Introductory lectures on Siegel modular forms doc

3 Fundamental sets of Siegel's modular group 275 Eisenstein series and the Siegel operator 54 8 Singular modular forms and theta-series 99 9 The graded ring of modular forms of degree tw

MATHEMATICS 20

Editorial BoardD.J.H Garling, D Gorenstein, T tom Dieck, P Walters

Introductory lectures on Siegel modular forms

1 W.M.L Holcombe Algebraic automata theory

2 K Petersen Ergodic theory

3 P.T Johnstone Stone spaces

4 W.H Schikhof Ultrametric calculus

5 J-P Kahane Some random series of functions, second edition

6 H Cohn Introduction to the construction of class fields

7 J Lambek & P.J Scott Introduction to higher-order categorical logic

8 H Matsumura Commutative ring theory

9 C.B Thomas Characteristic classes and the cohomology of finite groups

10 M Aschbacher Finite group theory

11 J.L Alperin Local representation theory

12 P Koosis The logarithmic integral: 1

13 A Pietsch Eigenvalues and s-numbers

14 S.J Patterson Introduction to the theory of the Riemann zeta-function

15 H-J Baues Algebraic homotopy

16 V.S Varadarajan Introduction to harmonic analysis on semisimple Lie groups

17 W Dicks & M.J Dunwoody Groups acting on graphs

18 L.J Corwin & F P Greenleaf Representations of nilpotent Lie groups and their applications

20 H Klingen Introduction to modular functions

22 M Collins Representations and characters of finite groups

Introductory lectures on Siegel modular forms

HELMUT KLINGEN

Professor of Mathematics University of Freiburg

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© Cambridge University Press 1990

First published 1990

British Library cataloguing in publication data

Klingen, Helmut Introductory lectures on Siegel modular forms.

1 Mathematics Automorphic functions

1 Title

515.7

Library of Congress cataloguing in publication data

Klingen, Helmut.

Introductory lectures on Siegel modular forms/Helmut Klingen.

p cm - (Cambridge studies in advanced mathematics; 20)

Anita, Christoph and Philipp

3 Fundamental sets of Siegel's modular group 27

5 Eisenstein series and the Siegel operator 54

8 Singular modular forms and theta-series 99

9 The graded ring of modular forms of degree two 112

12 Dirichlet series associated with modular forms

13 Analytic continuation and the functional equation 148

The theory of automorphic functions in one complex variable was createdduring the second half of the nineteenth and the beginning of the twentiethcenturies Important contributions are due to such illustrious mathe-maticians as F Klein, P Koebe and H Poincare Two sources may betraced: the uniformization theory of algebraic functions, and certain topics

in number theory Automorphic functions with respect to groups withcompact quotient space on the one hand and elliptic modular functions onthe other are examples of these two aspects In several complex variablesthere is no analogue of uniformization theory; the class of automorphicfunctions which can be considered becomes much narrower, and theunderlying groups are, in general, arithmetically defined

In the mid-1930s C.L Siegel discovered a new type of automorphic formsand functions in connection with his famous investigations on the analytictheory of quadratic forms He denoted these functions as `modular func-tions of degree n'; nowadays they are called `Siegel modular functions'

Next to Abelian functions they are the most important example of

automorphic functions in several complex variables, and they very soonbecame a touchstone to test the efficiency of general methods in severalcomplex variables and other fields Only recently, Hilbert modular func-tions have achieved a similar position due to the progress made in thatarea by K Doi, F Hirzebruch, F.W Knoller, H Naganuma and D Zagier,amongst others Siegel himself developed many powerful methods, and asteadily growing group of mathematicians increased the knowledge ofSiegel modular forms, or found similar types of functions such as Hermitianmodular forms, Hilbert-Siegel modular forms, or recently modular forms

on half-spaces of quaternions The need for a unified comprehensive ment of these different but related theories became obvious This wasrealized within the framework of arithmetically defined subgroups ofalgebraic groups and corresponding automorphic functions by the im-pressive work of W.L Baily, A Borel, R.P Langlands and I.I Pjateckij-3apiro around 1965 Since then, further progress has been achieved only

treat-partly in this generality, but also in much more concrete situations,

for instance A.N Andrianov's results on Hecke's theory or the work of

E Freitag, D Mumford and Y.-S Tai on the structure of the function fields

This book consists of lectures on Siegel modular forms that I havedelivered in steadily improved versions, first in 1968 at the Tata Institute

of Fundamental Research in Bombay, and afterwards on several occasions

at the University of Freiburg The audience in mind was composed ofstudents who had taken only a one-complex-variable course besides havingsome basic knowledge in algebra, number theory and topology In parti-cular, in order to understand this book, the reader needs no knowledge ofseveral complex variables, except perhaps the concept of a holomorphic

or meromorphic function The lectures were designed for a one-semestercourse with the intention of offering an easily accessible partial survey

of the elementary parts of an exceptionally active field in mathematics.Consequently the selection of topics can by no means claim to be complete;even essential subjects like Satake's compactification or Hecke's theorycannot be included in this book We restrict ourselves to the full modulargroup neglecting technical difficulties arising from subgroups The reader,however, should feel encouraged to deal with the more advanced parts

of the theory afterwards, using other books or the original literature

recommended in the text

Formulas, theorems etc are numbered separately in each section If areference to a previous section is cited, the number of the section is placed

in front of the number of the formula, theorem etc For example, (4.1) means

formula (1) of §4

Acknowledgements are due to Siegfried BOcherer and Petra Ploch fortheir valuable comments and detailed reading of the manuscript I am muchindebted to Ruth Muller for her careful typing of the manuscript and herpatience Finally, I would like to take the opportunity to thank CambridgeUniversity Press for the invitation and encouragement to write this book

The modular group

1 The symplectic groupThe symplectic group over Il8 is a subgroup of the general linear group,defined by certain algebraic equations, and appears from there as analgebraic group

v = 1, 2, imply

j[mimi] =j[mi][m2] =j[m2] =jDecomposing m into n x n blocks,

m=(

c

a bl

d ,

the condition j[m] = j is seen to be equivalent to

'ac,'bd symmetric, 'ad -'cb = 1

For n = 1, symplecticity just means det m = 1 In general, det m = ± 1 can

be derived from the definitions for arbitrary m e Sp(n, R) In fact det m = 1

is true, but the converse obviously does not hold for n > 1

It is well known how Sp(1, R) acts on the upper half-plane as a group ofbiholomorphic mappings To generalize this subject to arbitrary n, Siegel'shalf-space of degree n will be introduced

Definition 2

Let n be a positive integer Siegel's half-space of degree n consists of all

n-rowed complex symmetric matrices z, the imaginary part of which is positivedefinite,

H"={z=x+iyl'z=z,y>0}.

Considering the independent entries z,,, (k::5:, 1) of z as coordinates, H.becomes an open subset of C"("+1'2 Since for positive matrices y,, y2 and real A,,

AY1 +(1 -2)Y2>0 (0<A < 1),

W turns out to be a convex domain in C"O+"12 As a convex domain, H.

in particular is simply connected Now we have to use the concept of

a holomorphic function in several complex variables for the first time.Following K Weierstral3 we may introduce holomorphic functions ascomplex-valued functions which can be represented locally by power seriesexpansions Holomorphic mappings between two domains embedded incomplex number spaces are described by holomorphic coordinate-functions.Then we may state

Now let be any n-rowed complex column such that qc = 0 0

by the equation above, hence = 0 since y is positive The system of linearequations qc = 0 has only the trivial solution; therefore q is non-singularand m<z> is well defined Next, the symmetry of z* = m<z> is equivalent to

Here and from now on we use the brackets { } to denote the transformation

as a Hermitian form, i.e a{b} :_ `bab So z* E H and the map in question

is of the indicated kind It is an action of Sp(n, R) on H,,, since one verifiesimmediately that

(mim2)(z> = mi<m2<z>>, e<z> = z

for ml, m2 e Sp(n, (li), z e H and the unit element e of Sp(n, IR) As with every

action of a group on a set, the corresponding mappings

for fixed m E Sp(n, 68) are bijective These `symplectic maps' are phic, since they are rational, and biholomorphic, as the inverse map isperformed with m-' Of course, we could as well have used the fact thatevery holomorphic bijective map of a domain onto itself is biholomorphic

holomor-If we assign to each m e Sp(n, l) the automorphism (2) of H,,, we obtain

a group homomorphism

Sp(n, R) -+

of the symplectic group into the group of biholomorphic automorphisms

of H The kernel of this homomorphism consists of ± 1, since the identity

surjec-realization of H,,, namely as a bounded symmetric domain in the sense of

E Cartan This concept was created in order to guarantee the existence ofenough non-trivial biholomorphic automorphisms of a domain Note that

in several complex variables there exist domains for which the identity isthe only biholomorphic automorphism Such a domain would be of no use

in the theory of automorphic functions A domain is called homogeneous

if the group of biholomorphic automorphisms acts transitively; it is called

symmetric if to each point there exists an involution in the group of

biholomorphic automorphisms with the given point as a single fixedpoint E Cartan [13] proved that each bounded symmetric domain ishomogeneous I.I Pjateckij-Sapiro [55] discovered the first example of

a non-symmetric but homogeneous bounded domain in 1959 Boundedsymmetric domains were classified by E Cartan [13]; the larger class ofbounded homogeneous domains was investigated by I.M Gelfand, S.G.Gindekin, I.I Pjateckij-9apiro and E.B Vinberg E Cartan obtained fourmain types of irreducible bounded symmetric domains and two exceptional

ones that appear only for dimensions 16 and 27, respectively Here

irreducible means that the domain cannot be decomposed into the product

of two domains of the same kind One of Cartan's main types is relevantfor our considerations, it is a generalization of the unit-circle to severalcomplex variables

1-WW =1-(z+il)-'(z-il)(z+il)(z-il)`'

_ ((z + il)(z - il) - (z - il)(z + il)){(a - il)-'}

=4y{(z-il)-'}>0.

Thus the Cayley transformation maps H into D On the other hand, l - w

is non-singular for arbitrary w e D, for (1 - w)1; = 0 implies (1 - ww) {l; } _

Sp(n, R), respectively (D,,, act transitively on H,,, respectively D Both domains

are symmetric in the sense of E Cartan

Proof

Because of Proposition 2 it is sufficient to consider the action of J on D,,.First we show its transitivity Let w be an arbitrary point of D,,; we have tolook for an element m e dD which transforms w into any distinguishedpoint, for instance the origin Since 1 - ww is positive, there exists an n x nmatrix a with complex entries such that

which is equivalent to k{m} = k Therefore m e 4) by (4) and obviously

w = m <0> Concerning the symmetry, note that w H - w is an involution

of D which corresponds to an element of D and has the origin as a singlefixed point Since we already know that fi is transitive we obtain the sameproperty for any other point of D

On this occasion we prove a useful lemma, which allows us to strengthenProposition 3 as a corollary

The argument becomes a little involved since transformations as quadraticforms (denoted by [ ]) and as Hermitian forms (denoted by { }) appear atthe same time If w is a diagonal matrix with elements w1, , win themain diagonal, we may assume

A = ww,, (v = 1, ,n)

after reordering, which is an orthogonal transformation Then we take for

u the diagonal matrix formed with the elements

(w ) (v = 1, , n)

or 1, if w,, vanishes For arbitrary w first determine a unitary matrix u1such that ww{u1} = d2 Then q = w[u1] is symmetric and qq = d2 real

Therefore the real and the imaginary part of q commute and can be

transformed into a diagonal form simultaneously by an orthogonal matrixu2 Then q[u2] = w[u1u2] becomes diagonal too and u1u2 is unitary So

we have transformed w as a quadratic form into a diagonal matrix by theunitary matrix u = u1u2 Since such a transformation does not affect theeigenvalues of ww, we are back to the first case

Corollary

Let w1, w2 be two arbitrary points in D Then there exists an m e (D whichsimultaneously transforms w1 into 0 and w2 into a diagonal matrix t, thediagonal elements of which satisfy 0 < t 1 < < t < 1

By Proposition 3 we may assume w1 = 0 Take for t1, , t the squareroots of the eigenvalues of w2 w2 Then the lemma yields a unitary matrix

u such that w2 = t [u]; the mapping w i - w [u-'] is induced by the action

of d) and has the desired properties

The idea of considering the generalized unit-circle instead of Siegel'shalf-space has two advantages First, the boundedness is often good fortopological considerations, and second the action of (D may be extendedholomorphically onto the closure D of the unit-circle

Proposition 4

The mappings

w I-+ m <w> = (aw + b) (bw + a)-1 (m a

are topological automorphisms of D They are holomorphic on a

neighbor-hood of D (depending on m)

It is sufficient to verify

det(bw + a) # 0for all m e (D,,, w E b, Now let be a complex column satisfying

In the remaining part of this section we will mention some basic facts

of symplectic geometry which are used later The well-known model ofhyperbolic geometry in the upper half-plane can be generalized to Siegel's

half-space H by introducing a certain invariant Riemannian metric.

The geometrical properties were investigated originally by L.K Hua,

C.L Siegel and M Sugawara, and more recently by S Helgason,

A Koranyi, 0 Loos, S Murakami and J Wolf, amongst others tiating the symplectic mapping

From these two formulas we deduce immediately

d27y-1'(cz + d).

Hence the trace a(dzy-' diy-') defines a quadratic differential form indxk,, dy, (k < 1) over R, which is invariant with respect to the action ofSp(n, IR) Since this action is transitive by Proposition 3, we may check thepositivity of the differential form at a single point, for instance z = il.But there

a(dz dz) _ (dxkk + dyk) + 2 E (dxk, + dykl),

In Riemannian geometry the invariant volume element is defined as theEuclidean volume element multiplied by the square root of the determinant

of the quadratic differential form ds2 Since Sp(n, F) acts transitively on H",this volume element is uniquely determined by its invariance up to aconstant factor Therefore it is not necessary to compute the determinant

of ds2 explicitly if any invariant volume element is available by anotherargument Then only the determination of an inessential factor remainsopen To simplify the computation of Jacobians, note once and for all thatthe linear map

wHw[C]

from the space of n-rowed symmetric matrices w into itself has determinantdet c"'From this observation all the Jacobians which appear in this book

can be read off immediately So we obtain from (5)

detl7OZ = det(cz + d)-"-1for the Jacobian of any symplectic map; or, introducing real coordinates,

det y = Idet(cz + d)I" .

So, if dxdy = flk<, dxk, dyk, denotes the Euclidean volume element,then

det(1 - ww)for the volume element, after being carried over to the generalized unit-circle D by Cayley's transformation

2 Minkowski's reduction theory

The imaginary parts of the points z e H form the subspace

Pn={YIY>0}

of R"("+1)/1 consisting of all n-rowed positive definite matrices with realentries It is an open convex subset of 18"("+1)12 Moreover the ray originatingfrom the point 0 and passing through any point y e P lies completely in

P Therefore P is a convex cone with vertex at the origin Consider onthe other hand the subgroup

{maSp(n,R)Im=(a a01 ), aeGL(n,R)}

of Sp(n, U8), which is canonically isomorphic to GL(n, R) Then the action

of Sp(n, R) on H in Proposition 1.1 induces the action

GL(n, R) x P -+ (a, y) H Y[`a]

of GL(n, U8) on P,, Because of this connection we prefer to let GL(n, R)operate on P from the left If one assigns to a e GL(n, R) the map

Y[`a],

one obtains a group homomorphism of GL(n, O) into the group ofbijective maps of P,, The kernel is determined easily Inserting y = I intothe identity

By the considerations above we have found a close connection between

the action of Sp(n, R) on H and the action of GL(n, l) on P,, The

same situation holds for arithmetically defined subgroups of Sp(n, R) andGL(n, R), respectively For instance, we could have taken Sp(n, 7L) andGL(n,7L) Reduction theory is concerned with the action of GL(n, Z) on Pand therefore is expected to be a powerful instrument in the handling ofthe group Sp(n, 7L), which is Siegel's modular group However, having thusfound a good motivation for the subsequent investigations, one shouldpoint out that reduction theory was not invented for this auxiliary purpose.There are more important applications to the theory of quadratic formsand to geometry of numbers

Henceforth we call GL(n, Z) the unimodular group of degree n anddenote it by U It consists of all n-rowed matrices u with entries in Z anddet u = ± 1, called unimodular matrices The group operation is of coursematrix multiplication Our main objective will be the determination offundamental sets for the group action

of a particularly simple form The main results on reduction theory are due

to J.L Lagrange for n = 2, L.A Seeber for n = 3, Ch Hermite and

H Minkowski [52] for general n Other approaches to reduction theorywere discovered by H Weyl and OF Voronoi

Roughly speaking, a fundamental set is an irreducible complete set ofrepresentatives for the orbits of the underlying group action In our casethe group action is described by (1), and two points y, y* e P belong to thesame orbit if and only if there exists a u e U such that y* = y[u] The orbitsare nothing other than the equivalence classes of this equivalence relation

on P We try to determine a representative of each orbit by certain

minimization conditions For any point y in P first determine the integral

column u1 0 0 such that y[u1] becomes minimal Since y is positive, thisminimum is attained for finitely many ul which are primitive; fix one ofthose After having already determined u1, , uk-1, let Uk run over allintegral columns such that the n x k matrix

where 1 stands for the identity matrix of size k - 1, a is an integral and b

an (n - k + 1) -rowed unimodular matrix Therefore we obtain

R _ {y e P I y satisfies (i) and (ii)},

for all integral g with (gk, , 1 (1 < k < n),(1_<k<_n-1).

The conditions (i) and (ii) are called reduction conditions and the elements

of R reduced in the sense of Minkowski

In the following we call integral columns g with (gk, , 1

`k-admissible' for short Note that so far R has been described as a subset

of P by infinitely many linear inequalities Another useful remark isconcerned with the following fact If y e R and y* is obtained from y bydeleting the last n - k rows and columns, then y* a Rk (k = 1, , n) Thiscan immediately be seen from the reduction conditions

We have proved that P is covered by the images of R under the action

of Un, i.e.

P, = U u(Rj).

UEU

It is not difficult to deduce from the reduction conditions that each point

of P,, is covered by at most finitely many images of Rn; more precisely, thenumber

is shown to be finite for any fixed y e P Indeed, we infer from y[u] e R

by (i) that

Yi, = Y[uu-lei] ? Y[u1],

where e,, (v = 1, , n) denotes the with unit vector and u1 is the first column

of u Hence there are only finitely many possible values for u 1 Now assumey[u], y[v] e R and let u and v coincide with their first k 1 columns Thenafter replacing y[u] by y we have

y[v] a Rn, v =

with the (k - 1)-rowed identity matrix 1 The kth column of v-' is

k-admissible and we obtain from (i)

Ykk = Y[vv-'ek] Y[vk]

Therefore we have only finitely many choices for the kth column vk of v.After n steps we obtain (2) But we cannot see from our argument whetherthe bound in (2) may be chosen independently of y This is an importantquestion, which we will follow up later

Next we deduce some consequences of Minkowski's reduction

First we should mention that we have suppressed the double index of the

diagonal elements of y, writing yk instead of ykk for simplicity The essential

part of the proposition is the right-hand inequality in (iii), which is calledMinkowski's inequality We may describe the significance of the proposi-tion by the following remark The finitely many inequalities (i)-(iii) of theproposition are implications of the infinitely many reduction conditions inDefinition 1 On the other hand, we will see later that we do not lose theessential properties of R if we replace Minkowski's reduction conditions

by the inequalities of the proposition Against this background the tion to finiteness is the important point But note that linearity gets lostwith Minkowski's inequality

transi-Concerning the proof, (i), (ii) and the left-hand inequality in (iii) are trivial

Indeed, (i) follows from the reduction conditions of the first kind for index

k and the k-admissible column g = ek+l; then we take the 1-admissiblecolumns g = ek ± el in Minkowski's reduction conditions of index l toobtain

y[g] = Yk + Yi ± 2Yk1 ? Yi,

which is exactly (ii) The left-hand inequality in (iii) has a simple and known geometrical interpretation Write the positive matrix y as

To prepare for the proof of Minkowski's inequality we first considerHermite's lemma on the minima of positive definite quadratic forms Theminima in question are

µ(y) = min y[g],

e

where g runs over all integral columns different from zero; y is n-rowed andpositive definite For positive real numbers t we have U(ty) = tp(y) anddet(ty) = t" det y Therefore it seems reasonable to compare p(y) with

det yu"

Lemma 1 (Ch Hermite)

There exists a constant c2 = c2(n) depending only on n such that

,u(Y) <_ c2 det yii"

for all real n-rowed positive definite matrices y A possible choice is

affects neither µ(y) nor dety Therefore we may assume that y[g] becomes minimal for g = e1, so p(y)=y, Now use the elementary

identity

Y = \`9 q/

=(0 p 0)[(0

p 1 q)]' t = s - p-1 [q], (3)which is valid for arbitrary matrices decomposed into blocks of appropriatesize, p being non-singular Of course we observe behind this formula thewell-known method of `completion of squares' Apply (3) to y decomposed

Now we prove Minkowski's inequality by induction on n, where the case

n = 1 is obvious Let us consider any y e R and assume the assertion to

be true for smaller values of n

Case (i): The quotients of the diagonal elements satisfy

Yv+1

Yv

< yn (v = 1, ,n - 1)with some constant yn depending only on n to be specified later Then wehave

P[91 + P-'q92] < sk(k + 1)YkCombining the last three inequalities we get

Now we contrast this estimate with the first inequality (5) to concludethat

J < c2(n - k) det tl1("-k).

Yk+1 C 1 -k(k+1\

Then specify yn := n(n - 1)/4, for instance, to obtain the crucial upperbound

From the second group of inequalities (5) and (6) we obtain

Yk+l yn ynn-k)(n-k-1)/2Yk+1n-k

(i) Yk < tYk+1 (k = 1, , n - 1),

(iii) fjv_1 y,, < c1(n)tdety

Here and henceforth c1(n) denotes any fixed choice of the constant in

Minkowski's inequality

These are open subsets of P which exhaust P for t tending to infinity; i.e.Qn(t) c Qn(t') for t < t', and any compact subset of P is contained inQn(t) for sufficiently large t We have R c Qn(t) for arbitrary t > 1 byProposition 1

Another kind of auxiliary regions that we need is defined by the Jacobiandecompositions of positive matrices It is well known from linear algebra,

or can be derived immediately from (3), that any positive real matrix can

where d is a diagonal matrix and v is an upper triangular matrix with ones

in the main diagonal We call the d, (v = 1, ,n) and the elements vk, (k < 1)

of v `Jacobian coordinates of y'

Definition 3

For any positive integer n and positive real number t let

Q' (t) = {y e P I y = d[v] satisfies (i), (ii)},

where

(i) dk < tdk+i (k = 1, ,n - 1),

(11) I Vkl I < t (k < 1)

These again are open subsets of P which exhaust P as t - oo Note that

is characterized as a subset of P by finitely many linear inequalities

in the Jacobian coordinates

The regions Q (t) and Q;, (t) are equivalent in the following sense.Proposition 2

Let n be a positive integer

(i) For any real number t > 0 there exists a t, = t, (n, t) depending only on

n and t such that Q.(t) c Q;, (t,);

(ii) for any real number s > 0 there exists an s, = s, (n, s) depending only

on n and s such that Q'(s) c Qn(s, )

where the constant t2 depends only on n and t If on the other handy e Q;,(s)

we can deduce a similar result directly from (7) and Definition 3 Hence inboth cases y,, and dv are of the same order of magnitude

We prove statement (i) by induction on n, where n = 1 is obvious Assumethe assertion to be true for n - 1 and consider any y e Qn(t) In the Jacobiandecomposition y = d[v] we set

dety* < cl(n)Y tdety* =cl(n)ty <_ cl(n)t.

By the induction hypothesis applied to y* e Qn_1(t3) there exists a t4depending only on n and t such that

dk<t4dk+l (k=1, ,n-2), vk,I<t4 (1<k<1<n-1).Since y and d have the same order of magnitude and ty we may

add dn_1 < t4d So only the boundedness of co is left We obtain from (8)

n

Now v*-' is bounded as the inverse of a bounded triangular matrix The

boundedness of d*"1n follows from condition (ii) in Definition 2 and thefact that the yv and the d, have the same order of magnitude All the boundsdepend only on n and t.

The proof of the second statement is straightforward Let y be any element

of Q;,(s) Conditions (i) and (iii) in Definition 2 are satisfied for y as they areimmediate consequences of y belonging to Q;,(s)and the boundedness ofyt,/dv (v = 1, ,n) from above and below by positive constants Finallyfrom the Jacobian decomposition y = d [v] we obtain for the non-diagonalelements of y

the diagonal matrix made up of the diagonal elements y, , y of the

matrix y This is called the diagonalization of y

Since R c for t > 1 and c Q;,(t') for appropriate t' by

Proposi-tion 2, it is sufficient to consider Q'(t) Let y = d[v] be the Jacobian

decomposition of any y e and

x = d'12vd"1'2.

Then x is bounded by the reduction conditions in Definition 3 Since x is

a triangular matrix, x-1 is bounded too Hence

y =txx[dl/2]

has the same order of magnitude as d, i.e there exists a positive number

y1 = y1 (n, t) such that

Y1 1d<y<y1d

On the other hand, we have already seen in the proof of Proposition 2 that

d is of the same order of magnitude as yD,

721d<YD<y2d

Hence the lemma holds for y = Y1 Y2

To summarize our results, we have introduced three regions R,,, Q (t)and Q'(t) The last two of them are each contained in the other as inProposition 2 R is contained in Q (t), respectively for sufficientlylarge t The formal description as a subset of P varies in quality: thereare infinitely many linear inequalities for R, finitely many algebraicinequalities for Qn(t), finitely many linear inequalities in the Jacobiancoordinates for Q'(t) We know that the images of each of those threeregions (t sufficiently large) under the action of U cover all of P,, We nowturn to the question of whether there are only finitely many overlappings

It is because of this problem that we have introduced the auxiliary regionsand Moreover we will be able to analyze the geometricalstructure of R on this occasion We now state the most important theorem

This corollary is an immediate consequence of the theorem and the fact

that R and may be enclosed in Q;,(t') for appropriate t' We define afundamental set for a group action as a subset of the representation spacesuch that the images of the subset cover the representation space and thefiniteness condition of the corollary is satisfied In our case R,,, andQ' (t) for sufficiently large t are fundamental sets for the unimodular group

Any fundamental set contains at least one and at most finitely manyrepresentatives of each orbit The number of representatives is boundedindependently of the individual orbit Finally we use the fact that eachcompact subset of P can be enclosed in Q;,(t) for an appropriate choice of

t So we may state

Corollary (ii)

R,,, and Q;,(t), for sufficiently large t, are fundamental sets for the group

action of the unimodular group U on the space P of positive definitequadratic forms Each compact subset of P is covered by at most finitelymany images of these fundamental sets

Proof

The theorem will be proved by induction on n, where again the case n = 1

is obvious We distinguish two cases for n > 1

Case (i): Assume the existence of a number k, 1 < k < n, such that g splitsinto

diavi91v*1 -'di -'/2 orthogonal d1[vig1] = di [vi],

l 1*_'n orthogonal d2[v292] = d*[v2*],d'12v2gzv*2 2

Case (ii): To each k, l < k < n, there exists an r > k and an s < k such thatg,s 0 0 Since y, y* e the matrices y, y* have the same order of

magnitude as d, d*, respectively; consequently we have

The sequences d,, ,d and d, , are monotonically increasing up

to the factor t because y, y* belong to Q;,(t) Therefore we obtain

dk+1 < t2dk (k = 1, ,n - 1)with an appropriate constant t2 = t2(n,t) Now apply the same argument

to the equation

detg2y = y*[detgg-1]

Note that det g g-1 is again integral, with determinant less than or equal

to t"-' and of the relevant form forcase (ii) Hence we have

y[g] >- yk for all k-admissible g ±ek (k = 1, ,n), (10')

> 0 (k = 1, ,n - 1), (10")

.yk,k+1-called proper reduction conditions Thus R as a subset of P appears as

the intersection of countably many half-spaces, defined by the individuallinear inequalities (10) First we show that there exist points y e R suchthat all these inequalities hold with the > sign, henceforth called strictinequalities Indeed consider the countably many hyperplanes defined by(10) with the sign of equality and their countably many images under U.Choose a point y e P which does not lie on any of those hyperplanes.Determine u e U such that y [u] = y* is reduced in the sense of Minkowski.Then y* fulfils our requirements According to the next proposition such

a point is an interior point of R

We consider R as a subset of P from a topological point of view.

Obviously R is closed in P,,

Proposition 3

The interior and the boundary of Minkowski's reduced domain R in P are

the following subsets:

A,, = {y e P I y satisfies the strict inequalities (10)},

8R = {y e P I y satisfies (10), at least once with the sign of equality}.Proof

It is sufficient to show that the sets on the right are contained in A,,, OR.,respectively Let y e P fulfil the strict inequalities (10) If 2 is the lowesteigenvalue of y, choose 0 < s < 2 and a neighborhood V of y such that

y*-y> -E1, IYk -YkI <r, Ykk+i >0

for all y* e V Then

Y*[9] - Yk = Y[9] + (Y* - Y)[9] + Yk - Yk - Yk

>(2-e)`99-C-Yk.

The expression on the right does not depend on y* and is positive for allbut finitely many integral columns g Hence the inequalities (10), with theexception of at most finitely many k-admissible g, hold for arbitrary y* e V.Diminishing V afterwards, the exceptions may be included such that V iscontained in R Now let y e P satisfy (10) and at least one particularinequality with the sign of equality Since we have omitted the identitiesamong the reduction conditions, there are points of P arbitrarily close to

y for which this particular inequality fails Hence y is a boundary point.Next we are able to deduce a better understanding of the covering of P

by R and its images under the unimodular group Since ±u induce thesame automorphism of P,,, let us consider more precisely the action of

1} on P The next proposition states that overlappings may occuronly on the boundary.

Proposition 4

Let y, y* e y* = y[u] and ±1 u e U Then y and y* belong to theboundary of R

Proof

If u is a diagonal matrix we may assume for the columns ui = ei, ,

Uk = ek, but uk+i = - ek+1 From y* = y[u] we obtain

Yk,k+1 = -Yk,k+1

On the other hand, both quantities are non-negative by the reductionconditions Hence,

Yk k+i = Yk,k+l = 0

and y, y* e OR,, by Proposition 3 If u is non-diagonal, let ui = ±e1, ,,

uk_i = ±ek_l but uk ±ek Then Uk is k-admissible, and by Minkowski'sreduction conditions

Yk = Y[uk] ?

Yk-Interchanging y and y* and replacing u by u-1 we obtain yk >_ yk.Hence,

and y e OR,,, likewise y* E OR., by Proposition 3

F

Finally we are able to analyze the geometrical structure of R completely.Consider the set

{uc-It is a finite set by Corollary (i) of Theorem 1 The rows of the matrices in

V form another finite set, denoted by

Let y be a boundary point of R,, By Proposition 3, at least one of

the infinitely many reduction conditions (10) is fulfilled with the sign ofequality, for instance

y [g] = yk, g k-admissible, g ± ek.

From the construction of reduced forms by minimizing conditions at thebeginning of this section we immediately deduce that

(±el, , ±ek-1,g)

is complementary to a unimodular matrix u, such that y[u] a R,, Hence,

`g belongs to Note that the conditions (10") appear again as (11") So

we may state that for any y e OR at least one of the finitely many conditions(11) holds with the sign of equality In order to prove the propositionassume there is a point y e P satisfying (11) and not belonging to R Connect y with an interior point y* of R by the line segment

.1y+(l -A)y* (0<A.<1).

Then, on the one hand, the strict inequalities (11) hold for each point ofthis segment, since (11) are satisfied for y* in the strict sense and for

y in the usual sense Furthermore, the reduction conditions are linearhomogeneous inequalities On the other hand, there must be a boundarypoint of R on this line segment, and at that point one of the conditions(11) holds with the sign of equality as we have already seen This contradic-tion finishes the proof

So we have finally proved that the infinitely many original reductionconditions of Minkowski are consequences of a finite subset Summarizingour results we state the following

Among many important applications of Minkowski's reduction theory wemention only one, which is concerned with group-theoretical consequences

It was proved by M Gerstenhaber [25] and H Behr [6] under conditions

of wide generality that one may deduce a finite presentation of the ing group from the properties in the theorem Generators are the transforms

underly-which map R into its neighbors, called local generators; defining relationsare the local relations - the possible relations of the form u1 u2 = u3between the local generators For generalizations of Minkowski's reductiontheory see A Borel [11].

3 Fundamental sets of Siegel's modular group

In this section we return to the action of the symplectic group Sp(n, 01) onSiegel's half-space H as in Proposition 1.1 If G is any subgroup of Sp(n, R),

we denote by G the corresponding group of induced automorphisms

z H in <z) (m c- G)

of H The group G acts discontinuously on H - or, synonymously, 0 isdiscontinuous - if the family

{m<z)Ime G}

has no accumulation point in H,,, no matter what z e H may be Since H

is the union of countably many compact subsets, each discontinuous group

0 is enumerable On the other hand, G as a subgroup of the topologicalgroup Sp(n, R) is called discrete if there exists a neighborhood U of the unitelement I such that no other element of G is contained in U Obviouslythis is equivalent to the fact that no sequence (gv) of mutually distinctelements of G converges in G (or in the space of all 2n x 2n real matrices)

In our case we have

z* = mv<z) (v = 1,2, )converges to z* in H For the imaginary parts we obtain

y*-1 = y-1 {'(c,,z + d,,)}

by (1.6) Since y* -1 is bounded with respect to v and y-1 independent of v,

we can infer the boundedness of c,z + d,, Decomposing these matrices intotheir real and imaginary parts, c,, and d,, turn out to be bounded as well

Apply the same argument to

asimilar result for a,, and b,, Hence the sequence (m,,) is boundedand consequently there exists a convergent subsequence, proving G to benon-discrete

The most important example of a discrete subgroup of Sp(n, R) is

I' = Sp(n, 7L),

called Siegel's modular group We consider the action of F on H induced

by the former action of the symplectic group This section is devoted to thedetermination of different fundamental sets of F which are useful instudying the corresponding automorphic forms and functions For n = 1

we obtain the well-known elliptic modular group; keeping this specialcase in mind, the reader will certainly note that we are not consideringsubgroups, in particular congruence subgroups of the modular group.Indeed we shall restrict ourselves to the full modular group in order to keepthe whole theory as transparent as possible; subgroups will be mentionedonly sporadically

To construct fundamental sets for Siegel's modular group we introducethree subgroups of I',,:

{(o 1U0 i) 1 uunimodular}

{(a

Is symmetric, integral entries} (2)

u unimodular; s symmetric, integral entries (3)

tu-1 ) I

The first one is canonically isomorphic to the unimodular group; the secondone is the group of translations, a free Abelian group of rank n(n + 1)/2;the third consists of all elements of F for which the lower left blockvanishes, sometimes called integral modular substitutions

First consider the `second rows' (c, d) of modular matrices (a b). Ofcourse these lower blocks are matrices of n rows and 2n columns Looking

at the left cosets of r modulo the first subgroup mentioned above, werealize that one may multiply any second row (c, d) by an arbitrary u e U.from the left, getting another second row of a modular substitution Twosecond rows are called associated if they differ only by a factor u e U on

the left

Multiply c and d by an appropriate u e U such that y*-1 becomes reduced

in the sense of Minkowski Denote the diagonal elements of y*-1 by

y1, ,y, and the rows of c and d by c1, , c,,, respectively d1, , Then

we obtain

y* = y-1[`(c,x + d,)] + y[`ci] (I= 1, ,n). (4)

Since y > 0 and c,, d, are integral and not both zero, the y* are boundedfrom below by a positive constant independent of (c, d) On the other hand,

we have

det y*-1 < 12 det y_1,

and we may replace y*-1 in this inequality up to a constant by its

diagonalization because of Lemma 2.2 Hence the product of all the y* isbounded from above So finally we obtain that each y* is bounded fromabove and below by positive constants independent of (c, d) Then, lookingonce again at (4), only finitely many choices for c and d are left

Analogous to the one-variable case, we call det y the height' of the point

z in H From the transformation formula (1.6) for the imaginary parts andthe lemma above we realize that each F,,-orbit contains points of maximalheight Such points are characterized by the conditions

Idet(cz + d)I _> 1

for all m e r Furthermore, the height remains unchanged if one applies

an integral modular substitution So we can dispose of u and s in the

subgroups (1) and (2) such that y becomes reduced in the sense of

Minkowski and the real part x reduced modulo 1 Hence each r.-orbitcontains points of the following set F

Definition 1

Let n be any positive integer Siegel's fundamental domain is the subset

F = {z E H I z satisfies (i)-(iii)}

had already been introduced by C.L Siegel in his early papers [62]

by its diagonalization in accordance with Lemma 2.2 Furthermore thediagonal elements are increasing by Proposition 2.1 So it is sufficient toprove that the first diagonal element yl of y is bounded from below by apositive constant But this can be deduced immediately from condition (i)(for m = a

c

db)

,where

These decompositions are of type (1, n - 1) Condition (i) says that Iz1 I >_ 1,

which together with Ix11 <

i implies that yl >- /2.