex ture as projective algebraic variety. *

1.2.19 I.4.12.

point

space X carries

(1 is the order

of the

a struc-

image

of ra

Chapter II. Dimension

116 (Such

a structure

is unique

by the famous

comparison

theorem

Formulae

of Serre.)

Indication of a Proof: Ampleness means that two different points of X can be separated by a global section of a suitable power of the given line bundle. cl This can be proved in our case by means of PoincarC series. Hilbert Polynomial. Let M be a coherent sheaf on a compact complex space X. The Euler characteristic The

x(M)

:= g(-l)j

dimHj(X,

M)

j=O

is well defined because the cohomology groups are of finite dimension and vanish for sufficiently large j. Let t be an ample line bundle on X. 1. There exists a polynomial in one complez variable P such that

x(M

2. If r is suficiently

@ CBr) = P(r) .

large, r > 0, we have Hj(X,M@L@“)=O

if

j>O,

especially dim(M

@IL@r)(X)

if

= P(r),

r >> 0.

We now apply the theory of the Hilbert polynomial to

M = M2,

X = (Hn)*/F,

C = M2ro,

where rg is chosen as in 4.3 and 4.4. We obviously have

M 63 L@’ = M2+2rro

.

From the theory of the Hilbert polynomial we obtain the existence of a polynomial P with the properties 1)

P(r) = dim[P, (2 + 2rr0,. . . ,2 + 2rro)],

2)

P(O) =

r > 0,

.

x(M2)

The polynomial P has actually been computed in the previous sections by means of the trace formula. What we want to compute is the dimension of M2(X)

= [r,

(2,.

. . ,2)] .

$4 An Algebraic

Geometric

117

Method

This means that we have to get hold of the cohomology groups of MP. We want to apply duality theory and for this purpose we consider a desingularization* ILLX. Here X is a nonsingular biholomorphic mapping

connected

projective

variety

such that r induces a

= : r-‘(Xreg) + Xreg, where

Xres is the regular

locus of X.

We consider on X the so-called canonical sheaf Cz. The sections of are holomorphic differential forms of top degree n. In local co-ordinates h they have the form f(z)dzl

A---Adz,

We now consider the direct Grauert and Riemenschneider 4.5 Proposition.

images of K, states

x&J

if

= x(r*W

.

on X. An important

The higher direct images of lcx Rp?r,~~=O

4.51 corollary.

(f holomorphic)

result

of

on X vanish:

p>O.

*

We now investigate the direct image r&x. Let V c X be an open subset and ? c X its inverse image in 2. A section w E (x*X2)(V) is a holomorphic differential form on v. Restricting it to the regular locus of X we obtain a differential form ws on I&. We denote by U the inverse image of V in H” and by Us the inverse image of Vre9 in Hn. The complement U - Us consists of elliptic fixed points and is hence discrete (by our irreducibility assumption). The pullback of wc to Us extends holomorphicslly to U, because a holomorphic function in more than one variable cannot have isolated singularities (in case n = 1 we have Uo = U). The pullback has the form *

Such a desingularization exists by a general deep result of Hironaka. In the case at hand an explicit desingularization has been constructed by Ehlers [ll] generalizing results of Hirzebruch [30] who treated the case n = 2. We shall describe this construction in $5.

Chapter

118

f :U + C

Dimension

Formulae

holomorphic.

The function f is obviously a local automorphic We hence obtain an Ox-linear imbedding

and hence may identify tion we obtain

II.

.rr,Kx with

a subsheaf

form of weight

(2,. . . ,2).

of MZ. After this identifica-

4.6 Lemma. We have

r,Kx

= M;

(= sheaf

of local

cusp forms

of weight

(2,. . . ,2)).

Let f be a local automorphic form of weight (2,. . . ,2) on some open subset V c X and wg the corresponding holomorphic differential form on I&. We have to show that ws extends holomorphically to the inverse image v of V in X if and only if f is a cusp form.

Proof.

Remark. The differential and only integral

if for

each open

form wg on Vreg extends holomorphica2ly to v if subset W c V with compact closure in V the wo A& J Wr=g

converges. Proof of thesemark. T& criterion is obviously necessary, because the inverse image W of W in V has compact closure and hence

J

WAGi

i7

converges. To prove the sufficiency we first remark that c - VIeg is an analytic subset. As holomorphic functions (hence n-forms) always extend holomorphically over analytic subsets of codimension 2 2, we only have to prove the extension into the smooth points of codimension 1. Hence the assertion follows from the following criterion: Let E c C be the unit disc and f a holomorphic function such that

J

E”-lxE-(0)

converges.

Then

If(4l” f extends

dv

(dv = Euclidean

holomorphically

to E”.

volume

on En-’ x E - (0)

element)

$4 An Algebraic

Geometric

119

Method

This can easily be proved by means of the Laurent investigate the convergence of

I

expansion

in z, of f. To

WQA&i W r=g

we choose a suitable fundamental domain of the inverse image of W in Hn. We know that there is a fundamental domain which consists of the union of a relatively compact subset and a finite number of cusp sectors. It is sufficient to consider the cusp sector at 00. Now we have the following situation: On some set UC = {z E Hn, Ny > 0) the function f(z) is holomorphic and I?-invariant as well as regular at the cusp 00. It has to be shown that the integral I

v,,,r

m

I.wl”~w

converges for C’ > 0 if and only if f vanishes at 00. This can be proved easily by means of the Fourier expansion of f. This completes the proof of cl Lemma 4.6. of MO and Mi

We now compare the Euler characteristics short exact sequence

by means of the

O---+M;tMy--+M~/M;+O. The sheaf Mz/Mfj vanishes isomorphic to 43. We obtain

x(MJM;)

= h = number of cusp classes.

The Euler characteristic x(M2)

outside the cusps and the stalk at each cusp is

being additive, =

x(M;)

+

we obtain

h

(=

x(&l

+

h).

We now compute X(Kx). A s any nonsingular projective variety carries a K%hlerian metric, we may apply Hodge theory (s. App. III). The Hodge numbers of X are hp*Q := dimHq(X, 05) , where Q& denotes especially

the sheaf of holomorphic fl$=Kp

We have

x(K,)

= -j$l)Ph”rP. p=o

differential

forms

of degree p,

Chapter II. Dimension Formulae

120 From the duality

formulae hP,4

=

h&P

we obtain x(Xx)

.,

hPl9

=

hn-9,n-P

n = C(-l)“-‘hp,O p=o

)

where hf’f” = dimR$(X) 4.7 Proposition.

.

We have holo = 1 hnlo = ho’,, = dim[I’, hPso = 0

if

(2,. . . ,2)]0 ,

0

4.71 corollary. X(Kz)

= (-l)n

+ dim[I’, (2,. . . ,2)]0 .

4.72 Corollary. X(&z)

ProoJ We only have alternating differential back such a form we H”. But now we may

= (-1)”

+ h + dim[P, (2,. . . ,2)]0 .

to prove the third formula, i.e.: Each holomorphic form on X of degree p, 0 < p < n, vanishes. Pulling obtain a P-invariant holomorphic differential form on apply the corollary of 1.4.11. cl

In a later section we shall prove by means of analytic stein series dim[I’,

(2,. . . ,2)]s+h=dim[(2,...,2)]

continuation

if

of Eisen-

n>l

and dir@,

(2,. . . , 2)]s+h-lIdim[(2,...,2)]

if I’ is commensurable with a Hilbert modular group. One can show by means of the residue theorem that in the second case we also have an equality (n = 1). This gives us

$4 An Algebraic

121

Geometric Method

4.7s Corollary. If l? is commensurable with a Hdbert x(M2)

= (-l)n

modular group, we have

+ dim[I’, (2,. . . ,2)] - E:,

where

1 { 0

‘=

ifn=l ifn>l.

The arithmetic genus of 2 is defined as g := x(0,)

= arithmetic genus.

By duality we have

x(Q) = -&-qpg, = (-l)“x(&) > p=o

where gp = hpto = dimR$(z)

.

We obtain 4.74

Corollary.

The arithmetic

genus is given by

g = 1 + (-l)n

dim[I’(2,. . . ,2)]s .

The Final Formulae. Let a E Hn be an elliptic fixed point of l?. We define the contribution E(I’, u) by

where C = (cl,...,&)

are the rotation factors belonging to A4 (“A4 # id” means that the transformation induced by M is not the identity, i.e. M # (fE,

If K.is a cusp, the contribution Shin-&u L-series.

. . . , fE)

.)

L(I’, K) has been defined in $3 as a certain

4.8 Theorem. Let r c SL(2,R)” be a discrete subgroup such that the extended quotient (H”)*/l? is compact. we asnme that the first irreducibility

Chapter

122

assumption reducibility 1 + (-1)”

I.219 is satisfied, assumption I,,$.lZ. dim[I’,

(2..

II.

Dimension

Formulae

and, if Hn/l? is compact, also the second Th en the following formula holds

. ,2>]ll = (-l)nvol(H”/I’)

+ c

qr,

a> + c

qr,

ir-

tc) )

K

a

where a (resp. IC) runs over a complete set of representations of r-equivalence classes of elliptic fixed points (reap. cusps). This expression also equals the arithmetic genus g of a desingularization of X. A Simple Special Case. Assume that n > 1 is an odd number without elliptic fixed points. Then the formula simplifies g = 1 - dim[I’,

(2..

. ,2)]s

and that

I’ is

= -vol(Hn/I’)

So the genus is a negative number. As a consequence we have for example that the field of automorphic functions is never a rational function field under this assumption, because the genus would otherwise be 1.

$5 Numerical

Examples

in Special

Cases

The numerical evaluation of the invariants occuring in Shimizu’s function fi (r) is in general very complicated. In some cases it can be calculated explicitly. We collect some well-known results, mostly without proofs. For more details we refer to Hirsebruch’s paper [30].

The “main of a fundamental

term” of Shimizu’s domain of l? with

function respect

comes from the volume to the invariant measure

vol(I’)

dw=-& g2 dv = Euclidean

measure

= dzl

dyl . . . dx,

dy,, .

This volume has been calculated by Siegel [59] in the case of the Hilbert modular group. To be more precise, he expressed it in terms of the Dedekind C-function. Let K be a totally real number field of degree n. Its Dedekind C-function is defined as

k(s) = C(Na))-” , where the sum is extended over all integral ideals a (0 # a C 0). This series converges if the real part of s is greater than 1 and defines an analytic function in this half-plane. It has an analytic continuation as a meromorphic function into the whole s-plane with a single pole (of first order) at s = 1.

$5 Numerical

The

Examples

in Special

123

Cases

function

is invariant

under

5.1 Proposition. group

(dK

= d,“‘2~-““/2r(s/2)nCK(s)

k(s)

with

respect

the transformation

s --+ 1 - s. Siegel’s

The volume of a fundamental to the invariant measure

d&J = (4+“$

= discriminant

domain

(dv = Euclidean

of K)

result

is [59]:

of Halbert’s

volume

modular

element)

is vol(SL(2,o))

By means

of the functional

= 27-l)Q(-l) .

equation

(-1)%(-l) The

trivial

we obtain

= d$22-“r-2nCK(2).

estimation

gives us 5.2 Corollary. vol(SL(2,o))

Explicit

formulae

for CK( -1)

> 21-2nr-2nd$2

are known

.

in the case of a real quadratic

field

K = Q(h), where

a > 1 is a square-free

natural

number.

The discriminant

of K is given

by d=4a d= For the following given.

result

a

ifar2,3 ifar

1

mod4 mod4.

we refer to [30], p.192,

where

5.3 Proposition. Let

K = Q(h)

(a > 1, square-free)

further

references

are

Chapter

124

be a real quadratic field.

Then

II.

Dimension

Formulae

for a G 1 mod 4

andforaE2,3mod4

CK(-1)= &m(a) + 2.

ul(a - b2)).

c

l
al(x)

denotes

the sum of the divisors

of a natural

number x.

The Elliptic Fixed Points. Following a method of Shim+ Prestel succeeded in obtaining very explicit formulae for the elliptic fixed points of the Hilbert modular group SL(2, OK) of a real quadratic field K = Q(G) (u squarefree). In the following we describe some of these results, especially the cases a prime, a > 5. We first explain some notations. Let a E H2 be an elliptic fixed point of r = SL(2, OK). The order of a is the order of the group of mappings corresponding to Pa, i.e. e := order(u) = #l?,/{fE}. We denote by a, F) the number of P-equivalence classes of elliptic fixed points of order e. We recall the notion of the “type” of an elliptic fixed point: The stabilizer I’,/{fE} is a cyclic group. After the transformation 2 --f (2 - a)(z - a)-l

= w

it is generated by w+Cw, where c = (Cl,

(2)

is a pair of primitive roots of unity of order e. We c2 = cy

;

(el,

e2)

15

e2 < e.

We call = (1,

e2)

espe&,lly

have

$5 Numerical

125

Examples in Special Cases

the type of the elliptic

fixed point.

Special Cases 1)

e = 2: We have only one type, namely (l,l).

2)

e = 3: There axe two types: Type I: (l,l), Type II: (1,2), .

In the following results of Prestel +4 denotes

of the imaginary

the class number

5.4 Proposition.

(x ’ 1) field Q(G).

quadratic

Let a > 1 be a square-free natural number a>5,

(a,S)=l.

of order 2 and 3 GOT the Hilbert modular e Teal quadratic field K = Q(G). Their number AZ, As ia given by the following formulae There

e&at only

QTOUP

r

=

elliptic

%(2,0)

fixed points

Of th

1)

azlmod4:

Az = h(-4a) ,

2)

aG3mod8:

Aa = lOh(-a)

3)

aG7mod8:

Az = 4h(-a),

A3 = h(-3a) ,

Al = h(-12a)

fi(r)

We restrict

= dim[I’,

.

At = h(-12a)

In all three cases the fized point8 of order 3 (1,1) together with one of type (1,2)) *. We next determine the contribution and 3 to the rank formula

. . in pairs (one of type

occur

of the elliptic fixed points of order 2

(2r,.

. . ,2r)],

T>l.

to the case

rGOmod6, because this is sufficient, to determine the arithmetic dim[l?, (2,. . . ,2)]).

genus (equivalently

1) e = order (a) = 2 The contribution to the rank formula is

WW 2)

(=

E2(r,aN

=

&

+

,,‘(,

+

1)

=

i.

e = order (a) = 3 *

This is different if one also considers d divisible by 3 (see for example [30], p.237).

126

Chapter

((el, e2)

Type 1

Let C be a third

Type II

e2)

1 = 3

Dimension

Formulae

:

root of unity. The contribution

((el, Jfw+)

= (Ll))

II.

= (1,2))

is

:

1 1 (1 [)(I (2) + (1 C”)(l (

- <) > .

5.5 Lemma.

Let I? c SL(2, R)2 b e a discrete irreducible subgroup such that the extended quotient (H2)*/l? is compact. The contribution of the fixed points of order 2 and 3 to the rank formula for dim[I’, (2r,. . . ,2r)], r 3 0 mod 6, is given by 1)

e = order (u) = 2

qr,q 2)

= i/8.

e = order(u) = 3 Type 1

((el,

e2)

= (1, 1))

qr, Type II

((el,e2)

U) =

i/9

= (L2))

E(r,q

=2/9

The Contribution of the Cusps. We recall that the cusp classes of the Hilbert modular group l? = SL(2, o) are in l-l-correspondence with the ideal classes. If 6,:; U,CEO C

is a cusp, then a := (a, c) represents the corresponding ideal class. To transform K to 00 we choose a matrix ad-bc=l; which has the property ACO=K.

b,d E a-l

$5 Numerical

A simple

Examples

calculation

QP ; K 7 6>

in Special

shows

Cases

that

127

the conjugate

oC5-&=l;

group

A-rI’A

equals

cvEo,6Eo,j3Ea-2,yEa2

The translation module of this the group of unit squares:

>

is am2, and the group

group

.z unit

A = {E2, So in the case n > 1 the contribution given by

.

of multipliers

is

in 0) .

of the cusps to the rank

formula

is

where d(a) denotes the discriminant of a given ideal a which, in the totally real case, is nothing else but the square of the volume vol(a) (see App. I for the definition of d(a)). We recall that this expression does not change if one replaces equivalent ideal. It vanishes, of course, if there exists a unit E E o* with We hence

make

a by an

Ne = -1.

the

Assumption. Each

unit

E in o has positive

Under this assumption sgn Na merely generated by a. We hence may define $((u)) on the group of all principal ideals. by a totally positive element.

norm

(NE = +l).

depends upon the character

the principal

It is 1 if the principal

ideal

?it is the subgroup of the group of all principal consists of ideals generated by a totally positive element. so is defined on the factor group +

is generated

ideals, which character

OUT

(1, -1).

We denote group Z/7-i

by Z the group of all ideals of K. The are the ideal classes and those of I/‘?& ideal classes. Both groups are finite. We have n/7-&

(a)

:= sgn Na

Notation:

1c, := tipit

ideal

c 2/7-d

.

elements of the factor the so-called narrow

128

Chapter

Because Z/Nt

II.

Dimension

Formulae

is a finite abelian group, we may extend 1c,to a character x : z/7-6

--t s1 = {C E c, \(I=

1).

We have proved: 5.6 Lemma. Assume that each unit has positive norm. Then there esists a character x on the group Z of all ideals depending only on the narrow ideal class and satisfying x((a)) = sgn Na .

It is worthwhile the case if

asking whether 2/7-h

especially

x can be taken to be real. This is, of course, N z/x

x 7-l/7-& )

if the order of Z/7-t, i.e. the class number

h of K, is odd.

Using a character x described in Lemma 5.6 we may rewrite the contribution L(I’, 6) as follows: We recall that A is the group of all squares of units. This group is a subgroup of the group of all units of index 2n. This gives us

c sgn, a-a/,. INal

c 57!$=2n o:a-=/A

because all units have positive norm by assumption. If a runs over a complete system of representatives of aS2/o*, then .a2 = x runs over all integral ideals in the class of a2. This gives

c

E

= N(a2)X(a2)

F

f$$

,

a-=/o* where x runs over all integral ideals in the ideal class of a2. Here the Smite sum means the limit value of the L-series

x(x) c Jwa

which

converges

for s > 1 at s = 1. The expression

does not depend on the ideal b and hence equals the discriminant field K. We obtain for the contribution of our cusp -$dg2n(a2)

F

$f$

.

do of the

85 Numerical

Examples

in Special

129

Cases

If we make the further assumption that x is real, we have x(a2) and obtain for the sum of the contributions of all cusps $g2L(l, where

= x(a)2

= 1

x) )

L(l, x) denotes the limit value of the L-series L(s, xl = C

x(W(a)-”

,

SK0

where

a runs over all integral

ideals.

5.7 Proposition. Assume that the norm of each unit is positive. Assume fwthermore that the character in 5.6 can be chosen to be real (for example if the class number is odd). Then the contribution of all cusps to the rank: formula is

It is a well-known fact that this expression is always unequal zero. We finally consider a very interesting special case, namely the case of a real quadratic field K = &(fi), P prime The following beautiful formulae can be found in Hirzebruch’s paper [30], 3.10, where further comments and references are given: In Q(,/jS) a unit of negative norm exists if and only if p=2

or

pElmod4.

Hence precisely in this case the contribution of the cusps to the rank formula is 0. In the case p s 3 mod 4 the squares of ideals generate a subgroup of index 2 in the narrow class group Z/7-&. This implies that there is exactly one real character x with the properties in Lemma 5.6. By means of the decomposition law of the field Q(Jii) one obtains L(% x) = where the L, are the usual Dir&let formula gives

J5--4W-&)

L-functions. The so-called class number

L-4(1) = ?4-l&(-4) L-,(l) where h(-4)

?

= Fp-1/2h(-p)

(=class number of Q(i)) is one.

Chapter

II.

Dimension

for the arithmetic

genus

130

We now have the complete

formula

Formulae

g=l+dim[I’,(2,...,2)]0 =l+dim[I’,(2,...,2)]-h in the case K = Q(d), p prime, p > 5. In the remaining cases p = 2, 3, 5 the determination of the elliptic fixed points is also continued in the mentioned paper of Prestel [52]. (Actually those three special cases have been treated by Gundlach in an earlier paper [21].) 5.8 Theorem. Let p be a prime,

K = Q(&.

The arithmetic

genus

g=l+dim[P,(2,...,2)]0 =l+dim[I’,(2,...,2)]-h of the Hdbert modular surface with respect l? = SL(2, o) is given by the formulae

to the Hdbert

modular group

for p = 2,3,5,

g=l g = &+l)

+ !!p

+ qd

for p 3 1 mod 4,

g = &(-1)

+ ;h(-p)

+la(-i2p)

for p z 3 mod 8,

g = &(-1)

+h(-;2p)

p > 5,

for p 3 7 mod 8.

The Group I?. We define

r- := ivriV , NI=(;

;),Nz=(;

Tl).

The action of P- on HZ is equivalent to the action of I’ on the product of an upper and a lower half-plane (by means of the usual formulae). Consider an element a E K with a(r) > 0 ,

a(*) < 0 .

The matrix

has obviously the following property: The groups AI’-A-l

and

l?

$5 axe

Numerical

Examples

in Special

If there is a

commensurable.

131

Cases unit

a with

the above property,

both groups

axe equal. The structure

of the elliptic fixed points of I?- is dual with

that of l?.

The numbers of classes of fixed points of a certain order are equal, but the types are changed. In case of fixed points of order 3 precisely the But in the cases which we two types ((1,l) and (1,2)) are interchanged. considered the elliptic fixed points of order 3 occur in pairs (5.4). So in the formulae for the arithmetic genus g of I and g- of r- only the contributions of the cusps may differ. It is obvious that they precisely differ by a sign. This gives us the following beautiful result of Hirzebruch. (For sake of completeness we also include the special cases p = 2 and 3). 5.9 Theorem.

Let p be a prime. 9- - g = dim[I’, = dim[I’,

The difference

of the arithmetic

(2,. . . ,2)]e - dim[I’-, (2,. . . ,2)] - dim[I’-,

genera

(2,. . . ,2)]s (2,. . . ,2)]

equals 0

ifp=2 OT~OT pEl if P > 3 andpE3

h(-P)

mod4, mod4.

This result implies for example, that the field of modular functions with respect to l?- is not a rational function field if p > 3. Pinal Remark. One might conjecture that the main term in the formula metic genus is the term which comes from the volume of the fundamental following sense. Let rm c SL(2, R)” (n = n(m) may vary with m) be a sequence of groups commensurable modular group, such that for the different m,m’ with n(m) = n(m)) the T,! are not conjugate in SL(2, R)“. One then might conjecture vol(Hn/rm)+oo

1)

2)

g(r,)

-

(-wOwwm)

vol(Hn/r,)

4.8 for the arithdomain in the

with a Hilbert groups rm and

ifm-rco ~

o

ifm+oo.

This conjecture would imply that only finitely many conjugacy classes of groups r with rational function field exist. Well-known estimates for the class number of an imaginary quadratic field and the estimate 5.2 show that this conjecture is true for the sequence of the usual Hilbert modular groups of K = Q(@),p p rime (a. 5.8 for the formula of the arithmetic genus). The conjecture is unsettled even if one restricts to a fixed n > 2 and to a usual Hilbert modular group. For the case n = 1 see [60].

Chapter III. The Cohomology Hilbert Modular Group

of the

$1 The Hodge Numbers of a Discrete Subgroup r C sL( 2, W)n in the Cocompact Case In this

section

we compute

where

l? c

where

I has no elliptic

llxed

simple

invariant,

the volume

SL(2,R)

the Hodge

n is a discrete

namely

numbers

subgroup

points

with

all those

compact

numbers

of a fundamental

quotient

Hn/I’.

can be expressed domain

with

respect

In the

by means

case of a

to the invariant

measure. The

results

of this

We consider

section

open

are due to Matsushima

domains

D, c C

a,...,

of the complex function

plane

equipped

with

hi : D; --t We may consider

and Shimura.

the “product

a Hermitean

metric,

i.e. a positive

R+ = {CTE R 1I > 0). metric”

0

h= on the domain

D=D1x...xD,. Via

the identification C” ( 21 ,...,Gl)

-(21,y1,...

R2” ,Zn,Yn

>

C”-

Chapter III. The Cohomology of the Hilbert Modular

134 the associate

Riemannian

metric

is given

Group

by

h hl *.

9= 0

(See App. III, Sects. IX-XI.) Such a metric Sect. XII). We especially have the relation

has the K%hler

property

(A III,

A=20=2Ei. We make

use of this

relation

to prove

1.1 Lemma. If a, b are subsets

A( with

and if f is a P’-function

of (1,. . . , n} D (= D1 x . . . x D,), we have

our domain

a certain function

f dz, A dI&,)=

gdz,

A d&,

g.

(Recall: dz, = dz,,

A . . . A d.zap ,

where a = {al,.

..,c+},

15~1

<...
dZb) .)

(Simihdy

1.11 Corollary. A diflerential

form w=

is harmonic

Cf

a,adza

A ab

if and only if all the components

fa,b& A Gb are harmonic.

hoof.

It is easy to see that *(dz,

where From

A d&,)

E C”(D)dz,

A CT&,

a,6 denote the complements of a, b in (1,. the definition of 0, ii we now obtain

n (f dz, A fib)

= c

&,dz, a

. . , n}.

A dZi

on

$1 The

Hodge

Numbers

of a Discrete

135

Subgroup

and

q(f dz,

A &a)

= c

hZdze

A d&, .

ii This

information

together

We consider Poincare metric:

with

A = 20 = 2 n

H” of n upper

the product

is enough

half-planes

to verify equipped

1.1. with

Cl the

.

We have three

types

of motions:

1) The transformations z H Mz , M E SL(2, R)“. 2) The permutations of the variables. n} is some subset, we may define a transformation 3)IfaC{l,..., (Zl,...

,&)-(Wl,...,Wn),

where

Wi = --zi Wi

It can be shown ones.

that

We consider

=

a discrete

for i # a.

Zi

the group

for i E a ,

of motions

section

we make

(Actually

exists a subgroup

in their

H”/l? is compact. index

of 1) by a result

in the linear

Xpyq(I’) especially

Fly .

I’0 c I? of finite

2) is a consequence

We are interested

special

the assumptions:

1.2 Assumption. 1) The quotient 2) There

by these

subgroup r c SL(2,

In this

is generated

the so called hP*q = dim7+q(I’)

(We recall that Mgq(H”) d enotes of type (p,q) with Coo-coefficients,

elliptic

fixed points.

of A. Borel.)

spaces

= {w E Mkq(Hn)r

dimensions,

without

1 Aw = 0))

Hodge numbers .

the spaces of alternating see A III.)

differential

forms

Chapter

136

III.

The

Cohomology

of the

Hilbert

Modular

Group

1.3 Proposition. Under the above assumptions 1.2 we have 1) hPa9 = dim’HP~q(I”) < 00. 2) If w is a P-invariant harmonic differential form, we have

and obviously

the converse.

This follows from the fact that the quotient H”/I’, ture as compact KGhlerisn variety.

carries a natural struc-

1) is a consequence of general finiteness properties of linear elliptic differential operators and 2) is a simple application of Stokes’s theorem. We omit the details because we shall obtain a different proof of 1.3 by means of the theory of automorphic forms. We want to derive an explicit formula for the action of the star operator, and for this purpose we introduce a special basis of differential forms. Notation: Put wi =

dzi A dZi

Yf

’

more generally wa = wzl A . . . Aw,, where a={al)...)

a,},l~al<...
If a, b, c are three pairwise disjoint subsets of (1,. . . , n}, we set R(u,b,c)=dz,Ad&,Awc.

Obviously these elements form a basis of Msq(D), p=#a+#c,

where

q=#b+#c.

One advantage of the Q’s is that they are closed: aqu, b, c) = aqu, Another

b, c) = 0.

advantage is that the star operator can be easily applied to the

32%.

1.4 Lemma. Let (1,. . . ,n}=aubucud be a disjoint

decomposition.

We have &(a,

b, c) = CX(u,

b, d)

Hodge

where

C = C(a, b, c, d) is a certain

(The

where

Numbers

Subgroup

$1 The

constant

of a Discrete

C, which

number.

is not interesting

of a, b, . . . are denoted

the orders

137

Proof. We need some information the definition of the star operator.

for us, can be computed:

by the corresponding

about the pairing We have

< , > which

< dxi,dxj

> =<

dyi,dyj

>=

6ijhr1

< dxi,dyj

> =<

dyi,dxj

>=

0

< dzi, ~j

> =< dZi, dzj >= 2SijY:

< dzi,dzj

> =<

hence

dTi,&j

Greek

>=O

letters.) is used in

’

’

In general < qu, is defined

as the determinant

b, c), qi?i, a, 2) >

of a certain

m

x

m-matrix

(m = a! + p + 27).

If (6, a, E) # (b, a, c) this matrix contains less then m non-zero components. Its determinant therefore is zero. In the remaining case we obtain 0 yz . yi . det

The

2E(")

2E@) 0 0

star operator

0 0

0

0 0 0

0 0

2E(7) 0

2E(7)

is defined

= (-1)

. y; .

d+72Q+@+z7y;

by the formula

<*w,w’>wQ=wAw’. With

the information < CR(a,

Both

we obtained b, d), i-2(& i,ci)

sides are zero except

We now

case both

the pairing

> ‘WO = qu,

b, c) A R(ii, iG,2) .

= (b,u,d),

sides are (ycyd)-”

up to constant

obtain

1.5 Lemma.

it is easy to verify

when (i?,~,~)

and in the latter

about

Let w be a differential

form

w = fQ(u,

of the type b, c) .

factors.

cl

138

Chapter

We have = b) Z%J = c) a(*w) d) a(*,)

a) &

0H 0 H = 0 = 0

III.

The

Cohomology

Modular

Group

f is antiholomorphic in the variables zj, j E b U d. f ia holomorphic in the variables zj, j E a U d. in the variables zj,j E b U c. * f is antiholomorphic in the variables zj, j E a U c. * f is holomorphie

1.51 Corollary.

The relations

in the variables coming from a, an-

are equivalent with: w is holomorphic tiholomorphic in the variables coming variables coming from c U d. (A function p(z) of one complex P(Z) is holomorphic.) Proof.

of the Hilbert

from

variable

b and

is called

locally

constant

antiholomorphic

in the

if z H

We have dw=dfr\SZ(a,b,c)=O

iff Gaf

=Oforj

E bud.

This means - by the Cauchy-Riemann equations - that f is antiholomorphic in the zj’s (j E b U d). This proves a) and similarly b). For c), d) one has to use 1.4. 0 The

corollary

Cancellation

1.51 implies Rule.

If

a certain

w = fn(a,

cancellation

b, c) satisfies

rule. the equation

au=&=a~w=8+w=o, then the same is true

of

instead of w and conversely.

phic

We now want to transform “antiholomorphic variables” ones. For this purpose we consider the diffeomorphism bb

: H” -

H”

Z-

W = 0(,(Z),

zj

for j # b

-Zj

for j E b.

where Wj

=

into

holomor-

$1 The

The

Hodge

Numbers

of a Discrete

139

Subgroup

function

is holomorphic in all variables if fn(u, b, c) satisfies the condition 1.5. What does I’-invariance of w = fR(a, b, c ) mean for the transformed g(z)

To express

this we introduce

=

a)-d)

in

function

f(ObZ)?

the notation

(: ii>-=(2 ib)=(ii :1)(: :) (i !l). M I+ M-

Obviously

defines

an automorphism

M = (Ml,...

of SL(2,

, Mn) E SW,

R). If

R)”,

we define

N=Mb Nj = The

by

Mj

ifjgb,

Mj

if j E b.

groups

rb = {Mb 1M E I?} c SL(2,R)” satisfy the same assumptions general; the quotients Hn/r they carry different “analytic The

of w = fS2(a, b, c) means

r-invariance

I

=

1.2 as l? (but they are different from and H”/lTb are topologically equivalent structures”).

+ dj)2 n(cj~j

n(cjrj

jEa because

+ dj)2f(~)

for M E I’,

jEb

the forms Wi

=

dzi A ai Yi!

are invariant. For the function g(z)

=

f(abz)

we obtain

g(Mz)

=

n jEaUb

PJ

se a;l(Mabz)

= Mbz.)

(cjzj + dj)2g(z) for M E lYb.

l? in but

140

Chapter

III.

The

Cohomology

of the Hilbert

Modular

Group

Holomorphic functions g with this transformation property are special examples of automorphic forms as considered in Chap. I. Assuming a certain condition of irreducibility (1.4.12) we were able to show (1.4.13) that these functions vanish unless or

aUb=0

aUb=

{l,...,n}.

In the first case we have the constant functions, in the second case automorphic forms of weight (2,. . . ,2). We now obtain the complete picture of the Hodge spaces ‘FPq(I’). Th ere are two possibilities for a non-vanishing harmonic form w = fcqu, b, c) .

Case 1. a U b = 0. In this case we have w = const . wC .

These forms are actually invariant - not only with respect to our discrete subgroup - but with respect to the whole group SL(2, R)“, and they are harmonic as follows from 1.4 (and aR(u, b, c) = ~Q(u, b, c) = 0). We collect these “universally invariant” forms in the so-called universal part of the Hodge spaces ifp=q(andO
A fib,

and

g(z) = f@bz> is an automorphic form of weight (2,. . . ,2) with respect to lYb. 1.6 Theorem. Let I?

c SL(2,R)n b e a discrete subgroup which satisfies the assumption 1.2 (especially, that Hn/lJ is compact) and the iweducibdity condition I.4.1,% We have 1) in the case p + q # n

$1

The Hodge Numbers of a Discrete Subgroup

141

2) in the case p + q = n T-P(r)

[lTb,(2,. . . ,2)]. $ bc{l ,...,nl #b=q

E 7-g”

The dimensions of the spaces [rb, (2,. . . ,2)] have been computed in Chap. II by means of the Selberg trace formula in connection with an algebraic geometric method (to come down to the “border weight” (2,. . . ,2)). 1.61 Corollary. If ( in addition) I’ has no elliptic fixed points, of the spaces [rb, (2,. . . ,2)] do not depend on b. We have dim[rb,

(2,. . . ,2)] = vol(Hn/r)

+ (-1>n+r

where the volume is taken with respect to the invariant As a special case of 1.6 we obtain

the dimensions )

volume

element.

the spaces

v(r) = {wEAfIg

1Aw

= O}

of all harmonic I’-invariant differential forms of degree m. Notice. These spaces do not depend on the holomorphic structure on the underlying Riemannian metric. From the equation A = 20 we know that A is compatible (p, q)-bigraduation, hence

but only with

v(r) = $ w-(r). p+q=m

The dimensions

of these spaces are denoted by b” = dimtim =

c p+q=m

hpfq .

They are 0 if m > 2n (or m < 0). 1.7 Theorem. Under the assumptions 1) in the case m # n bm =

of 1.6 and its corollary

we have

( mT2 > if m is even, 0

if m is odd.

2) in the case m = n (&)

if n is even,

0

if n is odd.

b” = 2” . dim[I?, (2,. . . ,2)] +

the

142

1.71

Chapter

Corollary.

III.

The

Cohomology

of the Hilbert

The alternating sum of all the b” F(-l)jti

= (-2)n

. vol(H”/I’)

Modular

Group

is

.

j=O

Final Remark. groups. From

The numbers calculated the general Hodge theory

H”(r)

above are actually (App. III) follows

E Hm(Hn/l?,

dimensions

of cohomology

C)

(singular cohomology with coefficients C). It should be mentioned that the last formula (corollary of 1.7) is also a consequence of the Gaul3-Bonnet formula which expresses the Euler characteristic (= alternating sum of Betti numbers) by means of the curvature and the volume. If l? has no elliptic fixed points, one furthermore has

where

SF’ denotes

the sheaf

of holomorphic

$2 The Cohomology of a Cusp

p-forms

Group

on the analytic

manifold

Hn/r.

of the Stabilizer

Let

D c R” be an open We assume finite index

domain and I? a group of C”-diffeomorphisms of D onto that I? acts discontinuously and that I? has a subgroup which acts freely on D. We denote by

itself. ITo of

wL(w C” -differential the linear space of all l?-invariant We may consider the so-called “de Rham complex” ... -

ML(D)r

d

Mg’(D)r

(“complex” means d. d = 0) and the de Rham actually C-vector spaces) HP(r)

= HP((DJ))

forms

+

cohomology

= CP/BP )

of degree

p on D.

... groups

(they

are

$2 The

Cohomology

Group

of the Stabilizer

143

of a Cusp

where Cp = ker(M&(D)r BP = im( ML1 Of course

+ ( D)r

M~‘(II)~) --+

M&p)r).

we have Hp((D,r))=O

if

Notice. By the theorem of de Rham and the singular cohomology group convex) we have furthermore

there

pn.

is a natural isomorphism C). If D is contractible

HP(D/I’,

HP((D, I?)) S Hp(D/r,

H*(I’,

C) denotes

the group

cohomology

I’)) D is

C)

E HP(r,c) where

between HP((D, (for example if

’

of I? acting

trivially

on C.

We now assume that a discrete subgroup r c SL(2, R)” with cusp 00 is given. We want to compute the cohomology of the stabilizer roe. Recall that the stabilizer roe consists of transformations of the form

We have two types of differential forms all thEse transformations, namely 1) dq

A..

which

are closed

and invariant

under

. A dz,,

2) *+y.../\f+ Y,P

where a = (c-81,...,upL

l<ar

<...
The classes of the dy, /ya are not independent. invariant. Therefore

h/n -

and

-

The

Cy=,

Yn-1

the same class in the de Rham

2.1 Proposition. We obtain di$erential forms

cohomology

a basis of H”(r,)

group.

by the classes

of the following

a)m/y,

,

a c (1,. . . ,n - 1) , #a = m .

b)mln (h/,)/y,

A dxl

A..

log yi is

&/n-l

%+...+-

Yn define

function

. A dx, ,

a c (1,.

. . ) n-l},

#a=m-n.

Chapter

144

For the dimension

b,

There are several proofs the exact sequence

and the sponding

funny

(Betti

We hope one:

that

Cohomology

the reader

Perhaps

or equivalently

Hn/t -

Hn/I’,

Modular

Group

most

natural

the

one is to consider

spectral

sequence

corre-

.

appreciate

the following

proof

of 2.1 as a

of all transformations z+.s+b,

also operates

the

sequence

will

of the Hilbert

of the spaces we obtain

proposition.

spectral

PTOO~of 2.1. The group

The

numbers)

of this

Hochschild-Serre to the fibration

III.

Nc=l,

on

D= We may identify

D with

{z E H” 1 Ny=l}.

R2n-1

by means

of the mapping

R2W-1

D-

~-(~l,...,~,,~~gYl,...,~~gY,-l). =fb1

\

4 =e,-,

We especially may consider the de Rham cohomology of (D, I’,). back differential forms by means of the natural inclusion R 2n-1

Pulling

rD--tH”

gives a mapping HP((Hn, We make

use of the fact that

The

HR/r,

spaces

and

R x

D/I’,

L))

a homotopy

statement now follows

HP((D,

roe>> .

this is an isomorphism are obviously

D/r, is therefore

-

homeomorphic.

for all p. The

natural

imbedding

LI Hn/r,

equivalence and induces isomorphisms from the de Rham theorem.

in singular

cohomology.

The

$2 The

Cohomology

Group

It is possible to “imitate” integrating along t = vyl The

of the Stabilizer the homotopy . . . . . y, .

argument

directly

in the

de Rham

complex

by

formula h(2)

= f(z)dz,

where

5 g(z)

shows

145

of a Cusp

for example

that

the first

=

de Rham

f (W I

0

cohomology

group

of R vanishes.

is compact! We therefore can use Hodge theory to The quotient D/I’, compute the de F&am cohomology. For this purpose we choose any Riemannian metric g on D (2 Rzn-’ ) which is invariant under all transformations Z-+&Z+b,

NE: = 1

(not only under the given discrete subgroup I’,). Such a metric example the restriction of the PoincarQ metric on H”). We now consider

a whole sequence of groups,

G(l)=(‘;

+o

exists (for

namely

$),

Z=l,2,3

,...

The de Bham cohomology groups of Ia, and G(1) are of course isomorphic. Before we continue we make the following assumption Assumption. The group PM splits, i.e. if e is a multiplier, then the transformation z --+ cz is contained in PO0 (and not only some transformation of the form z ++ cz + b, as demanded in the definition of “cusp 00”). We shall show at the end of the proof how the general case can be reduced to the split case. In the split case G(I) consists z H

of all transformation

EZ + a/P

)

EEA

(group of multipliers

a Et

(group of translations

r,) of r,) ,

of

They contain all the transformations of G( 1) = Pm, because 12t c we denote by ‘H*(G(I)) th e sp ace of all harmonic m-forms on D which G( Z)-invariant , we obtain

IH”(G(I))

c

wy,)

t.

If are

.

Equality holds, because the dimensions are the same. The union of all fm2t is dense in R”. We hence obtain by a continuity argument that if

Chapter III.

146 is a r,-invariant do not depend

The Cohomology of the Hilbert

harmonic differential form, then the functions on x. The invariance under the multipliers z H

now

Modular

Group

fa,b(x,

u)

EZ

gives us fa,b # 0 +

We treat

the first

We now

b = Q)01 b = (1,.

case b = 0 (the

have a closed w =

case b = (1,.

differential

c

form

f,(u)du,

. . ,n}

.

. . , n} is similar):

of the type

(u = (Ul,...,%l-1))

aC{l,...pa-1)

and want tion

to show

that

these forms c

are cohomologeous G(dYa)lYa

to a linear

combina-

.

This means that the difference is of the form d2, where ij denotes a Pminvariant differential form on D. It is easily proved that no x-variables occur in L2. Hence we are faced with the problem of determining the de Rham cohomology of A acting on Fp;-r or, equivalently, of the lattice log A acting by translations on R”-l. The cohomology of a lattice L c FPm (acting by translation) can be determined by means of the same trick as above. Replacing L by (1/2)2L and using the usual Euclidean metric (g = E = unit matrix) one shows that each harmonic form which is periodic under L is necessarily of the form w=

Cadua

c aC{l

,...,ml

with constants C,. It is easy to see that the independent. If we transform du, by means of log : Rf

-

classes

of du,

axe linearly

FPn

yr---tu=logy, we obtain

the differential

forms

(dy,)/y,

instead

of du,.

Cl

We finally show how to avoid the assumption that !Zoo splits. First of all we may assume that at least for one multiplier c E A, e # 1, the transformation z H EZ is cant ained

in I’ oo, because

we may conjugate ZHZ+CX,

I’m by means

of a translation

$2 The

Cohomology

Group

of the Stabilizer

147

of a Cusp

where (Y = b,(l We now

- E)-r

.

claim

2.2 Remark. If IToo contains

at least one transformation Z.&oZ,

then there exists is of the form

a natural

of the form

co # 1,

number

d such

that

each transformation

of r,

z+-+ez+b, EE A where

PTOOf.

db E t

(group

of multipliers

(group

of translations

of I?,) of I?,).

Let z+-+Ez+b

be any transformation

of roe. We see that z-eaoz+b

and z are both obtain

contained

in rM.

(EZ + b) . E,,

Multiplying

one with

the inverse

of the other

we

b(1 - ~0) E t . We know

that

on t, especially

A acts by multiplication (1 -&o)t

The index fore there

has to be finite, exists a natural

because number

c t .

both groups are isomorphic d such that

to Z”. There-

dt c (1 - c,,)t and this

implies b E d-h.

Remark

2.2 shows

us that

the groups

G(l)=(l; ~)rk(~ pl)

El

Chapter

148

III.

The

Cohomology

of the Hilbert

Modular

Group

contain roe if 1 is divisible by d. This is enough to carry through the argument used during the proof in the split case.

Q3 Eisenstein Cohomology Let r c SL(2, R)n be a discrete the natural mapping

subgroup

such

(HR)*/r

that

is compact.

We investigate

h Hrn(l?)

-

Hm(rt2j

>

1

a3

j=l

where ~1, . . . , Kh denotes a system gruence group of a Hilbert modular which maps isomorphically to the theory is due to Harder [26].

of representatives of the cusps. In the case of a congroup we will construct a certain subspace of H”(I) image of by means of Eisenstein series. This

H"'(r)

We start with the basis of H”(I’,) Let w be one of the basis elements, i.e.

1) 2)

w=-

w = -4/a

&a 7 Ya

h dxl A..

which has been constructed in 52.

(m
UC{1 )...) n-l},

. A dx,

,

UC {l)...)

n-l},

(m2n).

YlZ

It is convenient to replace the differential forms in case 2) by a basis which fits better into our setting of Eisenstein series. 3.1 Remark. Assume

n 5 m < 2n.

The differential

dza A fit,

w=

forms

A dz b>

Ya

where aUb= are r,-invariant

{l,...,

anb=B,

n},

and closed. Their

classes

#a=m-n, define

PTOO~.We show that the differential forms dz, A CE’i,

A dzb

Ya

and *Adx Ya

1 A ...

A dx,

a basis of H”(I’,).

149

$3 Eisenstein Cohomology

are cohomologous upto a constant factor. This obviously follows from the fact that a differential form of the type $

A dxb A dy, , a # {l)...)

n},

bUc=

(1)“‘)

n},

defines the zero class if c # 0. This is trivial if c n a # 0. But if c n a = differential form yi* 2

is I’,-invariant,

AdXb

Adyct

,

0, we choose some index i E c. The where c’ = c - {i} ,

and its exterior derivative is up to a sign ha yc A dxb A dy, .

Let now w be one of our basis elements, i.e. Case 1: w = (dy,)/y,

,

&se2:~=~“~~~~Adzb

(aUb={l,...,

n},aC{l,...,

n-l}).

These forms are I’,-invariant, but we want to have I-invariant It looks natural to construct them by symmetrization: E(w) :=

forms.

c w 1M. MEr, \I’

We use the notation wIM:=M*w. We have (wIM)IN=wlMN, and hence w I M does not depend on the choice of the representative M. The main problem will be the question of convergence. The formulae dz 1M = (cz +d)-2dz, &I

M = (cz+d)-2&,

y 1M = I cz + d I-’ -Y , dy = (1/2;)(dz

- do)

show that the series E(w) can be expressed by Eisenstein series of the following type:

Chapter

150

We consider

is independent the Eisenstein

of j and that series = Ez,p(z)

c MEr,\r

considered E(w)

where p(z) is a linear More precisely

For the total

Cohomology

r is integral.

=

I, §5 we already

The

(Y, ,L?E Z” of integers.

two vectors

&J(Z)

In Chap. ma&)

III.

combination

of the Hilbert

We assume

We then may

qcz

Modular

that

(formally)

+ d)-“N(cZ

consider

+ d)+

the case ,f3 = 0. We now

Group

.

obtain

(for-

= P(Z) . WY of the Eisenstein

series introduced

before.

weights

we obtain: Case 1:

r=o

(w=~Ca+q,

Case 2:

r = 1

(w = c c, dZ$fia

From our assumption second case

A dza).

a c { 1, . . . , n - 1) we furthermore

obtain

in the

a#P. Up to now our consideration the question of convergence.

has been formal. We now have We have already proved that

to deal

with

c IJqcz +d)I-2r converges for all real r > 1. The same proof shows that this series does not converge if r = 1. In the first case (2~ = oj + /3j = 0) we are rather away from the border of convergence. The second case looks better. Here we are precisely at the border of absolute convergence (r = 1). Following an idea of Hecke we can define E,,p(z) in this case as follows: We first introduce for real s > 0 the series E,,p(z,

This

s) :=

series converges

c r,\r

N(CZ + fpN(cF

absolutely

+ cl)-@ 1 N(cz

+ d) 1-2s .

(in the case 2r = oj + pi = 2, s > 0).

$3 Eisenstein

One may

exists.

151

Cohomology

ask whether

If this happens

The series fined by

E,,~(z)

the limit

we say: admits

He&e

summation.The

value

of this

series

is de-

We keep the notation E,,&) where

the symbol

“=”

“ = ” c

N(cz

indicates

that

+ d)-V(cz

+ d)-fl

we applied

,

Hecke summation.

In the next section ($4: Analytic continuation of Eisenstein series) we deal with the question of Hecke summation. We shall show that in the case of a congruence subgroup l? (i.e., a discrete subgroup of SL(2,R)” which contains a principal congruence subgroup I’K[a] of some Hilbert modular group I’K. A deep Theorem by A. Selberg states that each discrete subgroup with a fundamental domain of finite volume and with at least one cusp is conjugate to a congruence subgroup.) Hecke summation always exists. We hence assume in the following that l? is a congruence subgroup. Let

be two vectors

of integers

b)

with

the properties

“j+/3j=2

Then

the Eisenstein Ea,B(~)

can be defined by He&e of the differential form

j=l,...,n.

fOT

series “ = ” c

N(cz

summation.

Cd=

+ d)-YV(cz

We now

dza A fia

+ d)-fl

return

to the symmetrization

A dz b.

Ya

We have w 1 A4 = w - N(cz

+ d)-“N(cz

+ d)+

,

Chapter

152

III.

The

where

Cohomology

of the Hilbert

Modular

Group

1 ifjEa “j =

2

ifjEb,

1

ifjEa

0

ifjEb.

Pj = The results form

about Eisenstein

E(w)

“ = ”

c

series described

above show that the differential

w 1M “ = ” w . c

N(cz

+ dpv(cZ

+ d)+

MEI-m\I’

can be defined by Hecke summation. 3.2 Proposition. the differential E(w)

“ =

”

Let w be one of the basis elements form c

c

wlM:=;li

1 N(cz

described

+ d) I-”

in 3.1.

w 1M

Then

(s 7 0)

h4am\r

MEr,\l-

exists.

We now have to investigate whether the differential form form E(w) is closed or not. A differential form

f(z)

dz, A dz,

y.

Adq,

(aubc

{l,...,n})

is obviously closed if and only if f( z ) is h o1omorphic in the variables coming from

b.

R.ecall:

In the next section ($4, Theorem 4.9) we prove the existence of a real number B such that adz> - VJY is holomorphic in all variables zj with /3j = 0. As a consequence, the differential form E(w) is closed if and only if the number B is 0. We also will obtain some information about the constant B: It is 0 if the number of all j such that

$3

Eisenstein

Cohomology

153

is less or equal n - 2. In our situation

we have

aj = pj = 1

e

jEa

and m=#a+n=2n-#b. This means that B is zero if #b22

or

m<2n-2.

3.3 Proposition. Assume n<m<2n-2. The

differential

form E(w)

“=”

1

w 1 M,

MEI’,\I-

where W=

c

c

a

dz, A &,

aUb=

A da ,

Ya

{l,...,n},

(which can be constructed by means of Hecke ries) has the following properties: 1) E(w)

m=#a+n

summation

of Eisenstein se-

is closed.

and w define th e same cohomology class in H”(l?,). 3) The image of E(w) in P(I’,) is 0 if Ic is a cusp inequivalent to 00. 2) E(w)

We know already that E( w ) is closed, because it depends holomorphitally on the variables coming from b. It is especially a holomorphic function of zn. The Fourier expansion of E(w) 1 M (4.9) may be rewritten in the form Proof.

E(w)jM-Aw=w.

C

b,(zl,.

. . ,z,-l)e2nignrn

where the coefficients bg are independent of z,. Integration a function of z, term by term gives us a function f(z)

=

b,(zl,.

C 9-O

,9”

. . ,zn-l)/2rig,.

>o

We now consider the differential form W’

where 6’ = b - {n}.

:=

f(z)dza;Ga

A dq,, ,

e2Rign*n

,

of the series as

.

Chapter III.

154

The Cohomology of the Hilbert

3.31 Remark. The differential form w’ is invariant

UT&T

Modular

Group

(MI’Mml),.

The proof is easy, because invariance under transformations ZH&Z+b by properties of the Fourier coefficients. The invariance of w’ will follow from that of E(w) 1M. We leave the details to the reader. can be expressed

We now obtain that E(w) 1M-Aw defines the zero class in

But from 4.9 we know that A = 1 if M is the unit matrix (hence E(w) N w in H”(I’,)) and A = 0 if the cusp rc = M-‘(m) is inequivalent to 00 (and hence E(w) N 0 in iY”(l?,)). This completes the proof of 3.3. cl By the method of transforming an arbitrary cusp to infinity one can easily generalize the results of Proposition 3.3 to arbitrary cusps: Let /c=M%o,

M E SL(2, K) ,

be an arbitrary cusp and w one of the basis elements described in 3.1. For the moment it is not necessary to assume m < 2n - 1. We may consider Eisenstein series with respect to the conjugate group FM = Ml?M--1 , especially j+(w)

“ = ”

c N:(rM)co\rM

The form l+(w)

(M

is again F-invariant. The one-dimensional space (@“(w)

1M) .4:

depends only on the P-equivalence class of IC. 3.4 Definition.

The space

w, 4 (m2 4

WIN.

§3

Eisenstein

155

Cohomology

is generated

by all differential

forms J@” C-4 I kf 7

where A4 E SL(2,

a)

K)

)

w is a basis element

of degree

b) The subspace

m) C W,

f0(r,

consists

m (9.1). m)

of all closed forms.

The image of &(I’, denote it by

m) in Hm(I’)

is th e so-called

i7g(r) = Im(Eo(l?,

Eisenstein

cohomology.

We

wyr)) .

m) -

In the following Kl,...,Kh

denotes

a complete

3.5 Proposition.

set of representatives Assume

of all cusp classes.

n 5 m 5 2n - 2. The natural

restriction

map

P(r) + 6 HyrKj 1 j=l

is surjective.

Proof.

The

we

have isomorphisms

proposition

We now come one-dimensional

is a consequence

to the border and generated

w=

of 3.3.

case m = by

d.q A . . . A d.z,-1

A d&

A..

. A dZ,ml

Yl * * * Yn-1

We choose

matrices

Mj

E SL(2,

K)

with

MjKj=oO We then

consider

the differential

(l<j
#J(i) := E”jrM;‘(W)

0

2n - 1. In this

1 Mj.

case H”(I’,)

A dz,

.

is

Chapter III.

156

The Cohomology of the Hilbert Modular

Group

We have (see 3.4)

E(l?,m)=&E

(j)

(m = 2n - 1) .

j=l

The

Fourier

expansion

of E(j)

= Aj + Bj/Ny

E(j) The

same argument

has the form

the forms

E(j)

a

are linearly

in contrast

.

Kj=m.

independent

and we obtain

2n - 1) = h .

dimE(I’, But

terms

as in the case m 5 2n - 2 shows

AjfO Hence

+ higher

to the case m < 2n - 1 the constant

Bj is not zero!

The reader who goes carefully through the formulae of $4 will see that Bj is different from 0. But we do not need this, because we can use another type of argument: If all the Bj were 0, the proof of 3.3 would show that the mapping P-l(r)

-

6

P--l(ry)

2 Ch

j=l

was surjective.

But

in $5 we will

see by means

Bj we consider E := C CjE(‘)

Such a linear

combination

is closed

if and only

h c j=l

We now

duality

= h - 1.

dimH2”-‘(I’) To get rid of the constant

of Poincare

CjBj

= 0.

obtain

3.6 Proposition. Assume m = 2n - 1. Let

linear

. if

combinations

that

$3

Eisenstein

157

Cohomology

be a set of representatives

of the cusp classes of I?. A linear combination

where

= Aj + Bj/IVy

+ higher terms,

is a closed form if and only if h

CjBj

c

~0.

j=l

In this case the image of E in H”‘(rKj Aj#O

3.7 Corollary. map

Assume

) is Aj - w 1Mj.

We have

KjmmmodI’.

e

m = 2n - 1. The dimension

of the image W of the

H”(r) - 6 Hm(rKj>= @ j=l

is at least h - 1. We have isomorphisms Eo(l?,m)‘-t

As we already mentioned

H&(r)

-

f W .

we shall prove by means of Poincark

duality

dimW=h-1. Hence the codimension

of the image will turn out to be 1.

We now define the Eisenstein

cohomology

3.8 Definition.

Hkis(r)

1)

2) ifm=22n

H&(r) orl<m
= HO(r) = 0

in the remaining

cases.

that

Chapter

158

III.

The

Cohomology

of the Hilbert

We shall see later that now in all cases the Eisenstein isomorphically onto the image of

P(r) -

Modular

cohomology

Group

is mapped

6 Hm(r,j ) . j=l

Another

justification

for definition

3.8 is the

Remark. One could also try to construct Eisenstein cohomology classes the case 0 5 m < n by means of symmetrization of the basis forms w=dy,/y, As we already

mentioned

(aC{l,...,n-1)).

at the beginning

qw,

in

s) = c

1 N(cz

of this

section

the series

+ d) l-s w 1 M

converges only if Res > 2. We hence cannot take the limit s + 0. But nevertheless this series has an analytic continuation into the whole s-plane (as a meromorphic function) and we can look at special values of se where E(w, SO) is closed or at values SO where E(w, s) has a pole and where the residue is closed. We could do this with the methods of $4. The reader who goes through the details will find that only in the case m = 0 a non-vanishing cohomology class can be found along those lines, Ho(r) is one-dimensional and generated by the class of a constant function (O-form). This constant function can be obtained as a residue of an Eisenstein series of the type

at s = 0.

34 Analytic We fix a totally

Continuation real number

field

of Eisenstein K

and an ideal

o#qco, We consider

the main

congruence

o=oK. subgroup

rK[d={MESL(2,o) Let I C SL(2, Eu,p(z,

R)”

s) = E,r,p(z,

1 MrEmodq}.

be In $3 we introduced s) =

c r,\r

Series

N(cz

the series

+ cd)-“N(cZ

+ cl>-@ 1 N(cz

+ d) 1-29 )

$4 Analytic where

Continuation

159

of Eisenstein Series

a, /I E Z” are two vectors

of integers

with

the properties

Q#P,

4

b)

2r := “j

+ p.j

of j (f or our

is an even number, independent sufficient). The series converges

purpose

r =

1 would

be

the behaviour

at

for 2r+Res>2.

In this

section

exists and the cusps.

we want

we will

to prove

also obtain

that

precise

4.1 Remark. Let I? c I” group

of finite

index.

be a subgroup Then we have

E,‘:ak

c

3) =

in the case r = 1 the limit

information

which

about

contains

E,&>

SL(2,

s)lW

R)n

aa a aub-

7

MEr\I”

where (f

the operator 1 M)(z,s)

. 1M = N(cz

is defined + d)--“N(cZ+

by d)-p

1 N(cz

This remark, which is of course trivial, allows of analytic continuation to the main congruence assume, unless otherwise stated, that

+ d) I+

f(Mz,s)

.

us to reduce the question group. From now on we

We determine a set of representatives of I’,\I’. A pair o) is the second row of a modular matrix (E SL(2,o)) unit ideal

(c, d) of integers iff it generates

(in the

cc, 4 = (1) * A necessary of course

condition

for being

the second

c=Omodq, But this condition may replace

(together

with

d=

row of a matrix

in l? = rK[qj

is

lmodq.

(c, d) = 1) is also sufficient,

because

we

Chapter

160

III.

The

Cohomology

of the Hilbert

Modular

Group

and hence assume b=Omodl--d

=+-b=Omodq. From the relation

ad - bc = 1 we then obtain

a E 1 mod q.

Two matrices (:

;),

(;I

;)Er

define the same coset iff there exists a unit e E o*

)

e-lmodq

such that c’ = EC )

d’ = Ed.

In this case we call the pairs (c, d) and (c’, d’) associate have the explicit form of the Eisenstein series &,&,

3) =

c (c,d)=l,(c,dh

CEO mod

q,del

N(cz + d)-TV@+ mod

1 N(cz

+ d) 1-2” .

q

The subscript (c, d)q indicates that the summation of representatives of associate pairs. This type of Eisenstein man.*

d)+

mod q. We now

series was considered

is taken only over a set

in 1928 already

by Klooster-

(In this connection we should mention that the method of HeckeKloosterman could not be generalized to other modular groups, for example the important Siegel modular group. In a deep paper, Langlands developed a method which gives analytic continuation of all Eisenstein series on semisimple Lie groups. But it would be even more complicated to extract our special case from Langlands’ paper than to give the direct proof following Kloosterman.) Before we start with the analytic continuation we have to introduce more general Eisenstein series: Assume that (besides our level q) a further ideal a is given. We do not demand that a be integral. For two elements CO

* Kloosterman easily reduced by Maa%.

Ea,

do E a

actually merely considered the case /3 = 0. But to this case by means of certain simple differential

the general operators

case can be introduced

$4 Analytic

Continuation

we define

the Eisenstein

Ga,p(z;

S;(CO,

do);

161

of Eisenstein Series series (of level

q) as

a) = Gz,,(z; s; (co,~o); a) = I qcz + d)-aN(cz + d)+ c

1 qcz

+ d) p

.

CECO mod qa mod qa,(c,d)q

d=do The

summation

is taken

(c, d) E K x K ,

over a set of representatives

(c, d) # (0,O) ,

with respect to the introduced associate if there exists a unit

out (for

that no condition z E H”) if 2r+2Res

and represents a + p = (2,.

an analytic .

.

)

E qa ,

CO

d-

E qa

do

relation: Two pairs (c, d), (c’, d’) are called E E o*, E = 1 mod q, with

d’=Ed.

c’=m, We point converges

c-

of pairs

of coprimeness

> 2 function

(2T

=

is demanded!

"j

+

on s (we

pj

E

This

series

22)

are interested

in the

case

2)).

4.2 Lemma. The group GZ+(2, K) = {A E GL(2, K) acts on the f(z,

s)

H

space generated by all Ga,p

R-vector

group

Because

by means

of the formula

(det A)2r+2S .f(Az,s)N(cz+d)-“N(cz+d)+

PTOO~. It is not difficult to find an explicit Eisenstein series. For the sake of simplicity The

1 det A > 0)

G1+(2,

K) is generated

1N(cz+d)

I-”

expression for the transformed we make use of the simple fact:

by the special

matrices

of the formula

one may even assume that a lies in a given ideal 4.2 is very easy. For example Ga,p(z

+ a; s; (CO,do); a) = GQ(z;

a. Now the proof

s;

(CO,

do +

coa);

a)

of Lemma

.

Chapter

162

G,,B(--Z-‘;

III.

The

Cohomology

S; (co, do); a) = N(z)“N(Z)~

Our next goal is to express of the G’s. For this purpose

of the

1N(z)

the Eisenstein

Hilbert

Modular

Group

12’ Ga,p(z; s; (do, --co);a).n

series E as a linear combination

we need the notion of a “ray class

mod q”.

Notation: 1 = group of all ideals of K, ‘FI = group of all principal

ideals.

A (not necessarily integral) ideal a E Z is called coprime prime divisor of q occurs in the prime decomposition of a.

to q, if no

We denote by %-I> c 1 the subgroup of all ideals which are coprime to q. We also have to define a certain subgroup R(q) of the group ‘H of principal ideals: A principal ideal belongs to X(q) i f an d only if it has a generator a! with the following two properties: a) cx > 0 (totally positive). b) The denominator of the ideal (CX- l)q-’

The usual proof of the finiteness that the group

is coprime

of the class number

to q, i.e.

h = #2/H

also shows

~tw-w is finite. Its elements

are the so-called

We also need the Mobius integral ideals. Let

ray classes

function

mod q.

p(a) which

is defined on the set of

a = p? . , . . . pz be the prime decomposition

of an integral

l-44 =

WY o

ideal a. One defines if all Vi = 1 otherwise,

p(0) = 1. The Mobius

function

has the basic property

c44 PaCl={01

ifq=o ifq#o.

$4 Analytic

Continuation

163

of Eisenstein Series

After these preparations we can give an explicit expression of E as linear combination of the G’s. Introducing the Mobius function we may get rid of the condition of coprimeness in the definition of E, namely

c p(a)N(c*+d)-w(c~+d)-~ 1N(cz+d) 1-28 -

~%,p(z,4 = c

The occuring ideals a are of course coprime with q (because d G 1 mod q). We obtain

a integral coprime

(c,d)+O,l) (c,d)E(O,O)

c

with

q

N(cz +d)-V(cz +cl>+ 1N(cz +d)1-29 .

mod q mod a,(c,d)q

We now fix a ray class mod q

and consider the contribution

of this ray class to Eu,p(z, s):

Ea,p(d; z, 3) = C

h+

aEd

a integral

(c,d)=(O,l) (c,d)=(O,O)

c

qcz+qYv(cz+cl>+1N(cz +d)1-25 .

mod q mod a,(c,d)q

Of course we have

Ea,&,4 =

Ea,p(d;z,s>. c dMqL)/Wq)

Now we fix an integral ideal in our given ray class A

wEA,

a0Co.

Then every other ideal a E A is of the form a=y-ao,

y>O.

:

164

Chapter III.

&,,@; z,4 = c

Group

The Cohomology of the Hilbert Modular

p(a)N(yp+S)

aEA

a integral N(c’z

c (c’,d’)r(O,l) (c’,d’)E(O,O)

+

d)-“N(c’z

+

cl’)+

1 N(c’z

+

d’)

I-2e

.

mod q eo,(c’,d’)q

mod

The ideals q and aa being the property

coprime,

we can find a pair CO(= 0) , do with

= (0, 1) mod q (CO,do) G (0,O) mod a0 . (CO,

We now

obtain

&,p(d;

z, s) = N(ao)2(p+g)

do)

c

~(a)nT(a)-2(‘+“).G,,B(z;

s; (CO,

and A C o* be a subgroup

of finite

do);

a~).

GA

a integral

4.3 Lemma. Let m C K be a lattice acting on m,

index

Axm+m (E, cZ)H Then if a runa

over a complete

system

EU. of representatives

of m-

(0) mod A

the series c

I w4

I--

,

0 > 1>

converges.

4.31 Corollary. The series

&A

a integral

defines an analytic finction

on the domain

Re s > 1.

Proof. We can choose the system of representatives such that (al,. . . , a,) is contained in a fundamental domain Q of A acting on Rn by (X,$-+X&.

Such fundamental domains have been determined. The series can then be compared with the integral 1 vol(m>

J rEQ,IN(z)l21

( iv(x)

I---b dx1 . . . dx, .

This proof gives a little more than stated in 4.3, namely

cl

$4 Analytic

Continuation

4.32 Remark. The

of Eisenstein

(Notations

165

Series

as in 4.3)

limit

lilh((T - 1) c’

1N(a) 1-O

exists (and is unequal to zero). For our purposes we do not need the deeper result of Hecke that (s - 1) c’

I N(a) r

has an analytic continuation as entire function into the whole s-plane. Analytic Continuation of the Eisensteiu Series G (as functions of a). We fist consider the simpler series fa,p(z; s; m) = C N(z + g)-aN(Z gem

+ g)-@ 1N(z + g) l--2s

where m c K is any lattice, for example an ideal. The function fLy,p remains unchanged if we replace

z+z+a,

aEm,

and hence admits a Fourier expansion fa,p(z; s; m) = me(“‘/2)S(p-a)

1

hg(y)e2”‘s(g”)

.

gEm*

Here m* denotes the dual lattice of m. The square root of the discriminant d(m) equals the volume of a fundamental parallelotope P of m. The Fourier integral gives the following expression for h,:

h,(y)= e(+9Sk--B) . = e(42)S(--8) J = 1Ny ys

J

I

fcl,p(z, s; m)e-2~is(gz)dx P

R” N(z)?IV(F)-p

( N(z)

ls2’ e-2niS(gz)dx

/iv(y”+q.

N(l - iz)-m’N(l

+ ix)-@

I N(l - ix) l-28 e-2rriS(gyr)dx.

R”

The integral splits into a product of n integrals of one variable. We first collect simple properties of this one-variable integral.

Chapter

166

III.

The

Cohomology

of the Hilbert

Modular

Group

4.4 Lemma. Put

such that

o+,B+Res>l. Then

the integral

m (1 - it)-W(l

h(y; (Y + s; p + s) :=

+ it)-s

1 1 - it I-”

emitYdt

J -cc3 converges (for arbitrary y E R). It has an analytic in fact morphic function into the whole s-plane, y # 0. Special values of h:

continuation as a meroas an entire function if

a)y=O: h(o;~+s;P+s) b)s=O(a,pEZ).

=

-

1)21-("+p+28)

Onehasfory>O NY; a; P) =

where Pa&Y) for example

+ P+2s qa+s)qp+s)

273a

is a certain

h(-Y;

polynomial

P; a> = e-ypa,p(y) in y which

Pcr,p(y>

= 0 if a 5 0

P@(y)

= (QT1)!Ya-’

Basic estimate for h: Ifs a constant C such that

varies

in a compact

,

can be computed

explicitly,

if (y 2 1 . set of the s-plane,

there

exists

I h(y; a + s; ,f?+ s) 1~: Ce-IYI12 . Proof.

If we replace t by -t,

we observe

h(y; a + s; P + s) = h(-y; P + s; a + s) and hence assume y 2 0. For the computation of the integral at y = 0 and for the analytic continuation as well as for the basic estimate we may assume p = 0, because the integral only depends on a! + s and ,f3+ s. I Computation of the Integral at y = 0. Integration by parts gives h(0; a + s; s) = S/(CY- 1). [h(O; a - 2 + s + 1; s + 1) - h(0; QS- 1 + s + 1; s)] ,

$4 Analytic

Continuation

of Eisenstein Series

167

if 01 # 1. The same recursion formula is satisfied by the r-expression in 4.4. It is therefore sufficient to treat the cases (Y = 0 and a! = -1. In both cases the transformation t2 + 1 = 5-l reduces

the integral

II Analytic

to an ordinary

B-integral.

Continuation.

The analytic continuation will follow from integration. We hence define the integrand (1 - ity(l+

a deformation

t2)-se-“ty

)

t , Im t < 0. (It looks the lower half-plane, beThe only problem is the

= e--sh3(l+ta)

.

We define log(1

t2)

+

= log 1 1+

t2 1 +i

arg(1

+

t2)

where arg(l+P)

:= arg(t

-7r/2 -37r/2 This

definition

1)

arg(1

2)

q(l

t2) = + t2> is

Let

t

3x/2,

< arg(t-i)L

7r/2.

be a point

three

- i),

properties:

0 if t E R . continuous on the domain

{tEC 3)

arg(t

<arg(t+i)<

has the following

+

+ i)+

1 ImtSO,

on the critical 1 <

it

;t$[l,co)}. line:

< 00

(it

E R) .

We have lim

arg(1

+ U”) = -7r

arg(1

+ U”) = 7r .

u-+t,Reu>O

and lim u+t,Reu
of

y > 0 )

not only for real but also for complex arguments promising to deform the path of integration into cause e-‘QJ is rapidly decreasing if Im t + -oo). definition of the complex power (1 + q--s

of the path

)

168

Chapter

III.

The

Cohomology

of the Hilbert

Modular

Group

We already mentioned that our integrand is rapidly decreasing if Im t ---t -oo. Hence we may deform the path of integration (real axis from -oo to +m>. We now decompose

our integral

into two parts:

a) The integral along the circle around -i is of course an entire function of s. b) We compute the jump of the integrsnd at the critical line if we pass it from the right to the left half-plane: The jump of the function (1 + t2)--s at a point

t on

(it ~]l,oo)) is - e--aia] = 2isin7ris.

the critical

axis

(1+ 1t 7)--s[p5 The contribution

of the two vertical 2i sin 7ris .

J

] t 12)-’ .

to our integral hence is

lines

(1 - it)-a(l+

(l+

] t 12)--se--itydt,

where the path of integration is the vertical line on the right hand side (starting from a point it0 , to > 0). This integral again defines an entire function of s. III The basic estimate is an immediate defines the analytic continuation.

consequence

of the formula

IV The special value of h at s = 0: F’rom the residue theorem positive y h(y; a!; p) = -2 ri

tF&s(l - it)-“(1

The residue is zero if a 5 0. If cr 2 1 it is (-i)-” a,-1 in the expansion (1 + it)-be--itY

= 2

a,(t

+ it)-be-ity

which

we obtain for

.

times the Taylor coefficient

+ i)” .

v=o All these Taylor coefficients are obviously products of e-v with certain polynomials in y. Their trivial computation completes the proof of 4.4. Cl As a consequence of Lemma 4.4 we obtain the analytic continuation of the series fa,a :

$4 Analytic

Continuation

169

of Eisenstein Series

4.5 Proposition. Let m c K be a lattice in K. The series

fa,j3(z; s; m) = C N(2 + g)-aN(T

+ g)+

I N(z + g) I-”

has the Fourier expansion

. c hg(y)e2”“s(g”) , gem*

vol(p)e(“‘/2)S(fl-a)

where h,(y) =I Ny /1-2s-2r ~hg(2?rgjyj;oj+s,~j+S). j=l

This Fourier series defines an analytic continuation of fa,~(z,s; m) as meTomoTphic function into the whole s-plane. The only poles come from the zero Fourier coeficient, i.e. fa,p(z; 9; m) - vo1(P)e(“‘12)S(B-“). r(2T

) Ny j1-2s-2r (27r)” +

r(aj

29

+

1)

* 21-2(‘+a)

s)r(@j

+

9)

1

is an en&e function of 9. (Recall:

2T

I= ‘Yj + @j E 22)

We now express the Eisenstein series G+(z;

s; (CO,

do);

a) :=

I

N(cz + d)-a.N(c~ + cl)-@ 1N(cz + d) 1-2s

c csamodqa d=dr, mod qa,(c,d)q

by means of the function fa,a. The contribution of all pairs (c, d) with c = 0 is zero if CO# qa and I c d=do

iv(d)-2’

1N(d) 1-29

modqa dq

if co E qa. The summation is taken over a set of representatives of all d=domodqa,

dfo,

with respect to the “associate relation”: Two elements d, d’ are called associate mod q if there is a unit E, E E 1 mod q, with d’ = cd. If we introduce the number 1 if COE qa 6 = 6( CO,qa) = 0 elsewhere,

170

Chapter

III.

The

Cohomology

of the Hilbert

Modular

Group

we obtain

Ga,p(z; s; (CO, do); a) =S -

c dsdo

+

1 N(d)

mod 4

C' CEC,,

fe,j3(cz mod

1--2(p+g)

qa

+

d0;s;qa).

qa

cq

We now replace the fa,a by their Fourier expression computed in 4.5. The volume of a fundamental parallelotope of m = qa is

where dK denotes the discriminant of K. From 4.5 we obtain c’ CECO

f&cz

mod cq

+ do, s; qa)

= n/(qa)drce(“‘/2)S(P-“)Ny’-25-2’.

qa

c=co c”g”d qa gc(qa)*

If we collect in G,,p all t erms with fixed cg we obtain the Fourier expansion Ga,p(z;q(co,do);

a)

=

c

~~(y,s)e~~~~(~+)

,

g-l*

where the Fourier coefficients are given by the following formulae a) g = 0: n q2r + 2s - 1) * 21--2(‘+8) s> = 6. n F(“j + S>ryPj + s) j=l

UO(Y,

.

c’ CECO

1 NC mod

ll-28-2r

qa

cq .

C’ dsd,

1 mod

N(d)

I-2(r+s)

+(2~)“N(qa)dKe(“i/2)S(B-‘Y)N?/1-29-2’.

qa

ds

b) g # 0: U,(y, S) = N(qa)dKe(““/2)S(B-(r)

c g=ed,dE(qa)* czco mod

. e2-(d.do)

. N

1-29-277

Y

1NC 11-2s-2r qa,cq

. fi j=l

h(2ngjYj;

“j

+

3;

pj

+

S)

.

$4 Analytic

Continuation

171

of Eisenstein Series

The sum is a finite one which can be estimated by a constant times a suitable power of 1 Ny I. So the interchange of the summation is justified. We now obtain: 4.6 Proposition.

The difference

of the Eisenstein

series

and its zero Fourier

coeficient G&Z; has an analytic

s;

4 - UO(Y,

(co,~o);

s)

continuation as entire function of s into the whole

s-plane.

Remark. If one makes use of the fact (which we did not prove) that the series c’

I N(d) r

mod qa dq

d=d,,

admits an analytic continuation as meromorphic function into the whole s-plane, we obtain: The series Ga,p, Ea,p admit analytic continuations into the whole s-plane as meromorphic functions. We now assume and

Z!T=(Yj+pj=Z

CK#P.

We want to investigate the Eisenstein series Gm,b if we approach the border of absolute convergence s = 0. Because we assume (Y # p, T = 1, we have 5 0 or

Ctj

/3j

5 0

for at least one j. This implies that 1 r(‘yj

+ S)

1 Or

r(Pj

+

S)

has a zero at s = 0. On the other hand the limit

exists (4.32). From our explicit formula for the zero Fourier coefficient we now obtain

Fyo ~o(Y, 3) = A + B/Ny , where A and B are constants. We collect the properties of the constants A and B which were needed in §3.

Chapter III. The Cohomology

172

4.7 Lemma. We

of the Hilbert

Modular

Group

have

lim ao(y, s) = A + B/Ny S-+0

with certain Teal numbeTs A, B. The constant j with Q!j = pi = 1 is less OT equal n - 2. From

the Fourier

expansion

B is .zeToif the numbeT of all

from the results 4.4 about

of Ga,p and especially

the function h(y; ol; /3) we now obtain: 4.8 Theorem. Let (Y, ,d be two vectors

a # /I

and

of integers

ckj + @j = 2

such that

(1 5 j 5 n) .

The limit lim G,,p(z;

3;

S-+0

exists

(CO,

a)

do);

and has a Fourier expansion of the following A + B/Ny

+

type:

c a,P,,B(gy)e-2”S(lgl’)e2”‘S(gZ) Sea* ,s#O

with 191 :=

IsnO*

(IslL-~~~

The coeficienta a, E C can be estimated by the suitable power of [Ngl. The functions Y-

are certain

WY)

of a constant

pTOdUCt

and a

. %B(Y)

polynomials.

We do not need the explicit form of the coefficient function P,,p(y). We only notice that the calculation of the special values of h(y; (Y; 0) = h(-y; 0; cx) in 4.4 shows 4.81 Remark. Assume @j = 0 for some j. the variable yj and moreover pa,&/)

4.82

Corollary. Assume lim

O-+0

=

0

if

Yj

Then

<

0

n 2 2. The Eisenstein

c CECO mod qa drdo mod qa,(c,d)q

is a holomorphic function of z.

P.-&y)

(Pj

=

does not depend

0).

series

N(cz + cJ)-~ IN(cz + d)l-“”

on

$4 Analytic

Continuation

of Eisenstein

Series

173

b e any congruence group with respect to of a totally real field K. Let (Y, p E Z” be two

4.9 Theorem.

Let I? c SL(2,R)n

the Hilbert modular group vectors of integers with the properties

Pj

a) “j +

= 2

(1 I .i L n> y

b)a#PThen

the limit

E,,jj(z) := liio c N(cz+ a)-QN(cZ + a)-@IN(cz + a)l-“” r,\r exists. If M

E GL(2,

K)

is a matrix with

totally

positive

determinant, the

function (E,,pIM)(z)

= N(cz

+ a)-“N(E

+ a>+

has a Fourier expansion of the following (E,,pIM)(z)

= A + B/Ny

- E,,p(Mz)

type

+ c

a,P(gy)e-2”s(lglm’)e2niS(g2)

,

gEtQ where

A, B denote

real numbers, the function

Y-

(NY) - P(Y)

is a polynomial and the numbers ag have moderate growth, i.e. they can be by the pTOdUCt of a constant and a suitable power of INgl. The constant B is zero if the number of all j with aj = @j = 1 is less OT equal

estimated n - 2.

The number

A is I if M

is the unit

matrix have:

is not equivalent to 00. we fuTtheTmoTe Assume

/3j = 0 for

some j.

Then

the function

J%,&) is holomorphic

but zero if the cusp M-‘(00)

B/NY

in zj.

We only have to put together what we did in this section: We expressed &,B as a sum of Ga,p (with real coefficients). We proved that the group GL(2, K) act s on the space which is generated by the Ga,p over R (4.2). Up to the statement about the constant A, Theorem 4.9 is hence reduced to the Ga,p. This last statement follows from the formula Proof.

174

Chapter

III.

which is easily verified Fourier expansion. The

for the limit

The

G,,p

Cohomology

instead

of the Hilbert

of &,b]M

by means

can be computed in the same way as in the case of holomorphic series of weight 2r > 2 (Chap. I, 5).

§5 Square

Integrable

Modular

Group

of the

Eisenstein

Cohomology

The results of $3 (including $4) will allow us to write each cohomology class of H”‘(I) as the sum of an Eisenstein cohomology class and the class of a square integrable differential form. The latter classes can always be represented by square integrable harmonic ones. The theory of square integrable harmonic forms runs similar to the case of a compact quotient. The method developed there ($1) will give the complete determination of Hm(I’).

We denote

by

the subspace of all cohomology classes [w’] which square integrable (closed) differential form w, i.e. w = w’ + o!d The form Of course

w” needs not to be square “square integrable” refers

=

(

integrable. to the Poincark 0 *. .

The

aim of this

5.1 Proposition.

section

is the proof

metric

) .

-2

0

by a

.

YT2

h(z)

can be represented

Yfl

of the following

two propositions.

Let Kl,..*,Kh

be a set of representatives

of the cusp classes. The Eisenstein cohomology

dejined in 13, maps isomorphically onto the image under the natural restriction map H” F)

-

&H”(L.,). j=l

$5 Square

Integrable

Up to now

Cohomology

175

5.1 has been proved

in the cases n 5 m 5 2n - 2.

5.2 Proposition. In the case m > 0 we have

Remark: In the case m = 0 Proposition

5.2 is not true,

one has

Ho(r) = l&(r) = H&(r) E C. (We have H&(P) = HO(P) by d e fi ni t ion and this definition is necessary if one wants to have 5.1. On the other hand the constant form w = 1 is square integrable because Hn/l? has finite volume with respect to the invariant measure w A* w. This implies H’(P) = H&,(P)). The proof of the two propositions depends on a good knowledge of the square integrable cohomology. The latter can be investigated by means of two important general theorems about complete Riemannian manifolds (which we explain in App. III without proofs).

A) Each square integrable cohomology square integrable differential form. B) Each square integrable harmonic We denote

class can be represented form

by a harmonic

is closed.

by %&l(r)

the space of all square The

two theorems

integrable

harmonic

A and B above

forms

of degree

give a surjective

m.

map

but in contrast to the cocompact case this map need not to be injective! The space Z,“,,(P) can be determined (because of B) in precisely the same way as in the cocompact case. We only have to check which of the harmonic forms occurring in $1 are square integrable. 5.3 Lemma. a) The universal are square integrable.

cohomology

classes (generated

by dzi A &i/y:)

b) Let f~ be a (holomorphic)

ailbert

w2,...,2)i

modular w = f(2)d.q

is square

integrable

form.

The

differential

A . . . A dz,

if and only if f is a cusp form.

form

Chapter

176

III.

The

Cohomology

PTOO~. a) Up to a sign w,, A *wa is the invariant has finite volume.

of the

Hilbert

Modular

volume element,

Group

but Hn/l?

b) One has w A *W =I f(z)

I2 -Euclidean

volume element ,

=, JWA*53

hence

where the brackets on the right denote the Petersson scalar product, introduced in Chap. II, 51. W e h ave shown that this converges if and only if f is a cusp form. 0 5.4 Theorem.

(Compare

1.6) Let I’ c SL(2, R)”

that the extended quotient of H”/I’ is compact compact. We have a “Hodge decomposition”

where

be a discrete but such

that

subgroup

H”/I’

such

is not

1) in the case p + q # n c

7-gg(r) = 7ig$ =

0

i

(wa =

ifp=qln

cwa

#a=p

dzcz, A Gz, y2 01

/\

elsewhere

. . . A dza$

%

),

QP

2) in the case p + q = n

7f;:(r) E a:;,

@ bC{l

(Recall: [I’,

~10

~r~,(2,...,2)io.

,...,nl #b=q

denotes the space of cusp forms.)

As a consequence of 5.4 we obtain 5.5 Lemma. Let w be a square integrable harmonic form of degree m > 0. If 00 is a cusp of l?, there exists a I’,-invariant form CYsuch that w=da!.

$5 Square

5.51

Integrable

Corollary.

177

Cohomology

Assume

of the two mappings

m > 0. The composite h

‘FI,“,,(r)

+

Hm(r)

+

$

Hm(rlCj)

j=l

is zero. 5.52

Corollary.

H&(r)

n H;,(r)

= (0)

if m > 0.

PTOO~. We show that a square integrable harmonic form defines the zero

class in H”(I’,),

where roe is the stabilizer of the cusp co.

1) Universal classes: The forms

axe r ,-invariant

(but not I?-invariant). One has

d(ai) = -dxiy: dyi = :Wi p t hence 4%

A w,, A . . . A warn) = (1/2i)waI A.. . Aw,,

ifm>l. 2) Classes coming from cusp forms: We may restrict ourselves to the case f(z)dzl

A . . . A dz,

,

where f(z)

= C

age2TiS(gr)

is a cusp form of weight (2,. . . ,2). We have a,#0 We integrate f(z)

*

g>o.

with respect to the first variable g(Z) = C

ag/(27rigl)e2”“s(gZ)

The form g(z)dz2

A . . . A dz,

.

Chapter III.

178 is I’,-invariant

The Cohomology of the Hilbert

Group

and one has d(g(z)dz2

A..

The proof of 5.5 actually property: We consider a sequence

. A dz,)

= f(Z)dZl

gives a little

more,

of Coo-functions

A..

. A dz,

namely

0

.

a certain

approximation

on the real line

‘Pk : R +[O,l], with

Modular

k=1,2

)...

the property 1 cPk@) =

1 0

iftk+l

and I cps>

IS 2 *

We define

+k : H* -

P, 11

by 4k(z>

In the notation

of Lemma

=

5.5 we now

+‘k(Ny).

consider

and ‘dk := We certainly

have

(pointwise convergence). Lemma 5.5 gives a little

5.5~ Remark. proximation

d((Yk).

With result:

O!k -

Q!

Wk -

‘d

But the explicit more, namely

the notations

construction

of Lemma

during

5.5 we have

the proof

the following

(UC = {z E H” 1 Ny > C} , C > 0)) where

,6 is any square

integrable

harmonic

form

of complementary

degTee.

of

ap-

55 Square Integrable

179

Cohomology

We leave the proof

to the reader.

For the proof of Propositions the Poincar15 duality.

5.1 and 5.2 we need a further

tool,

namely

which Recall (see App. III): The de Rh am complex has a certain subcomplex consists of all differential forms with compact support. The cohomology groups of this subcomplex are the cohomology groups with compact support which we denote by

One has a natural

linear

which

neither

is in general

mapping

injective

nor surjective.

H,O(I’) The

following

two theorems

duality):

1) ( Poancare’

Obviously

= 0.

are explained

(but

not proved)

in App.

III.

The mapping

(w.4 - JH”/rWAw’, w, w’ are closed differential forms, the first one with induces a non-degenerate pairing where

H,“(r) We especially

x Hz”-“(r)

c.

have dim Hr(l?)

2) TheTe

-

compact support,

exists

a linear

=

dim H2”-“(I’)

.

mapping

-

6: &H”(L+,)

H,“+‘(r)

j=l

such that the long

sequence h

. ..

is exact.

--f

H,“(r)

+

Hyr)

-

@ H”(l?Kj) j+l

-

H,“+‘(r)

-

...

Chapter III.

180

The Cohomology of the Hilbert Modular

Group

We use this sequence in the case m = 2n - 1 and obtain the exact sequence Pn--l(r)

-

0

P--l(rnj)

4

Hp(r)

+

P(r).

j=l

From

~,2yr) E Ho(r) E c H2yr) 2 H;(r) = 0

we obtain: The image of

H2n-l(r) ---+6 H2n--l(ry) j=l

is a subspaceof codimension 1. This completes the proof of Proposition 5.1 in the case m = 2n - 1. The cases m 2 n now can be treated by duality: From the surjectivity

of the restriction map Hyr)

-

$Hm(rKj) i

in the case n 5 m 5 2n - 2 and from the long exact sequence we obtain that

,m+l(r) q P+l(r) c

is injective in those cases. Dualizing this result we obtain: 5.54 Remark. The map

H;(r) --+ P(r) is surjective if 1 5 m < 72. The image of this mapping is of course contained in the square integrable cohomology. We obtain

wyr) = H;,(r)

if 1 5 m < 12.

From Lemma 5.5 we finally obtain that

Hm(r) -

&H”(rKj j j=l

is the zero mapping if 0 c m 5 n.

$5

Square

Integrable

This jusitifies

181

Cohomology

the definition

H&(P)

= 0 in these cases!

The proof of Proposition 5.1 is now complete. From 5.51 and from the long exact sequence we conclude furthermore that the square integrable cohomology is contained in the image of the cohomology with compact support if m > 1. Hence both are equal and the square integral cohomology is precisely the kernel of the restriction map (5.1) ( if m > 1). Now Propositon 5.2 follows from 5.1. Our next goal is to determine the kernel of the mapping

5.6 Lemma.

a) is injective

The natural

if m < 2n

b) is the zero mapping This

mapping

if m = 2n.

means

7i,m,,(r) if m am,

PTOOf.

=

{ 0

ifm

< 2n, = 2n.

Let

be a square integrable harmonic form whose cohomology class in Hm(I’) is zero. Prom the existence of the Poincare pairing it follows

J wAa=O, where (Yis a compactly supported closed differential form of degree 2n - m. We want to show that in the case m < 2n this implies w = 0, or equivalently

J wA*z=o. The convergence of this integral follows from the explicit description of the square integrable harmonic forms. The idea now is to approximate *sj by compactly supported closed forms. We now apply Lemma 5.5 to *G instead of w. We may apply this lemma to write *W as the derivative of a certain form in a small neighbourhood of an arbitrary cusp class. These differential forms can be glued together to one form QLby means of “partition of unity”. The result of this construction is a form p whith compact support such that

Chapter

182

III.

The

Cohomology

ij - /3 = dcu.By means of the approximation construction: There exists that

a) %-pk

a sequence of compactly

of the

Hilbert

Modular

Group

lemma 5.53 we may refine this

supported

differential

forms

@k such

= dak

The integrals as desired.

in the sequence vanish by assumption.

We obtain

w = 0 cl

Final Remark: We now have the complete picture of the cohomology and also of cohomology with compact support (by means of Poincare duality) and the square integrable cohomology. There is also the notation of the cuspidal cohomology. Let lattice

f : H” -----+ C be a continuous function which is periodic with t C R”. We call f a cusp form at 00, if the zero Fourier coefficient

R”,t

f(z)&

respect

to some

. ..&I

J vanishes

(this

coefficient

A I-invariant

is a function

differential

form

of y). w on Hn is called M E SL(2,

UIM, are cusp The

forms

a cusp K)

form,

if all the components

of

,

at 00.

cuspidal

part HcmUsp (r)

consists of all cohomology shown that each cusp form

The explicit

universal description

classes which may is square integrable,

forms w. , Q c { 1, . . . , n - 1) , are obviously 5.4 we obtain

fc:,,(r) if m = n and

be represented by a cusp hence we have

=

not cusp

form.

forms.

It can be

Prom

the

$ P, (2, . . . , a0 bc{l,...,n)

0 elsewhere.

36 The Cohomology

of Hilbert’s

Modular

Groups

We only have to collect the results of the previous sections to get a complete of the cohomology of the Hilbert modular group, more generally of congruence

picture groups.

$6 The

Cohomology

of Hilbert’s

Modular

183

Groups

The formula in the Betti and Hodge numbers involve several invariants of those groups like volume of a fundamental domain, number of elliptic fixed points of given type and certain L-series coming from the cusps. All these invariants can be computed in case of real quadratic fields.

In the following, l? denotes a congruence group, and ~1, . . . , ICY representatives of the cusp classes. In the Sects. 3,4,5 we investigated the restriction map. The most difficult part of the theory was the construction of an injective homomorphism

&H”(r.j) --t

space of I’-invariant

differential

forms

of degree

m

j=l

in the cases n 5 m < 2n. The

image

of this map

is the space of Eisenstein

series W,

m> -

As the Eisenstein series did not converge absolutely, we had to do the tedious job of analytic continuation ($4). Not all the Eisenstein series are closed differential forms. The subspace of closed forms has been denoted by E0(r,

m>

c

E(r,

m)

.

In case m = 2n - 1 this subspace has codimension l.In the cases n 5 m 5 2n - 2 both spaces agree.The natural map of &(I’, m) into the cohomology group of I’ is injective and hence defines an isomorphism to a certain subspace

Hg(r) c fvyr) . The case m < n could been forced to define

be treated

Hzm

= 0,

by means

if

of Poincar& duaIity.We

have

O<m
and

H~is= HO(r) (= C) . The

main

result

of this

construction

6.1 Theorem. The Eisenstein the image

is

cohomology

H&(r)

of the mapping

H”(r) --+ 6 wyrKj). j=l

maps

isomorphically

to

Chapter

184

It5 dimension

III.

The

Cohomology

of the Hilbert

Modular

Group

is

We now have to consider

0

ifO<m
h.(iIi)

ifnIm<2n-1

h-l

ifm=2n-1.

the space ‘H,“,m

of all square intgrable harmonic differential forms of degree m. By a general theorem they are all closed. Hence one obtains a mapping %&m

-

Hm(r)

-

In §5 we have proved 6.2 Theorem.

Assume m < 2n.The natural map ‘H&l(r)

-

fwr)

is injective. Its image is the so-called square integrable part of the cohomology. We denote it by Kpm In the cases m > 0 it coincides with the image compact support and we have in this case

of the cohomology with

The square integrable part of the cohomology can be determined by the method of Matsushima and Shimura ($1). One obtains a further decomposition

The universal part of the cohomology is generated by the harmonic and square integrable differential forms wa,

aC{l,...,

n-l}.

The cuspidal part only arises in the case m = n. It comes from holomorphic cusp forms of weight (2,. . . ,2), which belong to certain conjugate groups of

r:

xi,,(r)

2

@ bC{l

,...,nl

[rb, (2,. . . , ai0 .

$7 The

Hodge

We collect

Numbers

of Hilbert

these results

Modular

to obtain

formulae

bm = dimz of an arbitrary

congruence

b2” = 0.

2) Assume

0 < m < 2n.

for the Betti

numbers

Hm(I’)

group.

6.3 Theorem. The Betti numbers ailbert modular group are given 1) b” = 1,

185

Varieties

Then

of an arbitrary by the following

congruence formulae:

group

I? of Some

we have

b” = bu”niv+ b& + b&, , where n 4

Cniv

=

(

m/2>

0

if m is even

if

m is odd, ifO<m
c> bcmusp = where

0

ifm#n

Cp+q=m hk&

if m = 72,

hP,Q = c cusp bc{l ,.-PI

dim[rb,

(2,. . . ,2)]s

.

#b=p

$7 The Hodge Numbers Modular Varieties by Claus

of Hilbert

Ziegler

Introduction Let K be a totally real algebraic numberfield of degree n := dimQ K and let o c K denote the ring of algebraic integers. The Hilbert modular group I?K := SL(2, o) can be regarded as a discrete subgroup of SL(2, R)” acting

Chapter

186

discontinuously on the n-fold we shall consider congruence l? = I’K[a]

III.

The

Cohomology

of the Hilbert

(SL(2,

o) + SL(2,

Group

H. More generally

product of upper half-planes subgroups I’ c FK defined

:= Kernel

Modular

by o/a))

where a c o denotes some ideal. l? is of finite index in I’K, the cusps of I? are the elements of KU(m). Let (H”)* := HnUKU{co} and Xr := (H”)*/I’ be equipped with the topologies described in Chap. I, $2, then Xr is a compact a finite number of normal complex space. In the case n 2 2 Xr contains singular points, namely the classes of cusps

S := {[k] 1k E K u {cm}} together

with

the classes of elliptic

fixed

points

F := {[.z] ] z E Hn elliptic X := Xr\(S 2n. erties

u F) is a quasi-projective

fixed

complex

point}. manifold

Concerning the singularities of Xr, a resolution can be constructed (see Ehlers [ll]):

of real dimension

with

the following

x and a morphism 1) There exists a compact complex manifold spaces f := F + Xr 2) f]f-l(X) : f-‘(X) N + X is a biholomorphic mapping 3) f is proper is a divisor with normal crossings in x, 4) Y := f-‘(Xr\X) co-ordinates we have Y = {z ] zr . . . zk = 0, O
we can consider

X as an open

dense subspace

prop-

of complex

i.e. in local

of x.

The following results are due to Deligne (see Deligne [9]): Let X be a quasi-projective complex manifold and x > X a compactification of X, such that Y := y\X is a divisor with normal crossings in x. Let j : X of x denote the natural imbedding of X into x. We consider the logarithmic de Rahm-complex R,(Y)

:

. . . + 0 --t c+(Y)A

where

R$(Y)

C j,Qk

erated

by Q$

and the differential

some irreducible

local

denotes

the locally

component

The complex Q_(Y) is equipped Hodge filtration % and the weight

FP(Rg(Y))

f+(YpL

c+(Y)_e,

free Qrsubmodule

forms

... of j*$&

% for zi local defining equation of Y and 0?(Y) := A” R+(Y).

genof

with two natural filtrations, namely the filtration W. These are defined as follows:

:=

0

m
Q?.(Y)

7-n2 P

$7 The

Hodge

Numbers

of Hilbert

Modular

187

Varieties

and W,(sZE(Y)) is defined to be the locally % y differential forms of the form generated

free Orsubmodule

&k

dzil a/\-A...A-

k
Zik

Zil

with (Y holomorphic and the zij’s being irreducible local components Yi of Y.

of RF(Y)

’

equations

of distinct

The complex (R+(Y), F, IV) is biregular bifiltered and admits bifiltered acyclic resolution (K’, F, IV) (compare Deligne [9]):

a canonical

Km :=

c

0$(Y)

local

defining

@M$”

p’+q’=m

FJ’(K”)

:=

c

FJ’(R$(Y))@

Mp’

=

p’+q’=m

Wl(K”)

:=

c

fig(Y)

~3Mg’

p’+q’=m P’ 2P

c

W&(Y))

@IMg’

.

p’+q’=m

In the following Rham-complex.

we will

Now we obtain

H”(X,S-l,(Y))

refer

induced

11 Hm(l-K*)

to K’ as the

Especially

The

classical

Hodge

de

Hm(x,

R+(Y))

N Hm(X,

C) . The

key

F, W on H”(X,

C) are independent of the choice of the fT. F, IV[m] define a mixed Hodge structure on H”(X, C), functorially on X.

we get invariants hEq(X)

logarithmic

filtrations F, W on the hypercohomology groups and th us we are led to filtrations on the co-

homology of X by the isomorphism results of Deligne are: 1) The filtrations compactification 2) The filtrations which depends

differentiable

of X,

:= dimz theory

the so called

Gr$.Gr+Grp”+[a]

is contained

Hodge

numbers:

(H”(X,

C)) .

in Deligne’s

theory:

If X = x is compact, alent to the classical

then the mixed Hodge structure on H”(X, C) is equivHPTq(X), the @ Hodge decomposition Hm(X, C) = p+q=m Qq(X) are zero if p+ 4 # m and coincide with the classical Hodge numbers hP*‘J := dimz HPjq(X) in the remaining case.

We want to make use of Deligne’s theory to investigate the manifold X = Xr \(S U F); we restrict ourselves to the special case of l? acting freely on Hn. Then we have F = 0 and X = H”/I’. The cohomology H”(H”/I’, C) is

188

Chapter

well known

The

Cohomology

of the Hilbert

Modular

Group

(see §5,6); if m > 0 we have the decomposition

Hm(Hn/r,C)= which

III.

H"(r)=

has been described

H,m,i"(r)~H,mu,,(r)$H~s(r),

in §S.

In this section we shall study the filtrations on H”(H”/I’, C) = IP(l?) existing in case of no elliptic fixed points. We first have a look at the square integrable cohomology H&(I’) := HF& (I’) $ H=&(I’), before we turn our attention to the Eisenstein cohomology. Then we summarize our results and describe the Hodge numbers hf,$7(H”/I’). The appendix gives a short summary of the construction in Ehlers [ll], emphasizing the results essential for our purposes. Weightand Hodge Filtration on Km(r). The complex R+(Y) admits different acyclic resolutions: introduction) also j,Mj, is an acyclic resolution of natural imbeddings R?(Y) q K” it j,My induce C+(Y)

t

logarithmic de-RahmBesides K’ (compare 0$(Y). Therefore the quasi-isomorphisms

K’ + j,Mi

Hence: Hm(X,

R%(Y))

II Hm(rK’)

II H”(l?j,M~)

N

Hm(rM;r)NHm(r)11Hm(x,C) Because of that the isomorphism natural cochain map K’ of j*Mj,.

IP(I’K’) G We obtain:

7.1 Proposition. Each cohomology class [VI] E cully singular representative w’ E K”. ~;yyp;;t;;

;:i;

Hm(I’)

H"(r)

is induced

by the

contains a Zogurithmi-

is a asis of Km(r) represented by logarithmically w1 b . . . , [w,]r~*) is a basis ,***, w,, then ([W&K.,

of HyrK-). Weight- and Hodge filtration on Km(r) by (K’,F,W) on lP(ITK*) c IP(l?),

are defined as filtrations i.e.

induced

where 2” := Kernel(Km& Km+l) and Bm := Im(K”-l--h, K”). Thus a cohomology class [w] E IP(I’K’) is contained in W’(IP(l?K’)) if and only if it contains a representative w’ E W(K”). The same is true concerning the Hodge filtration. Considering the different (p, q)-types of differential forms we obtain filtrations F and @ on K’, resp. j,Mj,, which induce filtrations on the hypercohomology groups of these complexes. The filtration F on IP(I’K.) Z fP(l?)

$7 The

Hodge

is nothing imbedding

Numbers

of Hilbert

Modular

189

Varieties

else but the Hodge filtration,

but unfortunately

the natural

is not a filtered quasi-isomorphism. Therefore the filtrations F and @ generally do not coincide on IP(l?), i.e. the Hodge filtration on Hm(l?) is generally not induced by the (p, q)-grading of differential forms on Mj,. We want to determine the weight filtration fore we take a look at the decomposition

We first consider the square integrable image of the natural mapping

on H”(X,

C) E H”(r),

part of the cohomology

there-

H&(r).

The

is identically zero for m > 0, i.e. the representatives of H&(r) are locally exact at the cusps. Let [w] E HS&,(I’), then th ere exist open neighbourhoods K CC Vi of the CUSP classes (ICi)i=r...h with Vi f~ Uj = 8 for i # j and differential forms vi on Vi, such that dqi = w[Ui. Let ($i)i=r...k be a family of C”-functions on Xr with the following properties: 1)

SUPP

4i

C

vi

2) 4ilVi E 1 3) 0 5 4i 51 Then the differential form 77 = ~~=, $ivi is defined on Hn/I’, w - dq is a representative of [w] E H&(r). We have w’[Vi

= wIV~ - dq[Vi = wlVi - dqiIVi = 0

thus w’ :=

for i = 1. . , h ,

i.e. we have constructed a representative w’ of [w], which vanishes identically on an open neighbourhood V = VI U. . . UVh of the cusp classes,consequently w’ E 0 on an open neighbourhood of the divisor Y in F, i.e. w’ E W’(K”), therefore [w] = [w’] E VV”(IP(I’)). We obtain: 7.2 Proposition. z&l(r)

c ~“Pm(r))

To get some information about the Hodge filtration F of H&(r) we take a look at the (p, q)-types of our constructed representatives w’ E W”(Km). First we consider the canonical basis elements of the universal cohomology

Chapter III.

190

H,“,,,(r).

These

are given

The Cohomology of the Hilbert

by WI = wir A . . . A

Wik :=

wil,

I = {il )...)

and

Modular

Group

where i,} c {l)...)

n}

.

yfk

The

differential

dzik oik := i -

forms

are I,-invariant

and satisfy:

Yik

dZik A &ik y;k

Consequently we Wil A . . . A Wil = form of type (I, I some G E SL(2,K) sufficiently small of type (I, I - 1); SL(2, K) c SL(2,

=

wik

have d(oir A wi2 A . . . A wil) = d&i1 A wi2 A . . . A wil = WI, where CYI := oil A wi2 A . . . A wil is a differential - 1). Transformation of an arbitrary cusp k to {oo} by sh ows: The differential forms 7; satisfying dq = WI in neighbourhoods Vi of the cusp classes Ai may be chosen for th is one has to use the fact that G*wl = WI for G E R)n. We obtain:

W;

=

WI

-

d(&

4ili)

E

F’(K2’)

i=l

since $iqi

is of type

(1, I - 1). We have shown:

7.3 Proposition. Hzii,(I’)

C F’(H2’(r))

Next we shall consider the cuspidal part of the cohomology H,“,,,(I’). (Since H&,(r) = (0) f or m # n, only the case m = n is interesting). The canonical basis elements are given by w = f dz, A &a, where aiTb= 0,aUb= {l,... ,n} and f(aa(z)) E [I’“,(2,. . . ,2)]0. (see $5). First let 6 # 0. In this case we can use an argument similar to the one given above. Since f(ub(z)) E [rb, (2,. . . ,2)]s, f has a fourier series expansion f(z)

=

c

age2nih7yob(d)

get0

where ag # 0 only if g > 0. Since b # 0 we can choose f(z) with respect to z,. We obtain: g(Z)

=

_

c

%,2~ii(g,~b(z))

gEtO

2%

v E b and integrate

$7 The

Hodge

(Observe The

Numbers

that

of Hilbert

Modular

Varieties

191

gV > 0 for ag # 0, so this expression

differential

form

cx := g(z) dz, A &Q\{~}

da = d(g(z)

dz, A dam\+))

makes

sense).

is P,-invariant

= f(z)

dz,

and we have

A d&, = w

fourier series expansions Since f( 0b (z)) is a cusp form, f has analogous with respect to arbitrary cusps k. Like above we obtain: If w = f dz, A &b is a canonical basis element of H&,,(P) of type (p, q) with q # 0, then we can choose the differential forms Q satisfying dr)i = w in sufficiently small neighbourhoods of the cusp classes ki of type (p, q - 1). Therefore the representatives w’ E Ws(K”) of [w] E H,“,,,(P) constructed in 7.2. are elements of Fp(K”), 1‘f we base our construction on the qj’s described above. The case b # 8 is to be treated somewhat differently. The canonical basis elements w = f dzl A . . . A dz, with f E [I’, (2,. . . ,2)]s admit an extension as holomorphic differential forms on x. Thus w E W’(K”). (for a proof see Freitag [12], Satz 3.2). Since w is of type (n,O) we have w E F”(K”). A glance at the definition of the Hodge filtration on H”(P) shows:

7.4 Proposition.

@ $ P’>P

By determining

~rb,(2,...,2)lo c ~vvx

bC{l,...,n} p’=n-#b

the weight

filtration

show: ~-gs(ry-wo(wyr))

= {0).1f

of the Eisenstein cohomology we shall we assume this result for the moment

we have

This

together

with

(hy(H"/r)= 7.5 Proposition. FP(H&(r))

where

Propositions

The Hodge filtration = FP(HZiv(r)) KLW

FP(HZtiv(r))

3 and

4 yields

by a symmetry

argument

hgyH"/l?)):

= (0)

on Hs.&(I’) @ Fp(H~sp(r))

has the following 7

if p I m/2 if p > m/2 if m = n

~pucs,(r))

= ifmfn

form:

192

Chapter III.

The Cohomology of the Hilbert

Modular

Group

To determine the weight filtration of the Eisenstein cohomology I$&(I’), we investigate the geometrical and combinatorical structure of the Divisor Y C x. Especially we consider the spectral sequences S’ (see Deligne [9], 3.2.8.1): S’ : . . . +

Hm--k-2(pk+l,Ek+1)

d4 p--k(p&k) &+H

m--k+yp-1)

&l)

--f

. . .

The classical Hodge decompositions

H’(?‘,ak)

= $ H’(?,Q;&“)) i+j=l

define filtrations on HI@‘“, Ed). After suitable renumeration of these filtrations, (S’, F) becomes a biregular bifiltered complex, such that the morphisms dr are strictly compatible with the filtration. Then Grk(H’(S’)) E H’(GT,(S’)) hoId s, i.e. the induced filtrations on the homology groups of S’ are determined by

Fp(w(s*))

= $ H*(G&s*)) i>P

.

Deligne shows

H”-‘(S) and the filtration

2 Gr~(H”(I’K))

described above coincides with the Hodge filtration on Th eref ore we contemplate the associated graded com-

Gr~(H”(W)). plexes Gr&S”:

. . . + Hi-3(p3, $-$;“(s3)) % Hi-2(p2, fl;;“(c2)) 5 Hi-‘(PI,

fl;;‘(~l))

% H”(X,

“iv) + 0

We shall show: For 0 5 i 5 n we have dim G~~G~,W_i(H”+‘(I’))

-- dim

3 Hi+l(~n-i-l,~~~-i-l(~n-i-l)))

Kernel(Hi(~n-i,R~,-i(en-i)) (pn-i+l,

b(Hi-1

= dim H;;“(r) Consequently

ai-

+,-i+l(~n--i+l))

% H’(+‘,

R&JP-~)))

. G+Grg-,(H”(I’))

= H$,(I’)

for m 2 n and we obtain:

$7 The

Hodge

Numbers

7.6 Proposition.

of Hilbert

Modular

The weight filtration

193

Varieties

on H”‘(r)

has the following

= ~:dr) e ok = qu(r)

~~~4~v7))

~2n--m-lp(r))

form:

= Hm(r)

~owv7) = ~:dr) w-,(Hm(r))= c-u 7.7 Proposition.

We have

Hg.(r) c Pywyr))

.

order to prove the above propositions, we have to verify about the homology groups of the complex G’

the statement

In

G’

. . . 3 Hip-i,

n;“-i(En-i))

5 . . . 2 ,n-l(F,

fq’(El))

2 I-P(X$$) Since dimR Pn--j

--) 0.

= 2j for every j, we have

Hip’“+,

“;

_, (p-j))

11 pqp-j,

p-i)

,

n 3

Thus G’ may be written G’

. . . 5 jy2yp--i,$y

as: % . . .% H2(“-l)(p1,E1)

5 lT2y7c,

C) + 0

Without loss of generality we may assume that Y is the union of smooth divisors. For let us suppose the contrary; then there exists a subgroup I” c I? of finite index in I’, such that the divisor Y’ compactifying H”/I” is the union of smooth divisors (see appendix). If we consider the canonical projection x : H”/I” + Hn/l? and the induced mapping x* on the cohomology groups

~*:Hm(H"/r,C)-tHn(Hn/r',C)

)

we observe that x* is injective, respects the decomposition of the cohomology into Eisensteinand square integrable part, and moreover induces a morphism of the respective mixed Hodge structures. Consequently

must hold, if only I?’ satisfies the analogous statement. Thus let us suppose Y is union of the smooth divisors Dr , . . . , Dr. The choice of an order of

Chapter III. The Cohomology of the Hilbert Modular Group

194

these components of Y trivializes the sheaves ei, i.e. we obtain isomorphisms oi : ei 1~ C. The Yk split into disjoint connected divisors:

Fk= u ig) jCJk Especially Therefore

Y1 = Di b . . . b D,, i.e. Ji = { 1,. . . , r} we have the following decompositions:

fpi@+-i,

p-i)

11 jyZi(pn-i,

q N

@

H2’(5$‘,

and Y$, q N

@

= Dk. c

jEJ,-i

jEJn-i

and G’ becomes

jC Jn-i

j”EJn-im2

j’EJ,-i-1

We can use (#J,) x (#J,-r)-matrices (~pv)p~~,,v~~,-l to describe the mappings 3, where a,.,, E C may be regarded as a linear mapping a,, : C + C. Later on we shall show: After suitable choice of the isomorphisms H2’(Ycj), C) N C the a,,, have the following form: 1) a,, = 0, if YGTi $ YGFi-’

2) let Y&i

c YcF;i-l and let a) i 5 n - 2; then there are uniquely such that and

PC;;’ p;,‘-’

determined

indices Icj,

= Dkl f-l . . . n Dknai = Dkl n . . . n fi.k, n . . . n Dknmi

,

since this is valid locally and by construction of Y arbitrary intersections of the divisors D1, . . . , D, are connected (see appendix). We have: a,, = (-l)@ b) i = n - 1; then (a,,,)

is a (r x 1)-matrix (a,“)

The plex cusp. state

and we have

= (1,. . . ,I> *

above remarks enable us to compute the homology groups of the comG’. First let i 5 n - 2; in this case we only need to investigate a single If we take a look at the construction of C and Y in Ehlers [ll] we can (see appendix):

There is a 1:1-correspondence between the L-simplices of C/A and the components Y& of Y-k. Let VI,... ,v, denote the l-simplices associated to the components

DI, . . . , D, of Y; then the simplex

CTk E C/h(k)

spanned

by

$7 The Hodge Numbers of Hilbert Modular

195

Varieties

uik corresponds to the connected component D;r fl . . . ll Dik of Fk; if (vir,..., vik) $ C/A(‘), then Dil fl . . . fl Dik = 8. If we compare the mappings a of the complex G’ with the boundary operators 8’ of the simplicial complex C/A we observe: The complexes

Vii,...,

G’

. ..--+

$

Ca.

@

Ca.

j’EJ.,-i

jE Jn-i+l

@

c-t...

j”EJ,,-i-l

and k=l

Oi-1

k=l

k=lci+l

0;

are isomorphic. Hence H”-‘(G’)

~6

Hi(C/A,

C)

for

O
.

k=l

Remark.

C/A is not a customary simplicial complex in the senseof algebraic topology, nevertheless the homology groups of the complex c a’

. . . + @ CL@ Qi-1

vi

cl3 c Oi+l

coincide with the singular homology groups H;(E/A, C). To see this, we only have to observe that the complexes C resp. C/A together with their boundary operators a’ may be replaced by the customary simplicial complexes obtained by intersecting C resp. E/A with Al := {cc E Fly 121 e.0 2, = 1). Since H’(C) = 0 we have Hi(E/A,C) II Hi(A). To compute the group cohomology H’(A) we recall that A acts on D := {z E H” 1Ny = 1). We have H’(D)

= 0, thus H’(A)

since D/A 1~ (Sl)‘+-l dimHnmi(G’)

II H’(D/A,

C) N H”((S’)+‘,

C) ,

( see Chap. III, $2). Hence: = h. dimH”(A) = h

= ha dimHi((S1)+‘,

C)

= dim HEit*

for 0 5 i 5 n - 2. We have to add the treatment i = n - 1. If we cut off the complex G’ at

of the remaining case

Chapter III. The Cohomology of the Hilbert Modular Group

196

we obtain

the complex

-. : . ..%$C+o+o G jEJ1

’

Analogous to the case 0 5 i 5 n - 2 we can show:

H1(&) := H(% @ c + 0) N ($ IT’(*) jEJ1

j=l

Therefore we have: dim(Kerne1 ( $

C + 0)) = dim $

jCJ1

C = #Jr

iEJ1

and

dim@4

$

C f+ $

j’EJ2

C)) = #Jl - dim6

jEJ1

P-l(*)

.

= #Ji _ h

j=l

Considering the complex

we obtain dim(Kerne1 ( @ C % C)) = #Jr - 1 iEJ1

since LJ: 03 C + C is surjective. We already know: jEJ1

dim(Im( $

C% $

j’EJ2

C)) = #Jr - h

iEJ1

Therefore we have: amHl(G’)

= (#A

- h) - (#Jl

- 1) = h - 1 = dim&&‘(I’)

.

To complete our proof we have to verify that the mappings ~3,” : C ---f C resp. q a P” : IP(q C) + p(i+l)(jT-g-1,

$7 The

Hodge

Numbers

of Hilbert

actually have the form H2’(?(;;‘,C) cz C by

Thus it remains

44

=

described

to verify

and let [w’] = a,,([~])

kt(

Modular

above. We define the isomorphisms

the following

staement:

E H2(‘+l)(?~~;‘-‘,

[w])

0

197

Varieties

C)

C), then:

if S$;’

C

?(t;‘-‘,

if

$

pc;i-l

jYy,i

Let [w] E H2’(?$,

(Y :

where the sign is to be chosen as above

”

Let w be a differential form on ?(;y’, then Q,,([w]) = ~([w])]~~~J. We shall show that starting with w, the corresponding representative w’ of a([~]) has the required properties. The reader who looks carefully at the definition of the PoincarC-residues and at the construction of the spectral sequences S’ (see Deligne [l], 3.2.8.1) will establish that w’ may be constructed in the following way: 1) Extend w to wext on x . 2) Fori=l,..., r choose Coo-functions ti with the following property: Let z E Di, then there exists an open neighbourhood U(z) c r of z, such that ti = 0 is a local defining equation of Di in V(z). (The functions ti may be constructed via partition of unity out of local defining equations of Di). 3) Takewe”*~/\...h~,if~~~~~=Dil”...”oi~. This different:: form is d%ned in an open neighbourhood U of p(iy’. Multiplication with a suitable Coo-function 4 satisfying 4 G 1 in an open neighbourhood V CC U of $yi and 4 E 0 outside U yields a globally defined differential form

4) Take d(q5 - wext A % 5) Let ?(r;*-’

A . . . A 2)

= Dir n . . . fl Dya n . . . Al Dik.

,',jy-1

= (-l)“d($.

Then we have:

Wext /, F)

iu

198

Chapter

(that’s

III.

the PoincarC-residue). a([w’])

= (27ri)“-‘-l

The

Cohomology

of the Hilbert

Group

We have to compute: iin-i_l(-l)ad(4. J (v)

= (-1)U(27ri)n-i-1 where

Modular

J(Ir)c

lii

S, := {&I

= (-l)Q(2?ri)+i-l

p”-ixs

Iti,

= 0). $ile

wext(tior q;,

J

,(I

4. wext A 9,

= E}

J

= (-1)a(2ri)n-i

Wext A $5) *a

’

:

Ia

w = (-l)‘~a([wl) P(;; i

Moreover we have w’/~~~,~-’ complete.

= 0, if ?(“,yi @ %$;‘-‘.

Now the proof is cl

The Hodge Numbers of H”/I’. The Propositions 1 through 7 contain the whole information about the weight- and Hodge filtration on H”(P), i.e. we know the mixed Hodge structure (Hm(I’), F, W), which is equivalent to the Hodge decomposition

where .@$(I’)

is defined as Hgq(I’)

:= Gr$Gr$GrpW!;]

@“(I’))

Summarizing our results we obtain: 7.8 Theorem. We have 1)

2)

@O(r) = Ho(r) Hp(r) = univH$q(r)

63 cuspH;q(r)

@ EisH$(l(r)

where univHm/2~m’2(I’) m

= H&Q”);

cuspqyr)

=

cuspH~q(I’)

= (0)

@ bc{l ,...,nl p=n-#b,

EisHzn(r)

=

univHfiq(I’)

Hg,(r);

= (0)

otherwise

P,(2,...,2)io;

q=#b

otherwise EisHgq(I’)

=

(0)

otherwise

$7 The

Hodge

Numbers

of Hilbert

We may represent

Modular

this result

Varieties

199

by considering

the following

+q HZ”:,, ------I (rl,

+

++

H~,(fl.

(m=nbO 0

Hodge diagram:

imd

o”

++p 0-c

P

n

The Hodge numbers

IQ

= &q(H”/I’)

are defined as

h:Q := dim Hfiq(I’) Therefore

we obtain

7.9 Theorem.

from Theorem

1:

numbers of the Hilbert

The Hodge

modular

variety

are given

by: 4 b) C)

m=O m = 2n 0 < m < 2n

h;fO = 1 hfiq = 0 h$q = univ h%’ + cusp hfi’

: : :

+ Eish$’

with: univ hg’ cusphf)n9q = h&forp

where

hp”cusp ** --

+

c

q

= n,

= 0 otherwise cusphgq

= 0 otherwise

dim[l?, (2,. . . ,2)]0

bC{l ,...+I p=n-#b,q=#b

ifn<m<2n-1

EishQ’

= 0

otherwise

if m = 27-8- lcr Appendix We give a short summary of the basic definitions and results of Ehlers [ll] used in our context.

Chapter

200

III.

The

Cohomology

of the Hilbert

Modular

Group

1) Let t c R” be a free Z-module of rank n. Let ~1,. . . , q, E t be linearly independent. (~1 , . . . , q,) is called psimplex, if t/(wl,. . . , q,)Z has no elements of finite order. To each p-simplex (01,. . . , up) we associate the set 0 = (01 ,...,Wp)R+ := {CtiWi 1 ti > 0) Alsoweoftencallaapsimplex.The(vil,...,vi~)~+, called A-faces of (T. 2) A set C of simplices is called complex, (i)

0,~

E TZ *

int(cr) n &(a’) id(a)

where

=

1
properties

<pare hold:

0 for 0 # c’, 0 n 0’ e C,

= {C tivi 1 ti > 0)

St(r) := (~7 E C ) 7 face of 0) is a finite set.

(ii)

7 E C*

(iii)

r E C, dim?- < n +- r is face of a suitable

n-simplex

(T E C.

3) To each simplicial complex C we can associate a complex manifold Xn in the following way: Let 6, a’ E C’“’ := {n-simplices of C} and G,I, E Gl,Z the matrix defined by a” = G,I, . a; for arbitrary A E Gl,Z and z = (Zl, . . ..zn) E C” let

Then we define xc

where

the glueing relation

:=

u (C”),/N o&x(“)

is given by the mappings

9u’o : (C”), ---f (C”>d z -+ z’Go where Gz,, := (G;,rC)t . 4) Toeachr= (or,..., w,) E C(r) there is associated a connected submanifold F, c Xc of codimension r. In coordinates we have F, n (C”), = {z~zr=...=zr=O},if0=(wr ,..., w,,w,+r ,..., wn). 5, D ‘= Udimrcl F, is a divisor with normal crossings in Xc. For r = wr) E Cc’) we have: ( Wl,...,

F, = F,, n . . . n F,, But if (wr , . . . , 21~) # C(‘),

then

:

.

F,, n . . . n F,, = 0.

$7 The Hodge Numbers of Hilbert Modular Varieties

201

6) Let I’ c I?K be a congruence subgroup of some Hilbert modular group r~; let t be the translation lattice and A the group of multipliers of r. Then there exists a simplicial complex C having the following properties: (i) ICI := u (T = (R”,) u (0) oEC (ii) A acts on C, i.e. g E C, .5 E A j c0 E C (iii) 0 E C, E E A, E # 1 * dim(a n ~7) 5 1 When replacing r by a suitable subgroup I” c r of finite index, we can attain dim(o n ~0) = 0 in (iii). 7) There is an open neighbourhood U(D) c Xc of the divisor D and a biholomorphic mapping R:

?r : where

U(D)\13

N l&/t

UC := {z E Hn 1Ny > C > 0)

8) A acts discontinously and freely on U(D), such that: (i) x E D, e E A =+ EZ E D, i.e. A acts on D. (ii) n- : U(D)\D II UC/t is equivariant with respect to the natural action of A on UC/A. 9) Consequently U(D)/A is a complex manifold, Y, := D/A a divisor with normal crossings in U(D)/A. We have:

(U(D)\D)/A=(U(D)/A)\Y, is an open neighbourhood

=

of the cusp {co}.

f:U(D)/A-+Uc/Lu{oo}

uc/roo cH”/F We define a mapping

cXr

by f(Ym) := {oo} and by defining fj(U(D)/A)\Y, as the induced isomorphism r* : (U(D)/A)\Y, 11 Uc/I’oo. Then f yields a resolution of the singularity {oo} of Xr with the properties described in the introduction. 10) The action of A on U(D) is such that each E E A gives a rise to a biholomorphic mapping

If I” E I’ is chosen as in 6. with arlaa = (0) for E # 1, then F, rl Fe, = 0 for E # 1. To see this let 0 = (~1, . . . , r~,.) and .zcr = (vi, . . . , u:) and 7- = (WI ,...) wr,w: )... , wk); then r can not be a simplex of Ct2’), since that would contradict r n ET = (0). Thus:

F, n F,, = F,, n . . . n Fur n Fv; n . . . n Fv; = 0 Therefore

we have FJA

z F,

202

Chapter

III.

The

Cohomology

of the Hilbert

Modular

Group

for each u E C. Especially Fvi /A II Fvi for each l-simplex v;. Consequently Y, = D/A is the union of smooth divisors DI, . . . , D,; moreover arbitrary intersections of these divisors are connected, since for suitable vik we have: Vii, * - -, Dil n . . . rl Dik N Fvi, r-l. . . r-l Fui, II F, if r = (vir,...

,v&)

E d”)

and Dil

n.. . nDik = 8

if r $ Cck). Obviously we have a 1:l correspondence between the lsimplices vr , . . . , vr of Z/A and the smooth divisors D1, . . . , D,. The simplices (uir,. . . ,uik)R+ correspond to the submanifolds of codimension k given by Dil n . . . fl Djk, which coincide with the connected components Y& of Yk (see above).

Appendices

I. Algebraic We

Numbers

give a brief introduction,

proofs, to the theory of algebraic numbers.

number is called algebraic if it is a root with rational coefficients

A complex polynomial

c&an+... We may

without

+ma+ao

assume

that

ajEQforO<jIn,

=o,

the polynomial

polynomial

with

rational

P E &[z] is called

Gz#O.

is monk, i.e. a, = 1.

If a is a root of a manic polynomial integers we call a an (algebraic) integer. A manic

of a non-vanishing

whose

coefficients )

P(u)

coefficients of minimal

are rational degree

=o

minimal polynomial of a.

ALL The minimal polynomial of an algebraic number is uniquely determined. It is separable, i.e. it has no multiple root. The minimal polynomial of an algebraic integer has (rational) integral coeficienta.

Notation.

The degree of an algebraic number is the degree of the minipolynomial. The different roots of the minimal polynomial are called conjugates of an algebraic number a. We often denote them by

mal

&)

T-*-Y acn)

(u is one of them)

.

AI.2. The set a of all algebraic numbers is a subfield of C which contains Q. The set of all algebraic integers z is a aubring of a with the property

TnQ=Z.

Appendices

204

AI.3. Let P be a manic polynomial with algebraic coeficients (P E QI[x]). The roots of P are algebraic. If the coeficients are integral (P E z[x]), the Toots are algebraic integers. Number Fields. Let K be a subfield of the field of complex numbers (Q C K c C). We may consider K as a vector space over &. K is called an algebraic number field if the dimension of this vector space is finite. This dimension is called the degree of K and denoted by n = [K:Q]

= dimeK.

(One can always define the degree of an arbitrary field K with respect to a subfield k. [K : k] = d lrnk K 5 c~.) The elements of algebraic number fields are always algebraic numbers and each algebraic number is contained in some algebraic number field K. The smallest K which contains a is denoted by K = Q(a). Conjugate Fields. Let K be an algebraic number field of degree n. An imbedding of K into the field of complex numbers is a mapping cp:K with

--t

C

the properties

da) = a

1)

for a E & ,

p(a i b) = y+) i p(b). The image of K is a subfield of C isomorphic

to K, K

N b p(K).

AI.4. An algebraic number field K of degree n admits precisely imbeddings into the field of complex numbers. We usually

arrange

the imbeddings K + a-&),

K(j)

n different

in a certain order and denote them by c C , j=l,...,

n.

AI.5. If a is an element of K, the images under the imbeddings are conjugates of a. Each conjugate occurs under these images with multiplicity [K : Q(u)] = [K : &]/degree(a).

I. Algebraic

205

Numbers

(We have degree(a)

= [Q(a) : Q].)

Notation.

the n different

Consider

imbeddings

K-K(j)

,

Sa = Sqf$a)

l<j
= a(l) + . . . + aCn) )

norm:

= a (1) * . . . * a(n) .

Na = N,,g(a)

AI.& Bate and norm of an element a E K are rational algebraic integer they are rational integers.

number.% If a is an

The Discrimiiant. Let al, . . . , a, be elements of our algebraic number K (of degree n). The discriminant of (al,. . . , a,) is defined by d(al,...

field

, a,) = (det A)2 ,

where

Obviously A * A’ = (S(aiaj))lli,j
*

We obtain that d(al, . . . , a,) is a rational number (which does not depend on the ordering of the imbeddings). If all al,. . . , a, are algebraic integers, the discriminant is a rational integer. AI.7. The discriminant d(al,. . . ,a,) is 0 iff the elements linearly dependent over Q. We moreover have d(al,...

, an> = d(bl , . . . , b,)

if

al,. . . ,a,

are

Zal + . . . + Za, = Zbl + . . . + Zb, .

A lattice m in K is an additive subgroup of K which generates K as Qvector space and which is isomorphic to the group Z”. This means that we can find a Q-basis al,. . . , an of K with m = Zal + . . . + Za, . By AI.7 the discriminant d(m)

:= d(al, . . . ,QtZ)

Appendices

206

of a lattice

is a well defined rational

The Ring of Integers.

number,

different

from 0.

We denote by o=oK=KflZ

the ring of algebraic integers in our number field K (of degree n). AI.&

o is a lattice

in K,

i.e. K=Q-0,

a>

b)

0 = Zal + . . . + Za,

The discrimiiant

.

of K is by definition the discriminant of the lattice o, i.e. dK := d(o)

(= d(al,.

. . ,a,)>.

(The discriminant is a very important invariant of K. It can be shown that always d E 0 or 1 mod 4 and that for given d there exist only finitely many number fields with discriminant d, especially: n ---f co + d --t 00.) Units. The invertible elements of o are the so-called units of K o* = {E E 0 1 E # 0 ) .5-l E 0).

Special units of K are the roots of unity which are contained in K W={<EK

Of course o* is a multiplicative

1
Before we describe the structure of o* we have to define a further invariant r of K: An imbedding ‘p of K is called real if the image p(K) is contained in the field R of real numbers. We denote by ri the number of real imbeddings of K. The number of non-real imbeddings is even, because a H cp(a) is also an imbedding of K. We may therefore write n = 7-l + 2rz

rr = number of real imbeddings r2 = number of pairs (cp,p) of complex conjugate non-real imbeddings. We define the invariant r by r=ri+r2-1 A famous theorem of Dirichlet states

(5 n - 1).

I. Algebraic

207

Numbers

AI.9.

The

cydic

gTOUp.

group of roots The structure

of unity in an algebraic of the whole unit group 0

number field is given by

is a finite

* E WXZ’.

Special case: The field K is called totally real if it admits only real imbeddings. In this case we have T = n - 1 and, of course, W = { 1, - 1). It is often useful to select one representative from each pair {cp,~p} of complex conjugate non-real imbeddings and to arrange the imbeddings as follows: K +

RF’ x Cr2

a H

(a(l),

. . . ) c#-trq

.

Here ?

j

=l,...,T]

den0 tes the real irnbed~n~~~ a H

a(j)

)

1"1 < j 5 T1 +r2

the set of representatives of non-real imbeddings. The following formulation of the unit theorem is roughly equivalent with AI.9 (but more on the line of a proof). AI.10. An integer

a E o is a unit if and only if 1 Na

if

It ia a root of unity

[=I a(l) * . . . . c-An) I= 1.

and only if 1 a(l)

1=

. . . = 1 .(n)

1=

1.

The mapping

adefines

an isomorphism

(log I a(l) I7**., log I a(rl+rz) of 0*/W

{z E Rrl+rz I

I)

to a sublattice of the vector 21+...+2q+rz

space

= 0).

Ideals. A subset a c K, a # (0) of an algebraic number field is called an ideal if the following conditions hold: a) a is an additive group. b) aEa,bEo + a.bEa. c) There exists a non-zero element a E o with a. a

C

o.

Appendices

208

AI.11.

Each ideal a is a lattice in K, i.e. a=Zal+...+Za,,

where al, . . . , a,, is a Q-basis of K. We especially can define the discriminant d(a) of an ideal. Product of Ideals. of them 0, then

If al,. . . , a, is a finite system

,a,)

(al,...

= {ebjaj

of numbers

in K, not all

1 bj Eo)

j=l

is an ideal. An ideal is called principal

if it can be generated

by one element

a = (a). The product

of two ideals a, b is defined by a.b

=

ajbj finite

1 ajEn,bjEb

number

>

.

It is again an ideal and the formulae (a)a=aa=

(a5

1 2 E a}

,...,bf)

= (aibj)

(4 - 0) = Cab) and more generally (al,...

, am)@1

hold. From AI.11 it follows show even more:

that each ideal is finitely

AI.12. Let a be a non-zero element b with

element

generated.

But one csn

of an ideal a. There exists

a second

a = (a, b) .

AI.13. The set Z of all ideals of K is a group under the multiplication introduced above. The unit element of this group is the ideal o = (1). The inverse of an ideal a is given by a-l

= {x E K

1 za

c

o} .

I. Algebraic

209

Numbers

The set ‘FI of principal ideals is a subgroup of 1. The factor group 113-1 ia a finite (abelian) group. The order h = #Z/?-L is a very important invariant of K, the so-called class number of K. The elements of Z/‘H are called ideal classes. Prime Ideals. An ideal p of K is called prime ideal if it is integral and if it satisfies a-lisp,

a,bEo

+

asp

or

(p C o)

bEp

(i.e., o/p is an integral domain). A fundamental result of algebraic number theory states AI.14. Each

ideal

(of

acpf’.

OUT

number field K) can be written in the form . . . *Pk

7

j,EZforl
are pairwise distinct prime ideals. This representation is IfeTe pl,...,h unique up to permutation of the factors. a is integral iff all the exponents j, are non-negative. We note some immediate consequencesof AI.13, AI.14. 1) An integral ideal p c a is a prime ideal if and only if it is maximal: pcaco

*

a=p

or a=o,

or equivalently: o/p is a field. 2) For two ideals a, b the following two conditions are equivalent: a > b,

i> ii)

a-lb

is integral.

In this case we say that a divides b alb. Notice: In the case of principal ideals a = (a), b = (b) we have the usual notion of divisibility (u) 1(b) * a-lb E o.

Appendices

210

The Norm of Ideals. AI.15.

There

is a unique mapping (the norm of ideals)

N:2 with

the following

-

Q-(O)

properties

1)

N(a. b) = N(a).

N(b) .

N(a)

(E NJ.

If a C 0 is integral 2) If a = (u) is principal,

= #(o/a)

then

(N(U) = a(l) . . . . . CZ(~)).

N(a) =( ~(a) 1

3)

The Different.

On our number field K we consider the &-bilinear form K x K

--f

Q

,

(a, b) t.-+

S(u . b) .

It is non-degenerate, i.e. S(az) = 0 for all 2 E K

*

a = 0 .

Let m c K be a lattice. We define m* = {u E K AI.16. If m c K is a lattice,

1 S(uz)

E Z for all 2 E m}.

then m* ia also a lattice

of K.

We have

(m*)* = m.

If

m is an ideal,

then

m* is also an ideal.

We consider especially the dual lattice of the ring of integers. Its inverse is a distinguished integral ideal d = 0*-l of K, the so-called dierent AI.17. 4

b) 4

c o

of K.

We have a*a = d-’ d(a)

= o* ,

= dK. N(a)”

N(d) =I & 1 ’

,

I. Algebraic

211

Numbers

The notion of the dual a lattice t c R”, which

module is connected is defined as

t”={aERn

with

the notion

of the dual

of

1 &@forzEt). j=l

For sake of simplicity we explain this only for totally real number In this case we have n real imbeddings which give us

K +

R”

U H

(a@),

. . . , cl@))

fields

K.

.

AH& The image n c W’ of a lattice m c K is a lattice of R”. The image of the dual module m* is the dual lattice no. If we identify

m with

n we may write

this

simply

as

m*=mO. An Approximation

Theorem. Let t c R” be a lattice.

A multiplier

of t is

an element

E E R3 with

(R+ = {t E R 1 t > 0))

the property Et = t .

A multiplier

always

has norm

Let A c RF be a discrete possible rank, i.e.

1

(multiplicative)

subgroup

of multipliers

of maximal

A z z-l. This

means

that logh={(log&~,...,logE,)

is a lattice

in the hypersurface

defined

1 EEA} by

in R”.

Example: Let K be a number field of degree n which is totally real, which each of the n imbeddings of K in C is real. We hence have an

means that imbedding

K it R” a I-+ (a(l), . . . ) &))

.

Appendices

212

We denote by t the image of the ring of integers of K and by A the image of the group of all totally positive units in R”. By the Dirichlet unit theorem we actually have A S Z”-l . More generally one can take an arbitrary lattice m c K instead of o and a subgroup of finite index of A which acts on m by multiplication. Finally one can multiply the image of m by a real constant. It can be shown that each pair (t, A) arises in such a way. AI.19 Proposition. A c R;

of multipliers Each

1)

Let t C R” be a lattice which admits a discrete group of maximal possible rank n - 1. We then have

of the n projections t+R,

(1 I j I

UctUj

n) ,

is injective. 21

The image

oft

in Rnsl under

t + R“-1 is dense Proof.

each of the n projections

(cancellation

of one component)

in Rnml. 1) Assume that a is an element of t with the properties al =O, a#O.

We choose a multiplier e E A with the property E2 < 1,**-, &n cl. The existence of such a multiplier follows from the fact that {(logez,...

is a lattice in R”-l.

,logen)

I EEA)

Now a.?, ?2EN,

is a sequence of non-zero elements of t which tends to 0. This contradicts the discreteness of t. 2) The proof depends on the following trivial remark: Remark. Let t C R” be a lattice. There exists a constant each Euclidean ball of radiw T contains a lattice point.

T

> 0 such that

For the proof of 2) we now consider an arbitrary open non-empty subset U c R”-l. We have to prove the existence of a lattice point in R x U. We choose a multiplier E E A with &2 > 1,...,&>l.

213

I. Algebraic Numbers

It is sufficient to prove the existence of a lattice point in P.(R x U) for some natural number n. But for sufficiently large n this set contains a Euclidean ball of arbitrarily large radius. The proof now follows from the remark. q A Zeta Function. We again consider a lattice t c R” together with a group A c R; of multipliers of rank n - 1. A sign vector 0 = (al,. . . , cm) is a vector whose components are fl. There are precisely 2” sign vectors. Notation. t, = {u E t 1 au > 0). The group A acts on t,. AI.20

Proposition.

The series f&)

converges

ifs

=

c lN4-s a&/A

> 1. The limit .ly+(s

ezists and is independent

- W&)

of the sign vector CT.

We should mention that a deeper result of Hecke states that (s - l)fc(s) has an analytic continuation as holomorphic function into the whole s-plane. A proof of AI.20 can be given by comparing

JB where

B denotes a fundamental {zc E R”

pz!l-s

domain of A acting on 1 26 >o,

lNzl>

1).

are not quite trivial).

The limit lim S+l+

the integral

dq * * * c-kc:, )

We omit the details of this proof (which consequence of AI.20 is AI.21 Corollary.

the series with

c a:(t-{O})/h

sgn (Na)lN~l-~

ezis ts. This series occurs in Chap. II, $3 as “Shimizu’s

L-series”.

An immediate

Appendices

214

II. Integration We recall some basic facts about integration. Let X be a locally compact topological space with a countable basis of its topology. We denote by Cc(X) = {f :x

--f C continuous 1 supp(f) compact} ,

where supp(f) = closure of (3 E X 1 f(z)

# 0)

is the linear space of all compactly supported continuous functions on X. A (Radon) measure on X is a C-linear functional I: C,(X)

-

c

such that I(f)

2 0 if f 2 0

(i.e. f(z)

2 0 for all 2 E X) .

Notation.

Basic example: X = R”, I the usual Riemann integral. The functional I can be extended to a larger class of functions, the so-called integrable functions (in the senseof Daniell-Lebesgue) I 2(X,

dx) -

L(X,dx)

c

3 C,(X).

Before we formulate some of the basic properties of this integral we describe some derived notions. A subset A c X is called neglectible (with respect to dz), if the characteristic function 1 for x E A XA(x) = 0 for x $ A { is integrable (i.e. E ,C(X, dz)) and if its integral is 0. AII.1.

Properties

of neglectible

sets:

I) Each subset of a neglectible set is neglectible. 2) The union of countably many neglectible sets is neglectible. 8) If f is an integrable function and g is an arbitrary function which differs from f only on a neglectible set, then also g is integrable and the integrals agree.

215

II. Integration

4)

Assume

f 1 0. Then

the two properties f E L(X,dz)

a>

b)

{x E X

and

1 f(x)

# 0)

I(f)

= 0

is neglectible

are equivalent. A function f : X + 43 is called measurable if for each pair of functions c~,T,/JE Cc(X) the cutted function

f(x)ifv(x) I f(x) Lti(X> fw/J(4 ={o elsewhere

is integrable. A subset A c X is called measurable if its characteristic function is measurable. If it is integrable we call vol(A) =

J

x~(s)&T

the volume of A (with respect to dz). Notation. Let f : A + C be a function on some subset A of X. Assume that the function f(x) for z E A m = o for x # A I is integrable. We then say that f is integrable along A and write JA f(x)dx

:= J, f”(x)dx .

Special case: vol(A) = AII.2.

Stability

properties of measurable

JA

c&r .

functions:

1) Continuous functions are measurable. 2) Integrable functions are measurable. .?) open and dosed subsets are measurable. 4) Sum and product of measurable functions are measurable. 5) If (f,,) is a sequence of measurable functions which converges pointwise to a function f, then f is also measurable. 6) The maximum and minimum of two measurable functions are measurable.

Appendices

216

Stability

Properties

of Integrable

Functions.

AII.3.

The space L(X,cZx) of integrable functions is a linear space over C. The integral is C-linear. Let f be any measurable function and h an integrable function with

1f(z) 15 f

Then

is also integrable

h(x)

all 2 E X .

GOT

and we have

corollary. f E L(X, dx) *

1f IE L(X, dx) .

There axe two fundamental theorems about the compatibility with limits. AIL4

Theorem

of Beppo

of the integral

Levi:

Let h(x)

5

j-i(X)

be an increasing sequence of integrable of their integrals is bounded:

J

I

f f.

functions.

Assume

that the sequence

fn(x)da: I C.

X

Then

there

is an integrable

function f : X t

C such that

outside some negzectible set. We have

J

x f(x)dx

AII.5.

Theorem

=

;$I

J

x f&W.

of Lebesgue:

Let (f,,) be a sequence of integrable functions which a function f. Assume that there exists an integrable

1fn(x)

I 6 I h(x)

I

converges pointwise to function

h with

for all x E X and all n E N .

217

II. Integration

Then f is integrable,

and we have

J&

J

x fn(x)dx

=

JX

f(x)dx

.

Product Measures. Let (X, dz), (Y, dy) b e t wo spaces with Radon measures. If f : X x Y + 43 is a continuous function with compact support on the product space, then one easily shows: a) The

function x -

is contained b) The

f(x,

in Cc(X). function Y-

is contained

(Y fixed)

Y)

J

x f(X? YW

in C=(Y).

One can therefore the formula

define

a Radon

Jxxyf(x,

YP

measure

4i = J,

(the

product

[J,f(X’YW]

AII.6. Theorem of Fubini: For a function f : X x Y -+ C the following

measure)

dY.

three statements are equivalent:

1) f is integrable (with respect to dx dy). 2) There is a neglectible set S C Y such that

fy: x+c f&J = f(X> Y> is integrable for all y # S. The function Y-

.fx If(w)W

foryes

0

for y E s

is integrable. 3) The same condition as in 2) with the roles of X and Y interchanged. If these conditions are satisfied,then the formula

J

xxy f(x, y)dxdy = J, [J,f(X7YPq = J, [J,fkY)dY]

holds.

by

dY dx

218

Appendices

(The inner integrals have to be defined as 0 on some neglectible set.) Quotient Measures. Let I’ be a subgroup of the group of all topological mappings of X onto itself. We assume that r acts discontinuously, i.e. for any compact subset K c X the set of all

is finite. I’ is especially countable because X may be written as a countable union of compact subsets. The quotient space X/I’ (equipped with the quotient topology) is again a locally compact space with countable topology. Assumption.

The meawre dx on X is r-invariant,

i.e.

=Jxf(7+~ for YEr. Jxf(x)dx Under this assumption we are going to construct a quotient measure & on

x/r.

Convention: There is a one-to-one correspondence between (continuous) functions on X/I’ and I’-invariant (continuous) functions on X. We occasionally identify them. There is a well-defined mapping cc(x)

-

cwr)

The sum is locally finite. It is not very hard to prove the following facts: a) The mapping f I-+ F is surjective. If F 2 0, one may achieve f 2 0. b) We have F=O

+

Jxf(x)dx =O*

The properties a) and b) all ow us to define a Radon measure & on X/I’ means of the formula

1

x/r

F(x)&

=

J

x f(+x

by

.

This formula extends to the class of integrable functions. (The situation is similar as in the theorem of F’ubini.)

219

II. Integration

AII.7. A aubaet A c X/I’ is measurable (neglectible) if and only if its inverse image in X is measurable (neglectible). If f : X + c is an integrable function (with respect to dx) the series

F(x) = c fed -lgr converges outside a certain r-invariant x E S we have

set S. If we define

F(x)

= 0 for

F E C(X/I’,dZ) and

Fundamental Domains. A measurable subset F c X is called a fundamental domain with respect to l? if X = u +‘) -0 and if there exists a neglectible set S c F such that two different points in F - S are never equivalent with respect to r. AII.8.

with respect to & function f : X -+ 4: is integrable i an on y i i is integrable along F with respect to dx,

A r-invariant

~~:f%?~a~‘~,fhav~

1 f J,,,

The integral on the right-hand of a fundamental domain F.

t

f(x)fi

=

J, f(xPx

*

aide is especially independent

of the choice

Construction of Fundamental Domains. Let “N” be an equivalence relation on an arbitrary set X. If A is any subset of X, we denote by A the set of all elements of X which are equivalent to an element of A. Assume that a sequence AI,&.,... of subsets of X with the following two properties is given: a) X = A1 U AS U . . . b) Two different elements of an Aj are never equivalent.

Appendices

220

AII.9.

Under the above assumption the set F = Fl U Fz U . . .

where Fk = AK - (Al is a set of representatives

with

respect

u .x

Ak--l)

to the given equivalence relation.

We apply this trivial remark and construct a measurable fundamental domain. In all cases considered in this book the set of fixed points

S

= {a E X 1 lYa # {id}}

is a closed neglectible set. We may replace X by X - S and therefore assume that l7 acts freely (S = 0). We denote by Ur , Us,. . . a countable basis of the topology (i.e each open set is a union of Ui’s). We may assume that for all j lJj IT $Uj)

= 8

for

7 E I’ - {id} ,

because I’ acts freely on X. Let dx be a Radon measure on X with the following property. If A is a measurable set, then y(A) is also measurable for y E l?. This is for example the case if dx is l?-invariant. Under this condition the hull

d = u 64 -m of a measurable set is measurable. The construction AII.9 with Uj instead of Aj gives us a measurable fundamental domain. If Fo c X is a measurable fundamental set, i.e. x = u YPO) Y@ (not necessarily disjoint), we may apply the above construction to Uj n Fo instead of Uj, and we obtain AII.lO. tains

under the above assumption each measurable a fundamental domain.

fundamental

set con-

III. Alternating

Differential

221

Forms

III. Alternating

Differential

Forms

We give a brief introduction

into the theory of alternating

differential

forms.

We denote by Mp

= MP)

= {ac{l,...,n}

1 #a=p}

the set of all subsets a of (1,. . . , n} which consist of p elements. Their number is n (:=Oifpn). 0P Definition. A (n alternating) differential form w of degreep (shortly p-form) on an open domain D c R” is a system of functions (fakMp

3

f,:D---+C.

Notation. W’(D) := set of all differential forms of degree p. M&(D) := set of all (F-differential forms of degree p (i.e. all components fa are Co”-functions) (= 0 if p > n or p < 0). We may identify M’(D) with the set of functions on D (because MO consists of one element, namely the empty set). We have especially ML(D)

= C==‘(D).

We give the basic definitions of the calculus of differential forms: I. &P’(D) is an MO(D)-module:

(fa) + (sa) = (fa + sa) > f - (fa) = (f *fla) . II. The Total Differential of a Function. We may identify Ml(D)

= l~f’(D)~

M&,(D)

= C”(D)n.

= M’(D)

x . . . x M’(D)

We define the total differential by d : C”(D)

+

Ml(D)

df =(&..,$. One has 4

4f +d = df + &

Appendices

222

Ws) = f&l +!a

b)

df = 0 W

C>

f is locally

constant

.

Notation.

dXj := (0 ,...)

O,l,O )...) 0)

(This is the differential of the function pj(z) = zj). With this notation we may write df = 2 afdxj. j=l axj III.

The

Product. In analogy to the case p = 1 we define the for a subset a c { 1,. . . , n} of order p by

Alternating

p-form dx,

(dxa)a = { We may write an arbitrary

1 ifa=b, 0 ifa#b.

pform w = (fa) as w=

odxa.

Cf aEM,

Let a,bC

{l,...,

n} be two subsets of order p. We define a certain “sign” -1) .

~(a, b) E {O,l,

Case 1: ~(u,b)

= 0

if

an b #

0,

Case 2: Assume a II b = 8.

Let al,...,

aP be the elements of a in their natural order: a: al<...
and analogous b : bl < . . . < b, .

We have aub={al

,***,+,

b l,***,

bp 1.

The elements in the brackets are not necessarily in their natural order. We denote by ~(a, b) the sign of the permutation which arranges them in their natural order. We now define dx,

A dxa = ~(a, b)dx,ua

III.

Alternating

Differential

223

Forms

and extend this bilinearly to a mapping MP(D) (c

x MQ(D)

fadxa) A (c a

-

Mp+“(D)

gbdxb) = c b

,

f&b dx, A dxb .

a,b

It is easy to verify the following rules

fflw

1)

2)

= few

w A w’ = (-l)pqw’

if if

Aw

fEMO(D), w E MP(D),

w’ E Mq(D)

,

especially WAW=O

pisodd,

(w A w’) A w” = w A (w’ A w”)

3) 4)

if

if

(w~+w~)Aw=w~Aw+w:!Aw

,

wl,wz~M~(D),

WEM’(D).

Because of the law of associativity 3) we may define the alternating product differential forms. We obviously have

w1 A . . . A wp of several alternating

dx, = dx,, A . . . A dxa, a = {Ul

Y..,aPl

7

al < . . . < ap.

We may therefore write a differential form in the following form w=

adxa

c-f aEM,

=

fa, ,...,apdxq A.. . A dxcz,.

c l
Example: The alternating product of two l-forms (= differential forms of degree 1) is given by

(2

v=l

fvdxdg

p=l

g&d

c

=

(fvgp - fpgv)dxv

l
because dx, A dx, = -dx, IV. Exterior

Differential.

A dx,

(= 0 if v = p) .

We define d: M,&,(D)

+

MC’(D)

A dx, ,

Appendices

224

by the formula d(c

fadxa)

It is easy to verify the following

= c

dfa A dxa .

rules:

d(w+w’)=du+dw’,

1)

2) d(w A w’) = (du) AU’ + (-1)Pw A du’

3)

,

w E M&(D),

if

dw’=O.

w’ E M&(D),

d&)=0.

As a special case of 2) we get the formula d(wAw’)=dwAw’ We obtain

by induction d(w A dfi A . ..Adfm)=dWAdfrA...Adf.,,.

V. Transformation

of Differential

Forms.

Let

be a Cm-mapping between open domains D c R” , D’ c R”. We denote the co-ordinates of D by x = (21,. . . , x~) and those of D’ by , ym). We want to construct a mapping Y = (Yl,...

Mp(D’)

+ w cf

IMP(D) ‘p*w

such that the following “axioms” hold: 1) In the case of functions (p = 0) we have

P*(f >= f 0 P * 2)

Q*(w + w’) = ‘p*w + Q*w’ .

3)

Q*(W

4)

Q*(dW)

Of course such a mapping

A

w’)

=

= d(cp*w) is unique:

A

Q*W

Q*W’

.

(w E ML(D)). From 4) we obtain

cp*(dy;) = dpi = 2 j=l

@dzj axj

III.

Alternating

Differential

225

Forms

OF

where

Here g(cp, x)’ is the transpose

A very important

of the Jacobian -* *

~cpl/~Xn

...

a(Pmi~xn

case is p=m=n.

A differential

form of “top degree” p = n on D’ is of the form w = f dyl A . . . A dyn .

We have ‘p*w

= gdxl A...

r\dx,

with a certain function g. From the characterization of the determinant as an alternating multilinear form of the rows with certain properties one obtains g(x) VI. D

c

= det31cpyx)f(4x))

.

Complex Co-ordinates. We now consider the case of an open domain Cn. We may identify Cn with R2n by means of

(

21

,...,z,)

-

(Xl,Yl,...,%Y,)

and therefore apply the “real theory” of differential forms to this case. But it is frequently useful to use complex co-ordinates instead of real ones, for example dz, := dx, + i dyu , := dx, - i dy,

&, instead

of dxj = (dzj + &j)/2, dyj = (dzj - ~Ei)/2i

.

Appendices

226

Notation. dz, := dz,, A . . . A d.zap ,

a= {%..4p), MPlq(D) :=

witha,b~

{l,...,

lIal<...
c M’(D)dz, #a=p,#b=q

A dzb

n}.

We obviously have a decomposition

Mm(D) = c iw~q(D) p+q=m as a direct sum, i.e. the decomposition of an w E Mm(D) w=

c p+q=m

WPGl

7

WP&?

E Mp,qw

into a sum

7

is unique (as well as the decomposition of a single wp,q into its components, WPGl

=

c

fa,dh

A ab

.)

#a=p,#b=q

The alternating product respects this decomposition: w E MP”(D)

, w’ E kfP’jq’(D)

==+ w A w’ E kfp+p’aq+q’(~) ,

The exterior differentiation d does not respect the decomposition. To improve this situation, we write d as a sum of two operators a,3 which do respect the decomposition: We define

and

d/dZj := (a/&j

- ia/dyj)/2,

a/Zj

+ ia/ayj)/2

:= (a/&j

III. Alternating

Differential

227

Forms

af := -&Zj)dZj

)

j=l 3f

:=

.

e(aflaTj)&j

j=l

More

generally

we define

linear

mappings

a : My(D)

--t

M,gp’(D)

8: Mgf(D)

+

Mg’+l(D)

by

One can verify

a(fdza

A fib)

:= (af)

A dz, A &b

a(fdz,l

A fib)

:= @f)

A dz, A d?$, .

the rules d =

1) 2)

a(w A w’)

3)

aoa=o,

VII. domain

=

a+&

(8~)

A w’ + (-1)rw

BoB=o,

A i?w’ ,

aoB=-aoa.

Holomorphic Differential Forms. A differential D c Cn is called holomorphic if the following

a) w is of type

(p, 0), i.e. w = C

b) The

components

Notice: A Coo-function ferential

form w on an open two conditions hold

equations).

.fil,...,i,dzi,

fiI,...,i,

A . . - A dzi,

y

are holomorphic.

f is holomorphic Therefore a C”-(p,

iff af 0)-form

= 0 (Cauchy-Riemann w is holomorphic iff

3w=o.

Notation. W(D)

= {w E kP”(D)

1 w holomorphic}

.

We have d = a: S’(D) (8(x

VIII.

fadza)

+

w’+‘(D)

= c

afa A dz, ) .

Holomorphic Transformation of Differential p:D

--t

D’

Forms. Let

dif-

Appendices

228

be a holomorphic mapping between open domains D c Cn , D’ C C”. Prom the Cauchy-Piemann differential equations it follows that

Q*(dWi) = C(a(Pi/3Zj)dZj Q’(dEi) = ~(a(pi/azj)dzj. We therefore obtain mappings Q*

: n/rpSq(o’) Q*

: n’(D’)

MPaq(D)

(Q

holomorphic!) ,

-----f rip(D).

In the special case 72= m we have

cp*(dwl A . ..dw.J =j(cp,z)dzl where

A...Adz,

z) is the determinant of the (complex) Jacobian,

j(cp,

j(Q,

= det(&i/&i)

Z)

.

IX. Riemannian Metric. A Riemannian metric on an open domain D c I?” is a symmetric n X n-matrix 9 = (5%) of real CM-functions

7 gik : D + R

(1 5 i, k 5 n) ,

on D such that g(z) is positive definite for all x E D.

Let Q:D'

---tD

be a C”“-mapping between open domains D’ c Rm and D c R”. We define Q*(g) by Q*(g)(Y) := ~‘(QY Y)dQ(Y))~(Q, Y> (J(Q, x) Jacobian) . This is a symmetric real positive semidefinite matrix of CM-functions. It defines a Riemannian metric, if Q is a local immersion, i.e. g(cp, y) has rank m for all y E D. A diffeomorphism cp:D-=+D

is called

a

motion with respect to g if Q*!l = g.

Our basic example is the upper half plane equipped with the so-called Poincare metric

III.

Alternating

Differential

Forms

It is easy to prove that the fractional

z H

linear transformations

Mz,

M E SL(2, R) ,

are motions with respect to g. We want construct a certain pairing MP(D)

x Mp(D)

+

(w,w’) Here “pairing” pairing with

the following

b)

R” to

> .

f-g

f,gEM’(D).

if

g ik denotes the components

Aw,,wi A . . .Aw;

There

exists

a unique

M’(D)

>= det(<

If cp : D’ + D is a diffeomorphism <

Q*W,Q*W’

> = g'" y

of the inverse of g,

The proof is easy and straightforward. ements of the basis dxi, A . . . A dxi,, properties b) and c).

We have especially: motions.

w,w’

+

9 -1 =

4 <WI/l...

t---t<

x Mp(D)

< dXi,dXk

where

C

properties:

=

a>

g on D

M’(D)

1 near mapping.

means a M’(D)-b’li MP(D)

to use the metric

>t+,es=

The constructed

(gik). w~,w:

(wi,w:

>)l
Ml(D))

we have Q*

<

pairing

W,W’

>s

.

is invariant

dxl A..

with

respect

D c Rn with

. A dx:, .

This fundamental form is also invariant with respect to a motion Q, Q*Wg = Wg ,

if Q preserves the orientation, detJ((p,z)

.

One defines the pairing on the elextends it bilinearly and proves the

The so-called fundamental form on an open domain spect to a Riemannian metric g is defined as wg := + Jdets

E

i.e.

> 0 forallx

E D.

to re-

Appendices

230

(More

generally

we have for an orientation

preserving

diffeomorphism

‘p* : D’ + D p*wg

= wl+Pg .)

X. The Star Operator. Let D c R” be an open domain with Riemannian metric g. There exists a unique M” (D)-linear isomorphism + : W(D)

hl > Mn-P(D)

such that <*w,w’>wg

= WAW’

for all w E IMP(D)

,

w’ E AP-p(D).

The star operator is invariant with respect to orientation preserving motions 9 *(cp*w)

=

cp*(*w).

One has *(*w)

= (-1)PnSPW.

The codifferentiation 6: ML(D)

+

(= 0 if p = 0)

Mzl(D)

is defined by 6 := (-l)np+n+l

* d* .

The Laplace-Beltrami operator is A : M&(D)

-

MS(D),

A := d6+6d. Of course 6 and A commute with orientation preserving motions. (More generally: If y3 : D’ + D is an orientation preserving diffeomorphism we have p* o As = Apes .) XI. Hermitean Metric. A Hermitean metric is a complex matrix h with the property X’ = h. Each n x n-Hermitean matrix defines a real 272x 2n symmetric matrix g which is characterized by Z’h.z

=

a’ga,

where a’= (q,y1,...

7ZTl,Y7l

1.

231

III. Alternating Differential Forms

We say that g comes from the Hermitean matrix h. A Hermitean metric on an open domain D c C” is an n x n-matrix h of Cm-functions on D, such that h(z) is Hermitean and positive definite for all z E L). The associate 2n x Bn-matrix g is a Riemannian metric. We have det h(z) = +dm. The fundamental form may be written as wg = mdq

Adyl A...Adz,Ady,

= &--(det

h)dzl A &I A . . . A dz, A d!Z, .

Let now cp : DA

D, D’ c Cn open,

D’ ,

be a biholomorphic mapping, h a Hermitean metric on D’ and g the associate Riemannian metric. Then the pulled back Riemannian metric ‘p*g on D comes from a Hermitean metric, namely from

Here &((p,z)

denotes the complex Jacobian.

The Laplace-Beltrami

operator -A

= d*d*

+ *d*d

(the real dimension of C” is even) usually does not preserve the decomposition Mm(D) = c Mpsq(D).

p+q=m Therefore one also considers the operators

-•=a*8*+*8*a - ii =a*a*+*a*a which map MPfJ(D) into itself. This follows immediately from the following fact: In the case of a Hermitean metric the star operator maps (p, q)-forms to (n - q, n - p)-forms .+ : MM(D)

+

jtrfn--q+--p(D) .

XII. Kiihlerian Metric. A Hermitean metric h on an open set D c C” is called locally Euclidean at a point a E D if h(u) is the unit matrix and if the first partial derivatives of h vanish at a.

Appendices

232

Definition. A Hermitean metric h is called Kihlerian if it is locally equivalent with a Euclidean one, i.e.: For each point a E D there exists a biholomorphic mapping of an open subset U c C” onto an open neighbourhood V of a in D such that the pulled back metric cp*h is locally Euclidean at b = p-‘(a). For any Hermitean

This differential mations cp, i.e.

metric

form

h we may consider

is invariant

with

respect

=

cp*Q(h).

R(cp*h)

Remark. In the case of a KZhlerian

metric dS-2 =

The

converse

is also true

the (l,l)-form

to biholomorphic

transfor-

h we have

0.

but we do not need this.

Proposition. In the case of a KihleTian

metric

we have the identities

A=20=2Ii.

Corollary.

The Laplace-Beltrami M;(D)

We make the proof:

operator

A preserves

=

Mzp(D).

c p+q=m

use of this proposition in Chap. Let us consider the operator L

: MP,P

+

III,

the double

$1 and therefore

graduation

we indicate

MP+l,P+l

L(w)=RAw, where sition

R denotes the K&ler form (see above). are a formal consequence of the relations Lo*a* Lo*a*

-

-

*a*oL *a*oL

The

=

ia

=

-8.

identities

in the propo-

We leave this reduction to the reader. The advantage of the latter relations is that they involve only first order derivatives. From the definition of the

III. Alternating

Differential

233

Forms

K&ler property it follows that such a relation, which is invariant under biholomorphic transformations, has to be proved only in the case of the Euclidean metric h = E = unit matrix. In this case the relations can be verified by direct calculation. Example. Each Hermitean metric h on a (complex) l-dimensional domain D C C is Kihlerian.

The de Rham Complex XIII. Differential Forms on Manifolds. Let X be a topological space. We always assume that X is a Hausdorff space with countable basis of its topology. A differentiable structure on X is a family Qj

ZUj +

Vj 3 Uj C X open, Vj C Rn open,

of topological mappings with the properties

a>

x+4, Qj O(Pi1 : Qi(Ui

is a P’diffeomorphism

fI Uj)

+

Qj(Ui

n

Uj)

for all (i,j).

The space X together with a distinguished differentiable structure is called a differentiable manifold of dimension n. A differential form of degree p on X is a family w

=

(w)

Wi

7

E M’(Vi)

3

such that the formula (Qj

0 Qi’)*Wj

=

Wi

holds on Qi(Ui fl Uj). We denote by i@(X) the space of all pforms on X and by M&(X) the space of all COD-p-forms (i.e. all wi are Cm). We may identify a function f : X + 4: with the zero form (fi)

3

.fi

=

f

1 uiOQ:l

*

The function f is called Cm-differentiable if all the fi’s differentiable. We hence may identify M&,(X) and C”(X)

= {f:X+C

1 fist”}.

are C”-

Appendices

234

There are natural

mappings MP(X)

x MQ(X)

4

Mp+Q(x)

(w,w’)l-4wAw’, (W

A W’)i

:=

Wi

A W:

and M&(X)A

itdgyx>

(Ckd)i

:=

CJTWi

)

e

The sequence ... is the so-called

M&(X)

d > AIgyx>

de Rham complex

The de Rham cohomology C) are defined as HP(X)

(complex

groups

-

...

means: d o

d

(they are actually

=

0).

vector

spaces over

:= cyx)/Byx)

where

By the theorem

G’(X)

= ker(M,&(X)

BP(X)

= im(Mcl(X)&

of de F&am

there

exists

d b iL@l(X))

a natural

HP(X)--“--+ where

W(X,

C) denotes

the singular

M,&(X)). isomorphism

HP(X,

cohomology

C)

groups

with

coefficients

XIV. Real Hodge Theory. The Hodge theory is a powerful the de Rham cohomology groups in the case of a compact A Riemannian

metric

g on the differentiable

g = (gi) , such that the transformation

gi Riemannian

tool to compute manifold: X is a family

metric on Vi

formula (Cpj

is valid on pi(Ui

manifold

in C.

O Pil)*gj

n Uj). If a Ri emannian * : MP(X)

--f

(*W)i

:=

=

Si

metric is given, the star operator AP-P(X) *(Wi)

III.

Alternating

is well

Differential

defined.

235

Forms

We therefore

may define

A : ML(X)

+

(Au)~ The

kernel

One of the main

results

=

operator

M,&(X)

= Awi.

of A is the space of harmonic tip(X)

the Laplace-Beltrami

forms.

ker(M,&(X)

A b M&(X)).

of the real Hodge

theory

states:

Assume that X is compact. Then each harmonic form is closed. The natural mapping ‘HP(X) HP(X) is an isomorphism. Notice: If w is harmonic

then

*w is also harmonic.

We obtain

(for a compact

manifold!) w harmonic

XV. Integration

M

&J = 0 and d(*w)

= 0 .

of n-forms.

An n-form

w = fdxl

A . . . A dx,

on an open domain D c Rn is called integrable with respect to the Euclidean measure:

if the function

f is integrable

Notation. J,w

:= kf(x)dx+.dx,.

If

cp:D’ is an orientation

We hence

preserving

may generalize

+

diffeomorphism,

the notion

D we have

of an integrable

n-form

w and the value

to an arbitrary oriented differentiable manifold. Here “oriented” means that all transition functions 'pj 0 (pi1 are orientation preserving. A differential form w of arbitrary degree p on an oriented Biemannian manifold (X, g) is called square integrable, if the n-form (n = dimX) w A *c is integrable.

Appendices

236

XVI.

Some Results on Non-compact Manifolds.

Theorem. Let w be a square integrable and closed (dw = 0) C?‘-differential form on an oriented Riemannian manifold. TheTe exists a square integrable harmonic form wo such that w =

wo+&.

(6 some COO-differential form.) But in contrast to the compact case the form wg needs not to be unique and not each square integrable harmonic form needs to be closed. But there is a very remarkable Theorem. Let (X,g) each square

integrable

be an oriented complete Riemannian harmonic form is closed.

manifold.

Then

What does complete mean? Let a : [O, l] --f

D

be a Cm-differentiable curve in a domain D c FPnwhich is equipped with a Riemannian metric. The velocity of o at t E [0, l] is defined by G(t)

=

c

sij (+))&(t)~j(t))

*

l
The length of o is defined by

l(a) = l,(a) = 1’ &(&a. The notion of velocity and arc length is invariant with respect to diffeomorphisms cp: D’ -+ D if one replaces o by (Yo cp and g by cp*g. One may especially generalize these notions to the case of curves Q! : [O,l] +

x

in an arbitrary Riemannian manifold (X,g). connected (X,g) by means of d(a, b) := i;f

One defines a metric on a

1(o),

where (Y runs over all curves connecting a and b. (X, g) is called complete, if each Cauchy sequence with respect to this metric converges.

III. Alternating Differential Forms

237

XVII. Complex Hodge Theory. A complex analytic manifold dimension n is a topological space X together with a family

(PiZUi of topological

+

mappings

Vi

Ui C X open,

Vi

of (complex)

C” open

C

such that

a) X=UUi b) all the transition maps pj o tpi’ are biholomorphic. Each complex analytic manifold is also an oriented differentiable manifold of (real) dimension 2n. We may also consider the Dolbeault complex . . . --f and the Dolbeault

iw’(X)~

APQ+yx)

cohomology

+

...

groups

lP(X)

= cp+yx)/Bp~yx)

)

where Cpl’(X)

= ker(MPPq(X)A

BPtq(X)

= im(MP’q-l(X)L

Mp*q+‘(X)) it4p*q(X)).

Remark. There is a natural isomorphism IF(X)

Y HQ(X,QP,) ,

where Cl% denotes the sheaf of holomorphic p-forms on X. Similar to the real case we may generalize the local theory of differential forms on complex domains to the case of analytic manifolds. We especially may define the notion of a (p, q)-form and have the decomposition

kP(X)

=

$

MP+yX).

p+q=m

The same is true with

the subscript

“00”.

We now assume that a Hermitean metric h (i.e. a compatible family (hi) of Hermitean metrics hi on Vi) is given. Then the operators q , li are well defined. If h is K&lerian (i.e. all hi are Ktihlerian) we have A = 20. Notation. 7-P(X)

=

{w E MgJ(X)

1 q w=

In the Kiihlerian case we have 7-P(X)

=

$ w-(X). pSq=m

The main result of the complex Hodge theory is

0).

Appendices

23%

Theorem. Let X be a compact Kihlerian manifold. is %losed (i.e. &J = 0). The natural mapping 7-P(X)

-

harmonic form w

Each

IPyX)

is an isomorphism.

Notation. bm = dimJP(X) hPjq

=

m(-thBetti

number) ,

dimHPPq(X) .

In the case of a compact K%hlerisn variety we have 0” =

c h”lq, p+q=m

furthermore hP,‘l

=

h!7>P

=

h”-P,“-q.

The first relation follows from the fact that A is a real operator, especially Aw=O

M

Aw=O,

the second relation follows from Aw = 0 M

A(*w) = 0 .

XVIII. Differentiable Mappings. If X and Y are two differentiable (resp. analytic) manifolds, one defines in an obvious way the notion of a differentiable (resp. analytic) mapping $7:x Y. If w is an m-form (resp. (p, q)-f orm ) on Y, one defines the pulled back form ‘p*w on X. We obtain natural mappings . They

are compatible

with

cp* : P(Y) the de Rham

--f

W(X).

isomorphism

and

the mappings

Hm(Y,C)-Hm(X,C) of the singular

cohomology

groups

which

are usually

defined

A similar statement is true in the case of analytic holomorphic mapping we obtain a commutative diagram w

W

(o*w

HP*'(Y) Ill Hq(Y,Q;)

-

HP+'(X) Ill HQ(X,Q;).

--*

in algebraic

manifolds

X,Y.

topology. In the

case of a

III. Alternating

Differential

The mapping

Forms

239

in the second line is induced by the natural mapping

Let now D C Rn (C”) be an open domain and l? a group of C”diffeomorphisms (biholomorphic mappings) of D which acts discontinuously and freely (i.e. without fixed points). The quotient space

xr can be equipped the projection

with

a natural

:= D/l?

differentiable

p:D is locally diffeomorphic isomorphisms

The

superscript

l? means

structure

such that

-Xr

(biholomorphic).

(Mgq(Xr)

(analytic)

The

+

mapping

w H

‘p*w

defines

MGq(D)r).

that we take the subspace

of l?-invariant

elements.

Assume that a l?-invariant Ftiemanman (Hermitean) metric on D is given (i.e. all elements of l? are motions). Then we obtain a natural Ftiemannian (Hermitean) metric on the quotient Xr. The projection

D + is a local

isometry.

We obtain

isomorphisms

7dp(Xr) 7fp*‘(Xr) XIX.

Xr

= ‘Hp(D)r = 7f,pPq(D)r .

Poincark Duality. Let X be a differentiable

We define

the de F&am

cohomology

HP(X)

=

cyx)/Byx)

Cp(X)

=

space of all C”

BP(X)

=

space of all dw , w a C”

One can also define

manifold

of dimension

p-forms

w , CEW= 0, (p - 1)-form

.

the de Rham cohomology groups with compact support:

where C,P(X)

=

n.

groups

space of all C”

p-forms

w with

compact

support,

du=o.

Appendices

240

(“Compact support” means of course that there exists K C X such that the restriction to X - K is zero.) B,P(X)

= {ULJ 1 w a C”

(p - 1) -form with

a compact

compact

subset

support}

.

We have cw>

and hence obtain

a natural

c

The

-

in general is neither injective

theorem

of de Rham

between

the

without

compact

de Rham

states

that

cohomology

support)

with

7

linear mapping q-q

which

CP(W

HP(X)

t

nor surjective.

there

are natural

isomorphisms

W(X)

= HP(X,

C)

H,p(X)

Y H,p(X,

C)

groups

and

coefficients

the singular

cohomology

groups

(with

or

in C.

The Poincare duality theorem is usually proved in the context of singular cohomology. We express it in terms of the “de Rham cohomology”. First

we construct

a pairing HP(X)

We represent

x H,n-yX)

elements of HP(X)

+

c .

(resp. HF-P(X))

by differential

(resp. w’ E Cr-p(X))

w E Cp(X)

We can consider the n-form w A w’. It has compact integrable. We claim that the integral

forms

. support

and is hence

J WAW’ X

depends

only on the class of w or w’. This means for example

JX

o?GAw’=Q.

We have dr;, A w’ = d(3 A w’) and the assertion Stokes’s Theorem. we have

that

Let w be a C”

(n - l)-form

follows

from a special case of

with compact support.

JoLJ=o. X

Then

241

III. Alternating Differential Forms So Stokes’s theorem gives us the desired pairing HP(X)

x H,“-yX)

+

c I

The Poincark duality theorem states that this pairing is non-degenerate under certain assumptions. (A bilinear mapping vxw+c

(u,b)+-Ka,b>

for two vector spaces V, W is called non-degenerate if for each a E V, a # 0, there exists a b E W such that < a, b ># 0 and vice versa. The spaces V and W then have the same dimension.) We now assume that X is contained as an open subspace in a compact topological space x. We assume that the topological space ax

:= x-x

(with the induced topology of x) is also equipped with a structure as differentiable manifold. We assume furthermore that each “bo~undary” point a E aX admits an open neighbourhood U(u) and a topological mapping $0: U(u)

-

b v=

{CEE Rn ( IlLElI< 1) 3% 2 O}

such that

p(U(a)nX)

= vi = (~0

I ~,>o}

and such that the mappings U(u) n x

U(u)r-dX

-

vo )

--t

{~ER”-~

I (z,O)EV},

induced by cp are diffeomorphisms. (This means that X is the interior of a compact Cw-manifold with boundary.)

Poincar~

Duality.

Under the above assumptions on X we have

1) All the cohomology

groups HP(X)

are finite dimensional. 2) The pairing HP(X)

?

Rx-v

x H,n--P(X) --) c (w,w’)

cf

w A w’ JX

is non-degenerate.

We especially

dimHP(X)

have

= dimHESP(X)

.

Appendices

242

The proof of this theorem is usually reduced to a corresponding result in algebraic topology via the “de Rham isomorphism”. But is is also possible to give a proof in the context of differential forms. In this connection we mention another long exact sequence which is well known from algebraic topology. Under the same assumptions has a long exact sequence * ... -

q(x)

-

HP(X)

The nature of a is not important natural ones. Example: compact.

satisfies

as in the Poincard

+

duality

a P H,p+l(x)

Hyax)

in our application.

-

one

... .

All other mappings

are

Let I? be a discrete subgroup of SL(2, R)n such that (Hn)*/I’ is We assume that I’ has no elliptic fixed points. Then the quotient

the assumptions

of the Poincare

duality

theorem.

But the compactification by the cusps is not a manifold We have to modify this compactification. Recall that close to the cusp 00 the quotient l&/r, with

theorem

,

C > 0. We have a natural

&={eHn

boundary.

looks like

1 iVy>C}

topological

TL/r,-{toi

H”/I’

with

mapping

1 QC)XY,

where Y = (2 E Hn 1 Ivy=

q/r,.

This space carries a natural differentiable structure. We have proved that it is compact. Hence we may compactify UC/I’, by adding not only a single point but by adding 00 x Y.

&jr

L, [c~,~]xY.

We may repeat this construction for each cusp class and obtain a realization of H”/I’ as the interior of a manifold with boundary. This shows that the Poincark duality theorem can be applied to H”/I’. The spaces Hn/r, *

‘Exact”

means

that

the image

, ?Icp,

, [c,o~] x Y

of an arrow

equals

the kernel

of the next

one.

III. Alternating

Differential

are homotopy

243

Forms

equivalent.

We obtain

Hyax)

E 6 HP(H”,l?,;) j=l

where

ICI,...,

&h is a set of representatives

We therefore . ..+

obtain

an exact

iY,“(H”/I’)

+

sequence

Hm(H”/I’)

of the cusps. which

is used in Chap.

--+ 6 j=l

a All

the arrows

H”+l(Hn/I’) c besides

--t

... .

the a’s are obvious

ones.

H”(H”,l-,j)

III,

$5

Bibliography

Andreotti, A., Vesenlini, E. 1. Carleman Estimates for the Laplace-Beltrami equation on complex manifolds. Publ. Math., I.H.E.S. 25, 313-362 (1965) Ash, A., Mumford, D., Rapoport, M., Tai, Y. 2. Smooth compactification of locally symmetric varieties. Math. Sci. Press, Brookline, Mass. 1975 Baily, W.L. 3. Satake’s compactification of V,l . Amer. J. Math. 80, 348-364 (1980) Baily, W.L., Borel, A. 4. Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. 84,442-528 (1966) Bassendowski, D. 5. Klassifikation Hilbertscher Modulflilchen zur symmetrischen Hurwitz-Maa%Erweiterung. Bonner Math. Schriften 163 (1985) Blumenthal, 0. 6. Uber Modulfunktionen von mehreren Veranderlichen. Math. Ann. 56,509-548 (1903) and 58497-527 (1904) Cox, D., Parry, W. 7. Genera of congruence subgroups in Q-quaternion algebras. J. f. d. reine u. angew. Math. 351, 66-112 (1984) Cartan, H. 8. Fonctions automorphes. Seminaire No. 10, Paris 1957/58 Deligne, P. 9. Theorie de Hodge. I, II Publ. Math., I.H.E.S. 40, 5-58 (1971) and 44, 5-77 (1974) Dennin, J. 10. The genus of subfields of K(pR). Illinois J. of Math. 18, 246-264 (1984) Ehlers, F. 11. Eine Klssse komplexer Mannigfaltigkeiten und die Auflijsung einiger isolierter Singula&&en. Math. Ann. 218, 127-156 (1975) Freitag, E. 12. Lokale und globale Invarianten der Hilbertschen Modulgruppe. Invent. Math. 17,106, 134 (1972) 13. Uber die Struktur der Funktionenkorper zu hyperabelschen Gruppen. I, II J. f. d. reine u. angew. Math. 247, 97-117 (1971) and 254, 1-16 (1972) 14. Eine Bemerkung zur Theorie der Hilbertschen Modulmannigfaltigkeiten hoher Stufe. Math. Zeitschrift. 171, 27-35 (1980) Freitag, E., Kiehl, R. 15. Algebraische Eigenschaften der lokalen Hinge in den Spitzen der Hilbertschen Modulgruppen. Invent. Math. 24, 121-148 (1974) van der Geer, G. 16. Hilbert modular forms for the field Q(d). Math. Ann. 233, 163-179 (1978)

246

Bibliography

17. Hilbert modular surfaces. Erg. der Math. III/16 Springer-Verlag van der Geer, G., Zagier, D. 18. The Hilbert modular group for the field Q(m). Invent. Math. 42, 93-133 (1977) Gundlach, K-B. 19. Some new results in the theory of Hilbert’s modular group. Contributions to function theory. Tata Institute Bombay165-180 (1960) 20. Die Bestimmung der Funktionen zur Hilbertschen Modulgruppe des Zahlkorpers Q(A). Math. Ann. 152, 226-256 (1963) 21. Die Bestimmung der Funktionen zu einigen Hilbertschen Modulgruppen. J. f. d. reine u. angew. Math. 220, 109-153 (1965) 22. Poincaresche und Eisensteinsche &hen zur Hilbertschen Modulgruppe. Math. Zeitschrift. 64, 339-352 (1956) Hammond, W. 23. The modular groups of Hilbert and Siegel. Amer. J. of Math. 88497-516 (1966) 24. The two actions of the Hilbert modular group. Amer. J. of Math. 99, 389-392 (1977) Harder, G. 25. A Gauss-Bonnet formula for discrete arithmetically defined groups. Ann. Sci. E. N. s. 4, 409-455 (1971) 26. On the cohomology of discrete arithmetically defined groups. Helling, H. 27. Bestimmung der KommensurabilitLtsklasse der Hilbertschen Modulgruppe. Math. Zeitschrift. 92, 269-280 (1966) Hermann C.F. 28. Symmetrische Hilbertsche Modulformen und Modulfunktionen zu Q(m). Math. Ann. 256, 191-197 (1981) 29. Thetareihen und modulare Spitzenformen zu den Hilbertschen Modulgruppen reellquadratischer K&per. Math. Ann. 277, 327-344 (1987) Hirzebruch, F. 30. Hilbert modular surfaces. L’Ens. Math. 71, 183-281 (1973) 31. The Hilbert modular group, resolution of the singularities at the cusps and related problems. SBm. Bourbaki1970/71, exp. 396. In: Lecture Notes in Math. 244. SpringerVerlag (1971) 32. The Hilbert modular group for the field Q(&) and the cubic diagonal surface of Clebsch and Klein. Usp. Mat. Nauk 31, 153-166 (1976) (in Russian) Russian Math. Surveys 31 (5), 96-110 (1976) 33. The ring of Hilbert modular forms for real quadratic fields of small discriminant. In: Modular functions of one variable VI. Lecture Notes in Math. 627,287-324. SpringerVerlag (1976) 34. Modulflachen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe. Ann. Sci. E. N. S. 11, 101-166 (1978) 35. The canonical map for certain Hilbert modular surfaces In: Proc. Chern Symp. 1979. Springer-Verlag (1981) 36. Uberlagerungen der projektiven Ebene und Hilbertsche Modulflachen L’Ens. Math. 24, 63-78 (1978) Hirzebruch, F., van der Geer, G. 37. Lectures on Hilbert modular surfaces. Les Presses de 1’Univ. de Montreal (1981) Hirzebruch, F., Van de Ven, A. 38. Hilbert modular surfaces and the classification of algebraic surfaces. Invent. Math. 23, l-29 (1974) 39. Minimal Hilbert modular surfaces with p, = 3 and K2 = 2. Amer. J. of Math. 101, 132-148 (1979)

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Hirzebruch, F., Zagier, D. 40. Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus. Invent. Math. 36,57-113 (1976) 41. Classification of Hilbert modular surfaces. In: Complex Analysis and Algebraic Geometry. Iwanami Shoten and Cambridge University Press 43-77 (1977) KGller, F. W. 42. Zweidimensionale Singularitiiten und Differentialformen. Math. Ann. 206, 205-213 (1973) 43. Ein Beitrag zur Klassifikation der Hilbertschen Modulflachen. Archiv der Math. Vol. XXVI (1975) 44. Elementare Berechnung der Multiplizitaten n-dimensionaler Spitzen. Math. Ann. 225, 131-143 (1977) 45. Beispiele dreidimensionaler Hilbertscher Mannigfaltigkeiten von allgemeinem Typ. Manuscr. Math. 37, 135-161 (1982) 46. Uber die Plurigeschlechter Hilbertscher Modulmannigfaltigkeiten. Math. Ann. 264, 413-422 (1983) MaaP,. If47. Uber Grupen von hyperabelschen Transformationen. Sitzungsber. Heidelb. Akad. Wiss., 3-26 (1940) 48. Uber die Erweiterungsfahigkeit der Hilbertschen Modulgruppe. Math. Zeitschrift. 51, 255-261 (1948) Ma2sushima, Y., Shimura, G. 49. On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes. Ann. of Math. 78, 417-449 (1963) Meyer, C. 50. Die Berechnung der Klassenzahl abelscher K&per iiber quadratischen Zahlkijrpern. Berlin (1957) Mumford, D. 51. Hirzebruch’s proportionality in the non-compact case. Invent. Math. 42, 239-272 (1977) Predel, A. 52. Die elliptischen Fixpunkte der Hilbertschen Modulgruppen. Math. Ann. 177, 181-209 (1968) 53. Die Fixpunkte der symmmetrischen Hilbertschen Modulgruppe zu einem reellquadratischen K&per mit Primzahldiskriminante. Math. Ann. 200, 123-139 (1973) ResniLoff, H. L. 54. On the graded ring of Hilbert modular forms associated with Q(A). Math. Ann. 208, 161-170 (1974) Serre, J.P. 55. Faisceaux algkbriques coherents. Ann. of Math. 61 (1955) 56. Gdometrie algebrique et geometric analytique. Annales de 1’Institut Fourier 6, l-42 (1956) Shimizu, H. 57. On discontinuous groups acting on a product of upper half planes. Ann. of Math. 77, 33-71 (1963) Siegel, C.L. 58. Lectures on advanced analytic number theory. Tata Institute Bombay. 1961, 1965 59. The volume of the fundamental domain for some infinite groups. Trans. AMS 39, 209-218 (1936). Correction in: Zur Bestimmung des Fundamentalbereichs der unimodularen Gruppe. Math. Ann. 137, 427-432 (1959) Thompson, J. 60. A finiteness theorem for subgroups of PSl(2,R) which are commensurable with PSI(2,Z). Proc. Symp. pure Math. 37, AMS, Santa Cruz, (1980)

248 Vaaserstein,

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61. On the Group SL(2) on Dedekind rings of arithmetic type. Mat. Sbornik 89 (1972) (= Math. USSR Sbornik 18, 321-325 (1972)) Zagier, D. 62. Modular forms associated to real quadratic fields. Invent. Math. 30, l-46 (1975)

Index

algebraic integer 33, 203 algebraic number field 33, 204 alternating differential form 221 ff. alternating product 222 ample 115 arithmetic genus 121 automorphic form 47 local 114 Betti

number

142.

238

Cartan, criterion of 113 class number 37, 209 codifferentiation 230 coherent sheaf 114 ff. commensurable 35 complex space 113 ff. condition of irreducibility first 31, 89, 115 second 54, 115 cusp 12f., 24ff. boundary point 13 infinity 12, 24ff. cusp class 36 cusp form 47 cusp sector 19, 29 cuspidal cohomology 182,

Dirichlet unit theorem discontinuous 7, 21 discrete 7, 21 discriminant 123,‘205

34, 206

ff.

Eisenstein cohomology 2, 148 ff. Eisenstein series 6Off., 148ff., 158ff. analytic continuation 148ff. space of 64f., 183 elliptic fixed point 8, 30 elliptic matrix 8, 83 elliptic substitution 83 exterior differential 223 Euler-PoincarB characteristic 111, 116 factor of automorphy 44 finiteness theorem 66 ff. Fourier expansion 44 fundamental domain 19, 89, 219 fundamental form 229 fundamental set 18, 220 GGtzky-Koecher

184

Dedekind zeta-function 122 de Rham cohomology group 142, 234, 239 with compact support 239 de Rham complex 142, 233ff. de Rham, theorem of 143 desingularisation 117 different 210 differential form 221 ff. holomorphic 227 holomorphic transformation 227 on manifolds 233 transformation 224 dimension formula + Selberg trace formula

principle

51, 114

Hecke summation 151 Hermitean metric 230 Hilbert modular group 1, 32ff. Hilbert polynomial 116 Hodge decomposition 176 Hodge numbers 119, 133, 135, Hodge space 135 universal part 140 Hodge theory 234 complex theory 237 real theory 234 hyperbolic matrix 83 hyperbolic substitution 83 ideal class narrow Kiihler

36, 209 127

property

134, 231

185ff.

250

Index

kernel function Koecher principle ---+ Gotzky-Koecher

73

LaplaceBeltrami lattice 22, 205 dual 44

operator

Mobius motion multiplier

230

function 162 229 23, 211

norm 205 of ideals number field parabolic parabolic Petersson Poincark Poincare Poincare

principle

210 204

matrix 13, substitution scalar product duality 179, metric 135, series 58 ff.,

ray class 162 regular 45, 47 Riemann metric rotation factor

83 83, 105 68 240ff. 174 77

Selberg trace formula 73 ff., 79 cocompact case 81f., 89 contribution of cusps 89, 108, 110 contribution of elliptic fixed points 89, 110 error term 80f. main term 79 f. Shimizu L-series 109, 213 Shimizu’s polynomial 111 singularity 30 Sl(2.R) 5 square integrable cohomology 174 ff., 181, 184 star operator 230 Stokes’s theorem 240 total differential 221 totally positive 23 trace 205 trace formula + Selberg trace formula translation matrix 12, 22, 83 transvection matrix 83 universal cohomology upper half-plane 5

228 9, 89

weight

47

184

E. Freitag, University of Heidelberg; R. Kiehl, University of Mannheim

Eta/e Cohomology and the Weil Conjecture With a Historical

Introduction

by J. A. Dieudonne

Translated from the German manuscript by Betty S. and William C. Waterhouse 1988. XVIII, 317 pp. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Vol. 13) Hardcover DM 188,- ISBN 3-540-12175-7 Contents: Introduction. - The Essentials of Etale Cohomology Theory. - Rationality of Weil c-Functions. - The Monodromy Theory of Lefschetz Pencils. - Del&me’s Proof of the Weil Conjecture. Appendices. - Bibliography. - Subject Index.

This book is concerned with one of the most important developments in algebraic geometry during the last decades. In 1949 Andre Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. He himself proved his conjectures by means of an algebraic theory of abelian varieties in the onevariable case. In 1960 appeared the first chapter of the “Elements de Geometric Algebraique” par A. Grothendieck (en collaboration avec J. Dieudonne). In these “Elements” Grothendieck evolved a new foundation of algebraic geometry with the declared aim to come to a proof of the Weil conjectures by means of a new algebraic cohomology theory. Deligne succeded in proving the Weil conjectures on the basis of Grothendiecks ideas. The aim of this “Ergebnisbericht” is to develop as self-contained as possible and as short as possible Grothendiecks 1-adic cohomology theory including Springer-Verlag Berlin Heidelberg New York London Delignes monodromy theory and to present his original proof of the Weil conjectures. Paris Tokyo Hong Kong

G. van der Geer, University of Amsterdam

Hilbert Modular Sutiaces 1988. IX, 291 pp. 39 figs. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Vol. 16) Hardcover DM 148,ISBN 3-540-17601-2 Contents: Introduction. - Notations and Conventions Concerning Quadratic Number Fields. - Hilbert’s Modular Group. - Resolution of the Cusp Singularities. - Local Invariants. - Global Invariants. - Modular Curves on Modular Surfaces. - The Cohomology of Hilbert Modular Surfaces. - The Classification of Hilbert Modular Surfaces. - Examples of Hilbert Modular Surfaces. - Humbert Surfaces. - Moduli of Abelian Schemes with Real Multiplication. - The Tate Conjectures for Hilbert Modular Surfaces. - Tables. Bibliography. - List of Notations. - Index. Over the last 15 years important results have been achieved in the field of Hilbeti Modular Varieties. Though the main emphasis of this book is on the geometry of Hilbert modular surfaces, both geometric and arithmetic aspects are treated. An abundance of examples - in fact a whole chapter - completes this competent presentation of the subject. This “Ergebnisbericht” will soon become an indisSpringer-Verlag Berlin Heidelberg New York London P ensible tool for graduate students and Paris Tokyo Hong Kong researchers in this field.