On Siegel`s Modular Functions - School of Mathematics, TIFR

Transcription

Lectures on
Siegel’s Modular Functions
by
H. Maass
Tata Institute of Fundamental Research, Bombay
1954–55
Lectures on
by
Hans Maass
Notes by
T.P. Srinivasan
Tata Institute of Fundamental Research
Bombay
1954–55
Contents
1 The Modular Group of Degree n
3
2 The Symplectic group of degree n ....
19
3 Reduction Theory of Positive Definite Quadratic Forms
33
4 The Fundamental Domain of the Modular Group of Degree n 47
5 Modular Forms of Degree n
61
6 Algebraic dependence of modular forms
75
7 The symplectic metric
89
8 Lemmas concerning special integrals
101
9 The Poincare’ Series
109
10 The metrization of Modular forms of degree
125
11 The representation theorem
135
12 The field of modular Functions
149
13 Definite quadratic forms and Eisenstein series
169
14 Indefinite Quadratic forms and modular forms
189
iii
iv
Contents
15 Modular Forms of degree n and differential equations
197
16 Closed differential forms
217
17 Differential equations concerning ....
231
18 The Dirichlet series corresponding ....
259
The theory of modular functions of degree n is not quite new. The 1
conception of this theory is due to Siegel who gave a first introduction
of this in 1939 (Einfuhrung in die Theorie der Modulfunktionen n-ten
Grades, Math. Ann., Vol 116(1939)). Since then, numerous contributions have been made by various authors. My desire to give a course on
this special branch of mathematics arose out of more than one consideration. Siegel’s modular functions constitute one of the most important
classes of analytic functions of several variables, and we are able to look
forward to far reaching results out of this class. The modular functions
of degree n are connected with the manifolds of the closed Riemann surfaces of genus n in just the same way as the elliptic modular functions
are, with the manifolds of Riemann surfaces of genus 1. Moreover we
realise the excellent use of modular forms of degree n in the analytic theory of quadratic forms, first positive quadratic forms and then, indefinite
forms too. But in trying to extend this theory to the case of indefinite
quadratic forms, one meets with new types of functions defined by partial differential equations and one is led to a variety of unsolved problems in this direction. Finally, the researches of Siegel have becomes
representative of those on a general class of automorphic functions of
several variables.
Chapter 1
The Modular Group of
Degree n
Let f be a closed Riemann surface of genus n; dωi , i = 1, 2, . . . n, a
basis of the Abelian differentials of the first kind on f, and ψν , ψ′ν ,
ν = 1, 2 . . . n, a canonical system of curves which dissect f into a simply connected surface bounded by one closed curve. We ask for the
transformation properties of the periods
Z
Z
dων , pµν = − dων (µ, ν = 1, 2 . . . n)
qµν =
ψ′µ
ψµ
concerning a replacement of dων (ν = 1, 2 . . . n) by another basis dω∗ν 2
(ν = 1, 2 . . . n), and ψν , ψ′ν respectively by another canonical system of
curves ψ∗ν , ψ∗ν (ν = 1, 2 . . . n). We introduce the period matrices
P = (pµν ), Q = (qµν ), (µ, ν = 1, 2 . . . n),
and let P∗ , Q∗ denote the corresponding matrices in the changed system. Let Lµ , Lν′ be arbitrary oriented closed curves on f and Cµ , Cν′ be
arbitrary complex numbers. A homology
r
X
µ=1
Cµ Lµ ∼
3
s
X
γ=1
Cν′ Ln′ u
1. The Modular Group of Degree n
4
means that for every integrable function F, the equation
r
X
Cµ F(Lµ ) =
µ=1
r
X
Cν′ F(Lν′ )
holds.
ν=1
For every closed curve L , there exists, as is well known, a representation
n
n
X
X
L ∼
Cµ ψµ +
Cν′ ψ′ν
µ=1
ν=1
with uniquely determined integers Cµ , Cν′ (µ, ν = 1, 2 . . . , n). That is
to say that the curves of a canonical system represent a basis of the
homology classes of all closed curves. Thus we have in particular
X

ψ∗i ∼
(aiµ ψµ − biµ ψ′µ ) 





µ

X
(1)

∗′
′ 

ψi ∼
(−Ciµ ψµ + diµ ψµ )



µ
3
with integers aµν , bµν , Cµν , dµν (µ, ν = 1, 2 . . . n). Since the change from
the homology basis ψν , ψ′ν , to the basis ψ∗ν , ψ∗ ν′ will always be effected
by a matrix
which is unimodular, the determinant of the matrix M =
!
A B
with A = (aµν ), B = (bµν ), C = (Cµν ), D = (dµν ), denoted by
C D
|M|, is equal to ±1, i.e. |M| = ±1.
Furthermore,
dω∗µ =
n
X
dων tνµ (µ = 1, 2 . . . n)
ν=1
with a non-singular matrix T = (tµν )
Now we express the periods
Z
Z
∗
∗
∗
qiR = , dωR , PiR = − dω∗R
′
ψ∗i
ψ∗i
5
in terms of Pµν , qµν as follows.
Z
Z
X
p∗iR = − (aiµ dω∗R − biµ dω∗R )
µ
=−
=
X
(aiµ tµR
µν
X
ψ′µ
ψµ
Z
dων − biµ tνR
ψµ
Z
dων )
ψ′µ
(aiµ ψµν tνR + biµ qµν tνR ,
µν
q∗iR
=
X
µ
=
dω∗µ
+ diµ
Z
dω∗R )
ψ′µ
ψµ
Z
X
(−Ciµ tνR
X
(Ciµ pµν tνR + diµ qµν tνR )
µν
=
(−Ciµ
Z
dων + diµ tνR
ψµ
Z
dων )
ψ′µ
Rewritten in terms of the matrices, these relations give
P∗ = (p∗µν ) = (AP + BQ)T, Q∗ = (q∗µν ) = (CP + DQ)T.
As is well known,R a differential of the first kind is uniquely determined by its periods dω(µ = 1, 2 . . . n) so that |Q| , 0. The choice
ψ′µ
Q−1
T =
is therefore permissible and this leads to the relations P∗ =
−1
APQ + B = AZ + B, where Z = PQ−1 , and Q∗ = CPQ−1 + D = CZ + n.
If we change only the basis of the differentials and keep the canonical basis unchanged, the above relations lead to the normalized period 4
matrices P∗ = PQ−1 = Z; Q∗ = E (n,n) , the unit square matrix of order n,
as in this case we have
!
!
E (n,n)
o
A B
= E (2n,2n) .
=
M=
o
E (n,n)
C D
We denote by X, Y, the real and imaginary parts of the matrix Z and
by Z ′ the transpose of Z. From the theory of algebraic functions it is
6
known that Z is a symmetric matrix with a positive imaginary part (i.e.
the quadratic form Y ′ YY , for any real vector Y is always positive).
In symbols, these mean
Z = Z’ ; Y, 0.
(2)
It is obvious that in the general case (M arbitrary, unimodular; T
arbitrary, non-singular) we have
−1
Z ∗ = P∗ Q∗ = (AP + BQ)(CP + DQ)−1 = (AZ + B)(CZ + D)−1
and Z ∗ also satisfies the same properties as Z, viz. it is symmetric with
a positive imaginary part.
In the sequel we shall denote by Y the set of all symmetric matrices
with a positive imaginary part.
In order to obtain
the typical relations for the coefficients of the ma!
A B
trix M =
we consider the intersection properties of the canoniC D
′
cal system ψν , ψν (ν = 1, 2 . . . n), described by means of the notion of the
characteristic (Kronecker). For every pair of closed curves L1 , L2 on
f, there exists a number S (L1 , L2 ) called the characteristic of L1 with
regard to L2 , which satisfies the following conditions.
1) S (L1 , L2 ) is an integer depending only on the homology classes of
L1 and L2 .
5
We may now define S (L1 + L2 , L3 ) as S (L , L3 ) where L ∼
L1 + L2 .
2) S (L1 , L2 ) = −S (L2 , L1 ). This implies the additivity of S , viz.
S (L1 + L2 , L ) = S (L1 , L ) + S (L2 , L )
(3)
3) For every canonical basis ψν , ψν (ν = 1, 2 . . . n) of the homology
classes we have
S (ψµ , ψν ) = S (ψ′µ , ψ′ν ) = 0
S (ψµ , ψ′ν ) = δµν
the δ′µν S are the Kronceker δ′ S .
where
(4)
7
4) If L1 crosses L2 , r times from the right side to the left, and l times
from the left to the right, then
S (L1 , L2 ) = r − ℓ
(5)
as far as r and ℓ are computable.
As we only intend to sketch the way which leads to the typical relations for M, we do not prove here the existence of a characteristic
(cf:H.Weyl, Die Idee der Riemanns chen Flanche).
From the above properties it is clear that the characteristic defines
a bilinear form on the class of all closed curves and that the matrix
associated with this! bilinear form with respect to a canonical basis is
o
E (n,n)
.
I=
−E (n,n)
o
If we transform this basis by means of the matrix M, the matrix
associated with the above bilinear form with respect to the transformed
basis is clearly MI M ′. But M is so chosen that the transformed basis is
again canonical so that the matrix associated with the bilinear form with
respect to the transformed basis should again be I. Hence we conclude
that
MI M ′ = I
(6)
This clearly characterises M, since the matrix associated with the 6
above bilinear form with respect to a certain basis can be equal to I if
and only if that basis is canonical. As a consequence of (6) we obtain
the desire characteristic relations as
AD′ − BC ′ = E; AB′ − BA′ = 0; CD′ − DC ′ = 0
(7)
!
A B
A matrix M =
is said to be symplectic if it satisfies the
C D
relation (6) or equivalently the relation (7). We denote by S = S n the
class of all symplectic matrices. In view of the relations
I −1 = −I
M −1 = −I M ′ I
(8)
8
it is easily seen that S is a group, called the symplectic group of degree
n. The integral matrices M ∈ S form a sub-group, called the Modular
Group of degree n. We denote the modular group by M = Mn . We may
note that if one replaces M by M ′ in (7) one obtains
A′ D − C ′ B = E; A′C − C ′ A = 0; B′ D − D′ B = 0
(9)
These relations are equivalent to (7). For, from (8) we have
M ′ = I −1 M −1 I −1
= I −1 M −1 I
(8)′
Clearly I ∈ S . Since S is a group, the relation (8)′ signifies that
M ∈ S if and only if M ′ ∈ S which is precisely the content of our claim
that the relations (7) and (9) are equivalent. In view of (8) we also obtain
!
D′ −B′
−1
(10)
M =
−C ′ A′
7
With a view to fixing our ideas, we notice the following aspect
(cf. Siegel, Über die analytische Theorie der quadratischen Formen,
Ann.Math., Vol. 36(1935)). We define two points Z, Z ∗ ∈ Y to be
equivalent (with regard to M) if
Z ∗ = MhZi = (AZ + B)(CZ + D)−1
(11)
!
A B
∈M
for a suitable matrix M =
C D
Then our earlier discussion shows that to every closed Riemann surface, there corresponds a uniquely determined set of equivalent points
in Y . The converse is not always true. Let Y0 be the submanifold of Y
consisting of all sets of equivalent points Z, which appear from period
matrices in the manner described above. As Riemann has shown, Y0
is a complex analytic manifold depending on 3n − 3 independent complex variables when n > 1, while Y depends on n(n + 1)/2 independent
complex variables. By a module of f we shall understand a complex
valued function f(f) defined on all closed Riemann surfaces of genus n
9
and possessing certain analytic properties. For instance, every function
f(Z) meromorphic in Y and invariant under M is such a modul. We
call such an f a modular function of degree n, but later we shall give a
precise definition of this concept. The modular functions of degree n
constitute a function field. More over we shall prove that this function
field is generated by a special set of modular function
fo (Z), f1 (Z), . . . fR (Z), R = n(n + 1)/2
(12)
The degree of transcendence of this function field is k, that is to say,
the functions (12) satisfy one irreducible algebraic equation
Ao (fo , f1 . . . fk ) = 0
(13)
It is possible to find a system of algebraic equations
Aν (fo , f1 . . . fR ) = 0, ν = 1, 2
(n − 2)(n − 3)/2
(14)
which give a necessary and sufficient condition for Z to belong to Yo .
The field of modular functions will be changed into the field of the
moduls by these relations. The field of moduls has the degree of transcendence (n(n + 1)/2) − ((n − 2)(n − 3)/2) = 3n − 3. Since every closed 8
Riemann surface of genus n is biuniquely determined by the values of
fo , f1 , . . . fn , the manifolds of all closed Riemann surfaces of genus n represent an algebraic manifold by (13) and (14). The explicit computation
of these equations is still on unsolved problem.
We proceed to investigate the arithmetic properties of the matrices
M ∈ M. First
! we consider the more general group S . Let M ∈ S and
A B
. We call (AB) the first matrix row of M and (C, D), the
M =
C D
second. Two complex matrices P, Q are said to from a symmetric pair
if we have PQ′ = QP′ . We note that if (P, Q) is a symmetric pair and
|Q| , 0, then Q−1 P is a symmetric matrix.
A matrix A will be called integral if all the elements of A are integers. A pair of n-rowed matrices A, B will be called coprime if the
matrix products GA, GB are integral when and only when the matrix G
is integral. This in particular implies that both A, B are integral if (A, B)
is coprime.
10
9
Let (C, D) be a coprime pair of matrices and let U1 , a n-rowed and
U2 , a 2n-rowed unimodular matrix. Then the matrices C1 , D1 defined
by (C1 D1 ) = U1 (C, D)U2 are coprime too.
Indeed, GC1 , GD1 are integral if and only if (GU1C, GU1 D)U2 are
integral which in turn is true if and only if GU1C, GU1 D are integral
as U2 is unimodular. Since by assumption (C, D) is a coprime pair, the
above holds if and only if GU1 and consequently G are integral, and this
proves our contention.
We will now choose U1 and U2 so that D1 = 0 and C1 becomes a
diagonal matrix with non-negative integers as diagonal elements. The
choice is possible by the elementary divisor theorem. Then, of necessity
C1 = E, as otherwise we can always find a non integral matrix G such
that GC1 is integral, contradicting the fact the pair (C1 , 0) is coprime.
Thus we obtain
(CD)U2 = U1−1 (C1 , D1 ) = U1−1 (E, 0) = (U1−1 , 0)
which shows that the matrix (C, D) is of rank n. This says even more,
viz. that there exist integral matrices X, Y satisfying the relation
CX + DY = E
(15)
Indeed, we have only to define X, Y by
!
!
X
U1
= U2
Y
0
and then
!
!
X
U1
CX + DY = (CD)
= (CD)U2
= E.
Y
0
The converse is true too, viz. if (C, D) is a pair of integral matrices
for which the equation (15) is solvable for integral X, Y, then (C, D) is a
coprime pair. For, in this case if G is integral then GC, GD are trivially
integral, while if GC, GD are integral, then GCX, GDY are integral
which in turn implies that GCX + GDY = GE = G is integral.
!
A B
We now note that matrix rows of a matrix M =
∈ M consist
C D
of coprime pairs in view of our above inference and relations (7). They
11
are trivially symmetric pairs, again a consequence of (7). The partial
converse is also true, viz. every symmetric coprime pair (C, D) is a
second matrix row of some matrix M ∈ M. For, we can determine
integral X, Y such that CX + DY = E and then we have only to set
A = Y ′ + X ′ YC, B = −X ′ + X ′ Y D.
Then
AB′ − BA′ = (Y ′ + X ′ YC)(−X + D′ Y ′ X) − (−X ′ + X ′ Y D)(Y + C ′ Y ′ X)
= (X ′ Y − Y ′ X) + (X ′C ′ + Y ′ D′ )Y ′ X − X ′ Y(CX + DY)
= (X ′ Y − Y ′ X) + Y ′ X − X ′ Y = 0
and
10
AD′ − BC ′ = (Y ′ + X ′ YC)D′ − (−X ′ + X ′ Y D)C ′
= Y ′ D′ + X ′C ′ = (CX + DY)′ = E
which show that A, B, C, D satisfy (7) and consequently
!
A B
M=
∈ M.
C D
Let us inverstigate how far (C, D) determines M uniquely. Let M,
M1 be !two modular matrices
with the same second matrix row, say M =
!
A B
A B1
, Mi = 1
. Then
C D
C D
−1
M1 M
!
!
!
E S
A1 B1 D′ −B′
=
=
0 R
C D −C A′
where R = −CB′ + DA′ , S = −A1 B′ + B1 A′ .
Since M1 M−i ∈ M (M being!a group), it is necessary that R = E
E S
and S = S ′ . Thus M1 =
M where S is a symmetric matrix
0 E
!
E S
conversely also, if M, M1 ∈ M and M1 =
M for a symmetric
0 E
12
11
matrix S , then M and M1 have the same second matrix row as is seen by
direct multiplication. Let T be the !Abelian sub-group of M consisting
E S
of all matrices of the form
where S = S ′ . Then the class of
0 E
all modular matrices with different second rows provides a complete
representative system of the set of all right cosets of T in M.
We proceed to consider another sub-group A of M of which T will
be a normal sub-group. This group
! A consists precisely of all elements
A B
. The conditions (7) will then imply
M ∈ M of the form M −
0 D
that AD′ = E and AB′ = BA′ . The first condition means that D′ =
A−1 . Since A, D′ are integral matrices it is now immediate that they are
unimodular, say A′ = u and D = u−1 . Let S = Bu = BA′ = AB′ . The
second condition then implies that S , defined as above, is symmetric.
Thus A is the sub-group
! of M consisting precisely of the matrices of the
u′ su−1
where S is symmetric and u is unimodular. It is
form M0 =
0 u−1
easily seen that T is a normal sub-group
of A. !Then since the mapping
!
A B
A 0
which taken the matrix
∈ M to
is a homomorphism of
0 D
0 D
which T is the kernel, it follows that A/T is isomorphic to the group of
matrices
!
u′ 0
,
u − unimodular.
(16)
0 u−1
Let!us decompose M into! right cosets modulo A. The matrices M =
A B
A B1
belong to the same right coset of A if and
and M1 = 1
C D
C 1 D1
only if Mo M = M1 for some Mo ∈ A.
This implies that
u(C1 , D1 ) = (C, D),
u − unimodular ,
(17)
and hence CD′1 = uC1 D′1 = uD1 C1′ = DC1′
Thus we have
CD′1 = DC1′ .
(18)
13
On the other hand, if M, M1! ∈ M be any two matrices for which
∗ ∗
−1
(18) holds, then MM−1
1 = 0 ∗ so that MM1 ∈ A; in other words
M and M1 determine the same right coset of A. Thus (17) and (18) are
equivalent and either of them gives a necessary and sufficient condition
that M and M1 belong to the same right coset modulo A. Let us now
define two pairs (C, D), (C1 , D1 ) to be associated if
1. Both of them are symmetric co-prime pairs and
2. CD′1 = DC1′
This relation of being associated pairs is reflexive, symmetric and
transitive - in other words, and equivalence relation. We can now say
that in the coset AM lie exactly those matrices whose second matrix
rows are associated with the second matrix row of M.
Let {C, D} denote the class of all coprime symmetric pairs of matrices associated with (C, D). Let C be of rank r, O < r ≤ n. By the
elementary divisor theorem, we can can determine unimodular matrices
u1 , u2 such that
!
C1 0 ′
u1 C =
u , |C | , 0
(19)
0 0 2 1
Here C1 is a r × r square matrix.
In analogy with (19) we write
!
D1 D2 −1
u
u1 D =
D3 D4 2
12
(19)′
and determine the nature of Di , i = 1, 2, 3, 4.
Since CD′ = DC ′ , we have
!
!
!
!
C 1 0 D1 D3
D1 D2 C2′ 0
=
D3 D4 0 0
0 0 D2 D4
which gives C1 D′1 = D1C1′ and D3 = 0
We now contend that D4 is unimodular. For, in view of (15) we need
only verify that the pair (D4 , 0) is coprime. Let then GD4 be integral.
Of necessity, G is a matrix with (n − r) columns. We complete G with
14
zeros to a matrix (O G) with n columns. From (19) and ((19)′ ) we
obtain that (O G)u1 C and (OG)u1 D are integral. But (C, D) is a coprime
pair and consequently (u1 C, u1 D) is also coprime, so that G should be
integral. It is now immediate that (D4 , 0) !is coprime and consequently
E D2 ∗
u . Then u∗1 is unimodular.
D4 is unimodular. Let now u1 =
0 D4 1
Also
!
!
!−1
E −D2 D−1
C
0
E D2
1
∗
4
u1C =
u′2
u1 C =
0
D−1
0
0
0 D4
!4
C1 0 ′
=
u
0 0 2
and
u∗1 D
13
!
!
!
E −D2 D−1
E −D2 D−1
D1 D2 −1
4
4
=
u
u1 D =
0
D−1
0
D−1
0 D4 2
4
4
!
D1 0 −1
=
u
0 E 2
We will now show that the pair C1 , D1 ) is coprime. Let GC1 , GD1
be integral. Then (G O)u∗1C and (G O)u∗1 D are integral. But (C, D) is
a coprime pair and hence also (u∗1 C, u∗1 D). Hence we conclude that G
is integral. It is now immediate that (C1 , D1 ) is coprime too. We may
summarise our results into the following statement:
If (C, D) is a coprime symmetric pair with rank C = r, O < r ≤ n,
there exist square matrices C1 , D1 of order r which again form a coprime
symmetric pair, and unimodular matrices u1 , u2 such that
!
!
C1 0 ′
D1 0 −1
u1 C =
u ,u D =
u
(20)
0 0 2 1
0 E 2
We will now!fix u2 more precisely. If we replace u2 in (20) by a
u 0
matrix u2 3
where u3 is an r ×r unimodular matrix, we still obtain
0 E
the same form. This replacement amounts to changing Q to Qu3 where
Q consists of the first r columns of u2 . We shall call two matrices Q,
15
Q1 right associated if Q1 = Qu3 for some unimodular matrix u3 . Let
{Q} denote the class! of all matrices right associated with Q. A change of
u 0
u1 to u∗1 = 4
u where u4 is unimodular carries the pair (C1 , D1 )
0 E 1
into (u4 C1 , u4 D1 ) ∈ {C1 , D1 }. A proper choice of u3 and will transform
Q into a fixed matrix in the class Q and u4 the pair (C1 , D1 ) into a fixed
pair in {C1 , D1 }. This settles how for C1 , D1 and u1 characterise C, D in
(20). We may note that the matrices Q which enter into our discussion
are precisely those that can be completed to a unimodular matrix (Q R).
A matrix with this property will be called primitive. The elementary
!
E
−1
divisors of a primitive matrix are all equal to 1 as u2 Q =
shows.
0
Conversely, if all the elementary divisors of Q are equal to 1, then Q
is primitive. For,
! then we can determine unimodular matrices u2 , u3 so
E
that Q = u2
u . We write u2 = (Q1 , R1 ) where Q1 is r-columned and
0 3
obtain
!
u3 0
(Q R1 ) = (Q1 u3 R1 ) = u2
0 E
We use the notation Q = Q(n,n) to signify that Q is a matrix with n
rows and r columns. We will denote Q(n,n) simply by Q(n) . A parametric
representation of all classes {C, D} of coprime symmetric pairs C, D,
with rank C = r, 0 < r ≤ n is give by
Lemma 1. Let Q = Q(n,r) run through a complete representative system 14
of all primitive classes {Q} of n×r matrices and let (C1 , D1 ) run independently through a complete representative system of all classes {C1 , D1 }
of coprime symmetric pairs of matrices C1 = C1(r) and D1 = D(r)
1 with
rank C1 = r. To every Q we make correspond only one of the matrices,
say u2 , obtained by completing Q arbitrarily to a unimodular matrix.
Then we obtain a complete representative system of all classes {C, D}
of coprime symmetric pairs (C, D) where C = C (n) , D = D(n) and rank
C = r, in the form
!
!
C1 0 ′
D1 0 −1
C=
u ,D =
u ,
(21)
0 0 2
0 E 2
different choices of C1 , D1 , Q Leading to different classes.
16
Proof. Only the last part of the Lemma remains to be proved.
!
!
C 0 ′
D 0 −1
Let C = 1
u2 , D = 1
u
0 O
0 E 2
and
C∗ =
!
!
C1∗ 0 ∗′ ∗
D∗ 0 ∗−1
u
u2 , D = 1
0 E 2
0 0
and let {C, D} = {C ∗ , D∗ } i.e. to say
C ∗ D′ − D∗ C ′ = 0
(22)
Then we have to show that {C1 , D1 } = {C1∗ , D∗1 } and {Q} = {Q∗ } in
an obvious notation.
From (22) we get
!
!
!
!
C1∗ 0 ∗′ −1 D′1 0
D∗1 0 ∗−1
C1′ 0
u u
=
u u
0 0 2 2 0 E
0 E 2 2 0 0
or with the notation
′ ′
u∗2 u2−1
15
!
!
W1 W2
v1 v2
∗−1
, u2 u2 =
=
W3 W4
v3 v4
!
!
C1∗ V1 D′1 C1∗ V2
D∗ W1C1′ 0
We have
=
from which it follows
0
0
W3 C 1
0
that C1∗ V2 and W3C1′ and consequently V2 and W3 are zero (as C1 , C1∗
are nonsingular).
−1
Since u2∗ u2 is unimodular, the above implies that W1 is unimodular.
(n,r)
Let u2 = (Q R) and u∗2 = (Q∗ R∗ ) where Q = Q(n,r) and Q∗ = Q∗ . Then
(Q R) = u2 = u∗2
!
!
W W2
W1 W2
= (Q∗ W1 ∗)
= (Q∗ R∗ ) 1
0 W4
W3 W4
So that Q = Q∗ W1 . This in turn implies that {Q} = {Q∗ } which
is one of the two results we were after. Since Q has been assumed to
17
run through a representative system of the classes {Q}, it follows that
Q = Q∗ and therefore u2 = u∗2 . This in its implies that W1 = E and
V1 = E. Hence C1∗ D′1 = D∗1 C1′ or {C1 , D1 } = {C1∗ , D∗1 }. If we assume
that just one element is taken from each class {C, D}, then of course we
should have C1 = C1∗ and D1 = D∗1 .
It remains to prove that the class {C, D} does not depend on the manner in which Q is completed to a unimodular matrix. For this, we suppose in the above that C1 = C1∗ and D1 = D∗1 and Q = Q∗ . Then if
follows that W1 = E, W3 = 0, V1 = E and V2 = 0. We have to deduce
that {C, D} = {C ∗ , D∗ }, whatever R, R∗ be. Now
!
!
!
C1∗ 0 V1 V2 D′1 0
∗ ′
∗ ′
C D −D C =
0 0 V3 V4 0 E
!
!
!
D 1 0 W1 W2 C ′ 0
−
0 E W3 W4 0 0
!
!
C1∗ D′1 0
D∗1 C1′ 0
=
−
=0
0
0
0
0
This completes the proof.
Chapter 2
The Symplectic group of
degree n considered as a
Group of mappings
Let Z be a point of the manifold Y defined by! Z = Z ′ , Y > 0(Z = X +iY) 16
A B
and M, a symplectic matrix: M =
∈ S . We prove that the
C D
transformation
Z → MhZi = (AZ + B)(CZ + D)−1
(23)
is a 1 − 1 mapping of Y onto itself. For brevity, we put
P = AZ + B, Q = CZ + D
In view of (9) we obtain
1 ′
1
(P Q̄ − Q′ P̄) =
(Z ′ A′ + B′ )(C Z̄ + D) − (Z ′C ′ + D′ )(AZ̄ + B)
2i
2i
1
1
= (Z ′ − Z̄) = (Z − Z̄) = Y > O
2i
2i
Also the matrix CZ + D is non-singular so that the above mapping is
well defined. For, if z be any n-rowed complex column which satisfies
19
2. The Symplectic group of degree n ....
20
the equation Qz = 0, then Q̄z̄ = 0 and z′ Q′ = 0 so that
z′ Yz̄ =
1 ′ ′
(z P Q̄z̄ − z′ Q′ P̄z̄) = 0.
2i
But Y > 0.
Hence it follows that z = 0 and this in turn implies that Q is non-singular.
Let then Z1 = PQ−1 . In view of (9) It follows that
P′ Q = (Z ′ A′ + B′ )(CZ + D) = (Z ′C ′ + D′ )(AZ + B) = Q′ P
in other words, Z1 is a symmetric matrix.
Further
1
1 ′
1
(Z1 − Z̄1 ) = (Z1′ − Z̄1 ) = (Q −1 P′ − P̄Q̄−1 )
2i
2i
2i
1 ′−1 ′
′
′−1
′−1
= Q (P Q̄ − Q P̄)Q̄ = Q Y Q̄−1 > 0
2i
Y1 =
17
(24)
Thus it follows that Z1 ∈ Y ; in other words, the transformation (23)
takes Y into Y .
A simple computation shows that M1 M2 hZi = M1 hM2 hZii, Mi ∈
S . Since S is a group implies that the mapping Z → M < Z > is
actually onto and that this mapping has an inverse.
!
A B
We showed that |CZ + D| , o for any
εS. Specialising A, D
C D
to be 0, and B, −C to be E, the above implies that, for ZεY
|Z| , 0
(25)
Now for every Z ∈ Y , Z + iE also lies in Y so that we conclude that
|Z + iE| , o, Z ∈ Y
(25)′
We can then introduce the mapping
W=T <Z>
with T =
!
E −iE
i.e W = (Z − iE)(Z + iE)−1
E iE
Let K be the domain onto which Y in mapped by T . Then
W − iE = (Z − iE)(Z + iE)−1 − E
21
= {(Z + iE) − (Z − iE)}(Z + iE)−1 = −2i(Z + iE)−1
so that |W − iE| , 0 Also W(Z + iE) = Z − iE i.e i(E + W) = (E − W)Z
so that Z = i(E − W)−1 (E + W). This in particular implies that the
correspondence between Y and K is 1 − 1. Also is it easily seen that the
relations Z = Z ′ and W = W ′ are equivalent. Now
E − W W̄ = E − (Z + iE)−1 (Z − iE)(Z̄ + iE)(Z̄ − iE)−1
= (Z + iE)−1 {(Z + iE)(Z̄ − iE) − (Z + iE)(Z̄ + iE)}(Z̄ − iE)−1
= Zi(Z + iE)−1 (Z̄ − Z)(Z̄ − iE)−1
= 4(Z + iE)−1 Y(Z̄ − iE)−1 > 0
Hence z′ (E − W ′ W̄)z̄ is a hermitian form for any complex column z.
Thus W = T < Z > satisfies the relations.
W = W ′ , E − W W̄ > o
(26)
We claim that these relations characterise the elements of K. For, 18
if W is any matrix satisfying relations (26), then the matrix E − W is
non-singular, since the relation (E − W)z = 0 for any complex column z
implies along with (26) that z′ W = z′ and W̄z̄ = z̄ so that z′ (E−W W̄)z̄ = 0
and consequently z = 0. Hence the matrix Z = i(E − W)−1 (E + W) is
well defined, and precisely as in the earlier case it can be shown that
Z ∈ H and that W = T < z >. This allows us to conclude that the
domain K is characterised by the relations (26), viz. W = W ′ E − W W̄ >
0 for any W ∈ K. This domain is one of the four main types of E.
Cartan’s irreducible bounded symmetric domains. We may remark in
this connection that the domain Y is called the generalized upper half
plane and the domain K, the generalised unit circle.
We have seen that the symplectic substitutions (24) form a group
of 1 − 1 mappings of Y onto itself. By means of the transformation
W = T < Z > which is a 1 − 1 map of Y onto K, the above group
can be transformed into a group of 1 − 1 mappings of K onto itself. In
fact, which corresponds to the mapping Z → M < Z > of Y will be
given by W → M1 < W > where M < T −1 < W >>= T −1 < M1 <
W >>. In other words M1 < W >= T MT −1 < W > (each of the above
22
groups will be shown to be full group of analytic homeomorphisms of
the respective domain). Thus to the symplectic matrix M corresponds
the matrix M1 = T MT −1 and to the group S corresponds the group
S 1 = T S T −1 . Using (6) which characterises the elements of S we shall
obtain a characterisation of the elements
of Sl . M ∈ S is characterised
!
0
E
by M = M̄, M′ IM = I, I
With M = T −1 M1 T the above give
−E 0


M′1 T ′−1 IT −1 M1 = T ′−1 IT −1 

.
′ ′−1
−1
′−1
−1 

M T I T̄ M̄1 = T I T̄ 
1
19
By a simple computation we find that
!
1 iE iE
i
i
−1
T =
, −T ′−1 IT −1 = I, T ′−1 I T̄ −1 = H
2i −E E
2
2
where
E 0
H=
0 −E
(27)
!
Thus M1 can be characterised by
M′1 IM1 = I, M′1 H M̄, = H
We can replace the second relation, in view of the first, by
−1
−1
−1
IM−1
1 I H M̄1 = H or I H M̄1 = M1 I H.
!
0 E
With K = I −1 H =
we obtain finally the relations
E 0
M′1 IM1 = I, K M̄1 = M1 k
!
A1 B1
The decomposition M1 =
leads to the relations
C 1 D1
(28)
B1 = C̄1 , D1 = Ā1 , A′1 ĀI − C1′ C̄1 = E, A′1 C1 = C1′ A1
(29)
The relations (28) and (29) are clearly equivalent.
23
We now show that the domains Y and K are homogeneous, in other
words, that the groups S and S1 are transitive in their respective domains. It clearly suffices to prove for one, say K, and this in turn
only requires the determination of a substitution M1 which
takes the
!
A1 B1
, the relation
point 0 to any assigned point W ∈ K. If M1 =
C 1 D1
W = M1 < 0 > implies that W = B1 D−1
1 i.e B1 = W D1 . Then in view of
(29) we should have C1 = W D̄1 , A1 = D̄1 and D̄1 D1 − D̄1 W̄W D1 = E.
Consequently D̄′1 (E − W̄ ′ W)D1 = E or D1 D̄1 = (E − W̄ ′ W)−1 . The last
equation is certainly solvable for D1 as E − W̄ ′ W is positive Hermitian.
If D1 is obtained as any solution of the above equation, then the earlier
20
!
A1 B1
will
equations define A1 , B1 and C1 and the matrix M1 =
C 1 D1
have the desired properties.
We proceed to establish a result we promised earlier, viz that the
group of symplectic substitutions is the full group of analytic mapping
of the domain Y onto itself.
By an analytic mapping Z → Z × of Y onto itself, we mean a mapping with the following properties
i) it is topological,
∗ = Z ∗ (Z) is a regular function of the independent elements
ii) Zµν
µν
Zµν (µ ≤ ν) for µν = 1, 2 . . . η
iii) Zµν = Zµν (Z ∗) is a regular function of the independent variables
∗ (µ ≤ ν) of Z ∗ for µν = 1, 2 . . . η.
Zµν
Then every symplectic substitution Z → Z ∗ = M < Z > is clearly
and analytic mapping of Y . We prove the converse in the following
Theorem 1. Every analytic mapping of Y onto itself is a symplectic
substitution
In view of the 1 − 1 correspondence between Y and K by means of
the transformation Z → T < Z >, the above will imply a similar result
of the domain K- in fact, it suffices to prove the corresponding result for
24
K to infer Theorem 1. Since the group S1 is transitive with regard to K
we may even assume that the given analytic mapping
Wo → Wo∗
(30)
where Wo = (ωµν ) of K has 0 as a fixed point. Let W = tWo where t is a
complex variable and let the characteristic roots of the hermitian matrix
W0′ W̄ be r1 , r2 . . . rn where o < r1 ≤ r2 ≤ · · · ≤ rn .
It is well known that there exists a unitary matrix u such that

r1

′ ′
r2..
A Wo W̄o Ā = 

0
21

0 


rn
We then have

0
r1

r
2
A′ (E − W W̄)Ā = E − tt¯ 
rn

0



 > 0


if and only if tt¯rn < 1. But for t = 1 W = Wo and we know that
E − Wo′ W̄o . Hence rn < 1.It is now immediate that if |t| ≤ 1 then E −
W ′ W̄ > o and consequently W ∈ K. Then to each point W = t Wo
with |t| ≤ 1 there corresponds an image point W ∗ ∈ K. The mapping
: t → (two )∗ = W ∗ is regular function of the single variable t in tt¯rn < 1,
meaning each element of W ∗ is a regular function of t, so that there
exists a power series expansion for the elements of W ∗ in the form
W∗ =
∞
X
tK WK∗
(31)
K=1
which converges for tt¯rn < 1 and, a fortiori for, |t| ≤ 11 where WK∗
for each K is a matrix whose elements are functions (polynomials) of
the elements of Wo alone. One the other hand, the elements of W ∗ are
regular functions of the variables tωµν so that they may be developed
into a power series in tωµν (about the origin) converging for sufficiently
25
small values of the variables. But the ωµν occurring in Wo ∈ K can all
be shown to be uniformly bounded so that we need only restrict |t| to be
small while the ω′µν s may be arbitrary in their domain. Since a power
series expansion (for the elements 0 W ∗) is unique, it follows that (31)
again is the desired power series and that tK WK∗ is precisely the aggregate 22
of all terms K in it. Consequently, the series
Wo∗ =
∞
X
WK∗
(32)
K=1
obtained from (31) by specialising t, converges everywhere in K if we do
not spilt up the polynomials which are elements of WK∗ into their single
terms. Since E − W ∗W̄ ∗ > 0 for tt¯ = 1,we obtain by integration over the
circle tt¯ = 1 that
Z
dt
1
(33)
(E − W ∗ W̄ ∗ ) > 0
2πi
t
tt¯=1
R
where
by Adt with any matrix A = (aµν (t)) we mean the matrix
R
( aµν dt). Substituting for W ∗ the series (31) which converges uniformly
for |t| ≤ 1, termwise integration yields
E−
∞
X
WK∗ W¯K∗ > 0
(34)
K=1
the rest of the terms all vanishing. In particular we have
E − W1∗ W̄1∗ > 0
(35)
The elements of WK∗ are homogeneous polynomials of degree K in
ωµν (µ ≤ ν) so that the n(n + 1)/2 elements of W1∗ = (W (1)∗ µν ) are liner
functions of the independent elements ωµν (µ ≤ ν). Let D be the determinant of this linear transformation. Since W1∗ is the linear part of
the power series (32), the functional determinant of the n(n + 1)/2 independent elements of Wo∗ with respect to the variables ωµν (µ ≤ ν) at
the point Wo = 0 is also D. As the mapping Wo → Wo∗ is invertible it
follows that D , 0. Also if we replace the mapping Wo → Wo∗ by its
26
23
inverse, the determinant change into D−1 so that we can, without loss of
generality,assume that DD̄ ≥ 1.
Consider now the liner mapping ϕ : Wo → W1∗. Let K1 the image of
K by ϕ. We denote by ϑ(K1 ) the Euclidean volumes of K and K1 , the real
and imaginary parts of ωµν being the rectangular cartesian coordinates.
Then we have ϑ(K1 ) = DD̄ϑ(K) ≥ ϑ(K). But K1 ⊂ K so that ϑ(K1 ) ≤
ϑ(K). Hence it follows that ϑ(K) = ϑ(K1 ) and DD̄ = 1. We are going
to conclude from this that K = K1 . Our argument is as follows. Let
K∗ = Exterior K = (K̄)c and K∗1 = Ext K1 = (K̄)c the superscript ‘c′
denoting complements, and the bars denoting the closures. Since K1 ⊂
K and ϑ(K1 ) = ϑ(K) it is immediate that ϑ(Interior (K − K1 ) = 0) and
consequently.
Int (K − Ki ) = 0 (void). A point of K is then either a point of K1
or a limit point of K1 and so K̄ = K̄1 . We need only show then that the
boundary BdK = BdK1 . Being a linear map, ϕ is a topological map in
the large so that if it maps K onto K1 then it maps BdK onto BdK1 , Ext.
K onto Ext. K1 , and Bd. (Ext. K1 ) onto Bd. (Ext K1 ). Thus
Bd.K1 = ϕ(Bd.K) = ϕ(Bd.K∗ ) = Bd.K∗1 = Bd.(K1 )c
= Bd.(K̄)c = Bd.K∗ = Bd.K.
We have incidentally shown that ϕ maps Bd.K onto itself.
We prove in Lemma 2 that any complex symmetric matrix can be
represented in the form Wo = u′ Pu, u-unitary and P, a diagonal matrix
with diagonal elements p1 , p2 . . . pn where p22 , ν − 1, 2 . . . n are the char

 2 0  

acteristic roots of Wo W̄o . Then E − Wo W̄o = u′ E −  pν
 ū so that

0
Wo ∈ K if and only if p2ν < 1, ν = 1, 2 . . . n. The boundary points of K
are therefore precisely those for which p2ν ≤ 1, ν = 1, 2 . . . n, the equality holding for at least one ν. Since ϕ(Bd.K) = Bd.K it is immediate that
|E − W1∗W̄1∗ | = 0 for W0 ∈ Bd.K. In other words, |E − W1∗ W̄1∗ | considered
a polynomial in pν , ν = 1, 2, . . . n vanishes if p2ν = 1 for some ν. Hence
n
Q
it is divisible by (1 − p2ν ). Considering the degree of |E − W1∗W̄1∗ | in
ν=1
24
the
p′i s
we have |E − W1∗ W̄1∗ | = c
n
Q
ν=1
(1 − p2ν ), c − a constant. The choice
27
of P as 0 leads to the determination of c to be 1. Hence
|E − W1∗ W̄1∗ | =
n
Y
(1 − p2ν ) = |EWo W̄o |
(36)
ν=1
1
Replacing Wo in (36) by √ Wo , in view of the linearity of ϕ : Wo →
λ
W1∗ we obtain
|λEWo W̄o | = |λE − W1∗ W̄1∗ |
identically in λ and therefore that Wo W̄0 and W1∗ W̄1∗ have the same characteristic roots. Lemma 2 (proved below) will then imply that
W1∗ = u′ Wo u
(37)
for some unitary matrix u. We stop here to prove
Lemma 2. Every complex symmetric matrix

 Wo admits a representation
0 
 p1
 2

p2
 pν being the
of the kind Wo = u′ Pu with u′ u = E, P = 
.
..


0
pn
characteristic roots of Wo W̄o .
Proof. Wo W̄o being hermitian, there exists a unitary matrix u1 such that
−1
Wo W̄o = u′1 P2 ū. Then F = u′−1
1 Wo u1 is symmetric and satisfies the
relation F F̄ = P2 . Writing F = F1 + iF2 (Fν being real), it follows that
F1 F2 − F2 F1 = 0. Hence by a well know result, an orthogonal matrix Q
can be found which transform F1 and F2 simultaneously into diagonal
matrices Q′ F1 Q and Q′ F2 Q. Then R = Q′ FQ is also diagonal. If rν ,
ν = 1, 2, . . . n be the diagonal elements of R the relation RR̄ = Q′ P2 Q
implies that rν r̄ν (ν = 1, 2, . . . n) are identical with P2ν (ν = 1, 2, . . . η).
Then we can find a unitary matrix u2 such that R = u′2 pu2 . Now Wo =
u′1 Fu1 = u′1 Q′−1 RQ−1 u1 = u′1 Q′−1 u′2 pu2 Q−1 u1 . Taking u = u′1 Q−1 u′2 25
we obtain Wo = u′ Pu. Clearly u is a unitary matrix.
We revert now to the proof of the main theorem. From (34), we have
E − W1∗ W̄1∗ − WK∗ W̄K∗ > 0
(38)
28
for K = 2, 3, . . ., and any Wo . Choosing in particular Wo = ucis with
o < u < 1 and s = s′ = s̄ we get from (37) that
W1∗ W̄1∗ = uu′ eis uuū′ e−is ū = u2 eis e−is = u2 E
But Wo W̄o = u2 E and form (38)
(1 − u2 )EWK∗ − W̄K∗ > O for K > 1, o < u < 1.
Letting u → 1 this gives WK∗ = 0 for K > 1, Wo eis , S = S ′ S̄
Since the elements of WK∗ are analytic functions of the elements of
S and they vanish for real values of these elements, it follows that they
vanish for complex S too. Since the mapping S → Wo = eis maps
a neighborhood of 0 onto a neighborhood of E so that WK∗ = 0 in a
neighborhood of E, so that WK∗ = 0 in a neighborhood of for K > 1,
P
we conclude that WK∗ = 0 identically for K > 1, and Wo∗ = WK∗ = W1∗ .
Thus the mapping Wo → W0∗ = W1∗ is linear. If now we show that in (37)
the matrix u does not depend on Wo , then W1∗ = u′ Wo u = M1 < Wo >
!
u′ 0
where M1 =
∈ S 1 and our theorem would have been proved.
0 u−1
We proceed to establish this.
Let
X
W1∗ = W1∗ (Wo ) =
ωµν Aµν
(39)
µ≤γ
and introduce
W1 = W1 (Wo ) =
X
ωµν Āµν
(40)
µ≤γ
26
where Aµν are constant matrices. Then W̄1∗ = W1 (W̄o ) and W1∗ W̄1∗ =
u′ Wo W̄o ū = EWo W̄o = E Hence W1∗ (Wo )W1 (Wo−1 ) = E for Wo such
that Wo W̄o = E. In particular then, this is true for Wo = eis where
S = S ′ = S̄ Since W1 and W1∗ depend analytically upon their arguments,
this relation holds for complex symmetric S too, that is to say identically
in Wo , by an earlier argument.
For convenience, we shall usually write W2 instead of Wνν and Aν
instead of Aνν .
29
Let

o
ω1

ω2
Woo 

o
ωn



 Wo1 = Wo − Woo .
Employing the Taylor Series Expansion in the neighborhood of
Wo1 = 0 we find
−1
−1 −1
Wo−1 = (Woo + Wo1 )−1 = Woo
(E + Wo1 Woo
)
−1
−1
−1
= Woo
− Woo
Wo1 Woo
+ ···
Using the linearity of W1 and W1∗ we therefore get
E = W1∗ (Wo )W1 (Wo−1 )
−1
−1
−1
= (W1∗ (Woo ) + W1∗ (W01 ))(W1 (Woo
) − W1 (Woo
Wo1 Woo
)+ ···)
Comparing the terms of degree 0 and 1 we obtain
−1
W1∗ (Woo )W1 (Woo
)=E
−1
−1
W1∗ (Woo )W1 (Woo
Wo1 Woo
)
(41)
=
−1
W1∗(Wo1 )W11 (Woo
)
(42)
P
The first equation means nµ,ν=1 ωµ ω−1
ν Aµ Āν = E identically in ων ,
ν = 1, 2 . . . n
This leads to
Aµ Āν = 0, µ , ν
(43)
Let the ‘u’ which enters in (37) corresponding to Wo1 = 0 be denoted 27
by uo
From (37) and (39) we get
n
X
ων Aν = u′o Woo uo
ν=1
From this, in view of (43) we infer that
n
X
ν=1
ων ω̄ν Aν Āν = u′o Woo W̄oo ūo and consequently
30
|λE −
n
X
ν=1
ων ω̄ν Aν Āν | = |λE − Wo W̄oo |
identically in λ. Choosing ων = 1 and ων = 0 for µ , ν the above gives


o o


|λE − Aν Āν | = λE −  1  

o
o
so that the characteristic roots of Aν Āν are 1, 0, 0 . . . 0. In virtue of
Lemma 2, we can assert the existence of a unitary matrix ν such that


o 
1

 o


′

A1 = ν 
ν
o 


o
o
Without loss of generality we can assume ν = E, as in the alternative case, we need only consider the mapping ν̄Wo∗ν̄ instead Wo∗. Then
Aν Āν = Aν A1 = 0 for ν > 1 by (43) which means that for ν > 1
!
o
o
Aν =
o B(n.1)
ν
28
Successive application of this argument shows that we may assume


o



o


 (ν = 1, 2 . . . n)
1
Aν 


o 


o
P
Then we shall have nν=1 Aν = E and
W1∗ (Woo ) =
n
X
ων Aν = Woo
W1∗ (Woo ) =
n
X
ων Aν = Woo
ν=1
ν=1
and
31
From (44) we then conclude that
−1
−1
W1∗ (Wo1 ) = Woo W1 (Woo
Wo1 Woo
)Woo
Now
W1∗ (Wo1 ) =
X
ωµν Aµν
(44)
µ<ν
and
−1
−1
Woo W1 (Woo
Wo Woo
)Wo =
X ωµν
Woo Āµν Woo
ωµ ων
µν
(45)
A comparison of (44) and (45) yields
Aµν =
or
1
Woo Āµν Woo
ωµ ων
ωµ ων Aµν = Woo Āµν Woo
(µ < ν).
Since this holds identically in Woo , setting Woo = E, this in particular implies that Aµν = Āµν i.e Aµν is real, and further Aµν = aµν (eµν + eνµ )
with real aµν ,where eµν denotes the matrix with 1 at the (µ, νth) place


o



and 0 else where. Since Aµ  1  we have in particular aνν = 12 .


0
P
Now W1∗ = µ≤ν ωµν Aµν = (a∗µν ωµν ) with real a∗µν and a∗µν = 1. In
view of (37), the expression |W1∗ ||Wo |−1 has a constant absolute value.
On the other hand it is a rational function of ωµν . Consequently it is
a constant. Since the product ω1 ω2 . . . ωn appears in both the determinants with the factor 1, it is immediate that |W1∗ | = |Wo |. Since the
matrix (±δµν ) is unitary, we can assume that a∗1ν ≥ 0 for ν > 1. The 28
term (ω1 ωµ ων )−1 ωω1µ ωµν has in |W1∗ | the coefficient 2a∗µν (1 < µ < ν)
and in |W0 |, the coefficient 2. Hence a∗µν = 1(1 < µ < ν). Also the term
(ω1 ων )−1 ωω21ν has in |W1∗ | the coefficient, −a∗2 1ν and in |Wo |, the coefficient −1. Hence a∗1ν = 1 for ν =≥ 1. We already know that a∗νν = 1.
Thus we conclude that W1∗ = (ωµν ) = Wo . In other words we have
shown that with the aid of appropriate symplectic transformations, any
analytic map Wo → Wo∗ of Y onto Y can be reduce to the identity map
so that the analytic map we started with must itself be symplectic. This
completes the proof of theorem 1.
Chapter 3
Reduction Theory of Positive
Definite Quadratic Forms
By the reduction of a positive definite quadratic from we shall under- 29
stand the reduction of the corresponding matrix Y = Y (n) > 0. Investigations in this direction are needed if we wish to construct a suitable
fundamental domain for the modular group of degree n in Y . Let Y
be the domain of all real symmetric matrices Y and P the domain of
all Y > 0 in Y . Two matrices Y, Y1 ∈ Y are said to be equivalent if
Y1 = Y[u] = u′ Yu for some unimodular matrix u. We consider only the
classes of equivalent positive definite matrices, and the theory of reduction consists in fixing in each class a typical element, called a reduced
matrix, satisfying certain extremal properties.
Let Y > 0 be a given matrix in Y . Choose a primitive column
ũ1 such that Y[ũ1 ] = ũ′1 Y ũ1 is a minimum for all primitive (integral)
columns. Such a ũ1 clearly exists. Consider now all integral column
vectors ũ such that (ũ1 ũ) is primitive and choose ũ = ũ2 so that Y[ũ2 ]
is a minimum. We can further assume that ũ1 Y ũ2 ≥ 0 as otherwise −ũ2
will serve the role of ũ2 and satisfy this. Continuing in this way, let
ũ1 , ũ2 , . . . , ũr be already determined so that in particular (ũ1 , ũ2 . . . ũr ) is
primitive. Then we choose ũr+1 such that
i) ũ′r yũr+1 ≥ 0
33
34
3. Reduction Theory of Positive Definite Quadratic Forms
ii) (ũ1 ũ2 . . . ũr+1 ) is primitive and
iii) y[ur+1 ] is a minimum among those ũr+1 satisfying (i) and (ii).
30
In this way we obtain a unimodular matrix u = (ũ1 , ũ2 . . . ũn ) with
some extremal properties and we call R = y[u] a reduced matrix
We shall now obtain explicitly the reduction conditions on the elements rµν of a reduced matrix R
Let uK be a unimodular matrix which has the same first (K − 1)
columns as u. Such an uK can be represented by
!
E A
uK = u
0 B
where E = E (K−1) , A integral and B, unimodular.Let YK be the Kth column of u−1 uK . Then the first column of B is formed just by the (n−K+1)
last elements gK , gK+1 . . . gn of YK so that these elements are coprime.
Conversely if gK , gK+1 . . . gn by any n − K + 1 elements which are coprime, there exists a unimodular matrix B with these elements precisely
constituting the first column and consequently, a uK too. Since uYK is
the Kth column of uK we obtain
Y[uYK ] = R[YK ] ≥ rKK ≡ rK , K = 1, 2 . . . , n;
ũ′K Y ũK+1 = rKK+1 ≥ 0, K = 1, 2 . . . η.
This proves
Lemma 3. In order that R = (rµν ) should be a reduced matrix, it is
necessary and sufficient that
R[YK ] ≥ rK , rKK+1 ≥ 0, K = 1, 2 . . . n
(46)
where YK denotes an arbitrary integral column whose last n − K + 1
elements are coprime.
While the necessity part has been shown above, the sufficiency is
immediate since R = R[E] and this is clearly a reduced matrix by virtue
of the conditions (46).
35
31
In case YK = ±nK the Kth unit vector, the equation K[YK ] = KK
holds identically in K so that one of the condition (46) will now be innocuous. We may therefore assume that the YK in Lemma 3 is different
from ±nR .
Lemma 4. Every reduced matrix R = (rµν ) satisfies the following in
equalities
rR ≤ rℓ , K ≤ ℓ
(47)
r1 r2 . . . rn < c1 | R | where C1 = C1 (n),
(49)
rℓ ≤ 2rKℓ ≤ rℓ , K > ℓ
(48)
viz. a positive number depending only on n.
While (47) is immediate from Lemma 3 by choosing YR = nℓ (ℓ ≥ K
to prove (48) we have only to set YK = nK ± kℓ (ℓ < K) The proof of (49)
will be by induction on n. Clearly (49) is true for the case n = 1. Let
Rℓ denote the matrix which arises from R by deleting the last n − ℓ rows
and columns. It is clear that Rℓ > 0. Then by Lemma 3, it will follow
that Rℓ is reduced too. The induction hypothesis will now imply that
r1 r2 . . . rn−1 < C2 | Rn−1 |, C2 = C2 (n)
(50)
We denote by DKℓ the (n − 2) rowed sub-determinant of Rn−1 obtained by deleting the row and column containing eKℓ . Then DKℓ is the
sum of (n − 2)! terms of the type rν1 1 rν2 2 . . . rνℓ−1 ℓ−1 rνℓ+1 ℓ+1 . . . rνn−1 n−1 ,
Majorising each term with the help of (48), we get
±DKℓ rℓ < C3 r1 r2 . . . rn−1 , C3 = C3 (n).
Consequently, in view of (50) we have
±DKℓ | Rn−1 |−1 < C2C3 rℓ−1
R
H
We now define H by n−1′
H
rn
and set r = rn − R−1
[H
]
n−1
!
36
!
!
Rn−1 0
E R−1
H
n−1
| R |= r | Rn−1 |.
Then R =
0
r
0
1
Let now ξ ′ = (z′ xn ) be a row with n variable elements.
We have
!
!
!
Rn−1 0 E R−1
H
z
n−1
R[ξ] =
0
r
0
1
xn .
!
!
−1
Rn−1 0 z + Rn−1 H xn
=
0
r
xn
32
2
= Rn−1 [z + R−1
n−1 H xn ] + rxn .
−1 so that by virtue of (47)
The elements of R−1
n−1 are ±DKℓ | Rn−1 |
and (48) and (50) we have
X 1
rnK rnℓ
rℓ
< C4 rn−1 , C4 = C4 (n)
R−1
n−1 [H ] < C 2C 3
(51)
Also rn = r + R−1
[H ] < r + C4 rn−1 so that r1 r2 . . . rn < C2 | Rn−1 |
n−1
rn−1
rn < C2 1+C4 r | R |. We will now show that rn−1 < C5 r, C5 = C5 (n)
and then we would have proved (49).
Let
n−1
C6 = 4(n − 1)2 , C7 = (2n − 2)n−1 = C6 2
(52)
Let K be determined such that
rℓ+1 < C6 rℓ
33
(53)
for ℓ = n − 2, n − 3, . . . K + 1, K but not for ℓ = K − 1 (The statement
will have the obvious interpretation in the border cases corresponding to
K = 0 and K = n − 1). Let xν + aν xn be the νth element of z + R−1
n−1 H xn .
For each integer x′n in the interval 0 ≤ x′n ≤ C7n−K we determine a set
of n − K integers x′ν , ν = K, K + 1 . . . n − 1 such that 0 ≤ x′ν + aν x′n < 1
for each ν. Thus corresponding to the C7n−K + 1 possible choices for x′ n
we obtain C7n−K + 1 points in the (n − K) dimensional Euclidean space,
all lying in the half open unit cube 0 ≤ Yν < 1, ν = K, K + 1, · · · n − 1.
If we divide this cube into equal cubes each of whose sides is of length
37
C7−1 , there will be C7n−K such cubes and consequently, by a well known
principle, one of these cubes contains at least two of the above C7n K + 1
points. Their difference is clearly a point with coordinates of the form
xν + aν xn , v = K, K + 1, · · · , n − 1 where x′ν s are integers,
| xν + aν xn |< C7−1 , 0 < xn ≤ C7n−K
(54)
In other words we have solved (54) with integral xν , xn , ν = K, K +
1, · · · n − 1. We can assume that xK , . . . xn are coprime. Now we choose
integers x1 , x2 . . . xK−1 such that
1xν + aν xn |< 1, ν = 1, 2, . . . K − 1
(55)
From Lemma 3, we have R [ξ] ≥ rK . On the other hand, the relation
rK > C6 rK−1 and the relations (52) - (55) entitle us to conclude that
2
R[ξ] = Rn−1 [z + R−1
n−1 H xn ] + rxn
< (K − 1)2 rK−1 + (K − 1)(n − K)rK−1 C7−1 +
+ (n − K)2 C7−2C6b−K−1 rK + rC72(n−K)
≤ (K − 1)(n − 1)C6−1 rK + (n − K)2 C6−K rK + C72n−2 r
Thus rK < C8 r, C8 = C8 (n) and by (53), rn−1 < C5 r. This completes
the proof.
Having thus settled the arithmetical properties of reduced matrices,
we proceed with their existential nature. We have already seen in the
beginning of this section that given any matrix Y = Y (n) > 0 there exists
a unimodular matrix u such that Y[u] is reduced. In other words, for any
matrix Y > 0 there always exists an equivalent reduced matrix R. Such
a matrix R is by no means unique. However, the number of matrices R
equivalent to a given matrix Y − Y (n) > 0 is finite and this number is 34
bounded by an integer which depends only on n. It is our aim now to
establish this result.
Given any quadratic form Y[ξ], by the method of completion of
squares, it can always be rewritten uniquely as
Y[ξ] = d1 (x1 + b12 x2 + . . . b1n ×n )2 +
38
+d2 (x2 + b23 x3 + . . . b2n xn )2
+.........
+dn x2n
and, Y > 0 if and only if the di′ s are positive. This shows that any matrix
Y > 0 has a unique representation in the form
γ = D[B]
(56)
where D is a diagonal matrix (δµν dµν ) with the diagonal elements dνν ≡
dν , all positive; and B is a matrix (bµν ) with bµµ = 1 and bµν = 0 for µ >
ν. A matrix B = (bµν ) whose elements satisfy the above conditions will
be referred to as a triangular matrix. Assume now that Y is a reduced
matrix R = (rµν ). From (56) we will have
rℓ = dℓ +
ℓ−1
X
dν b2νℓ , ℓ = 1, 2, . . . , n
ν=1
and | R | = d1 d2 . . . dn .
In view of (49), this implies that
n
1≤
rℓ Y rν
≤
< C1
dℓ ν=1 dν
(57)
and consequently, with the help of (47) we have for K ≤ ℓ
0<
35
rK
dK
< C1 ≤ C1
dℓ
rℓ
(58)
We use these to prove that in the case of reduced matrices Y represented in the form (56), the b′µν s have an upper bound depending only
on n. The proof is by induction on u. Assume then that ±bPℓ < C9 .
C9 = C9 (n) for p = 1, 2, . . . K − 1 and ℓ > p. By means of the relation
rKℓ = dK bKℓ +
K−1
X
p=1
d p b pK b pℓ ,
39
Our assumption will imply in view of (57), (58) that for ℓ > K,
±bKℓ ≤
K−1
X
dp 2 1
rK
+
C < C1 + (n − 1)C1 C92 = C10 (n).
2dK p=1 dK 9 2
Thus assuming the result for p = 1, 2, . . . K − 1, ℓ > p, we have
established it for p = K, ℓ > p and by the principle of induction, this
completes the proof. We have now proved
Lemma 5. Let D = (δµν dµν ) be a diagonal matrix and B = (bµν ) a
triangular matrix such that γ = D[B] > 0 is reduced. Then
dν < C11 dν+1 , ν = 1, 2, . . . n − 1,
(59)
±bµν<C11 , µ < ν, C11 = C11 (n),
The possible converse to this is false, viz. if D∗ , B∗ be two other
matrices whose elements satisfy (59) (with the same constant C11 ) and
D∗ is diagonal while B∗ is triangular, we cannot conclude that R∗ =
D∗ [B∗ ] is a reduced matrix. In this direction, however, we have
Lemma 6.
∗
∗
≡ dµ∗ > 0; B∗ = (b∗µν )
), dνµ
I f D∗ = (δµν dµν
a triangular matrix and G integral matrix with | G |, 0 such that
D∗ [B∗G] is reduced, then the elements of G all lie between two bounds
which depend only on µ and n − µ being a common upper bound for the
d∗ ν
, ν = 1, 2, . . . , n − 1, bµν (ν > µ) and | G | and
absolute values of ∗
d ν∗+ 1 ∗
n being the order of D or b .
Proof. We again resort to induction, this time on it. For n = 1 the lemma
is clearly true. Since D∗ [B∗G] is reduced have by (56), D∗ [B∗G] = D[B] 36
for some diagonal matrix D and triangular matrix B. Let G = (gµν ),
−1
B∗GB−1 = Q = (qµν ), B∗ = (βµν ) Then we have D∗ [Q] = D and
P
D[Q−1 ] = D∗ and therefore dℓ = nK=1 dK∗ qzKℓ (ℓ = 1, 2, . . . , n) Consequently,
40
dK∗ q2Kℓ ≤ dℓ , K, ℓ = 1, 2, . . . n.
(60)
−1
Since G = B∗ QB and B, B∗ are triangular matrices, we have
gKℓ =
n X
ℓ
X
βKλ qλλ ℓλℓ ; K, ℓ = 1, 2, . . . n
λ=K λ=1
Hence
dK∗ gzKℓ = dK∗ (
n
n X
X
βKλ qλλ ℓλℓ )2
λ=K λ=1
ℓ′∗ s
By assumption the
are bounded, by µ so that the β′ s which are
′∗
rational functions of the ℓ s are also bounded, and by Lemma 5, the ℓ′ s
are bounded, in both the cases the bound depending only on µ and n.
We can therefore write
dK∗ g2Kℓ < µ∗ (
ℓ
n X
X
qχλ )2 dK∗ , µ∗ = µ∗ (n)
χ=K λ=ℓ
X
< µ ( qχλ qχ′ λ′ dK∗ ), χχ′ ≥ K
∗
Majorising qχλ qχ′ λ′ by (q2χλ +q2χ′ λ′ ) and q2χλ dK∗ (χ ≥ K) by q2χλ dχ∗ , with
P
the aid of (59), we get dK2 g2Kℓ < µ∗1 (q2χλ dχ∗ + q2χ′ λ′ dχ∗ ′ ) and using (60),
we have finally,
dK∗ g2Kℓ < µ1 dℓ Kℓ = 1, 2, . . . N
(61)
37
(µ1 and the µ′ν s that occur subsequently in the course of the proof are
positive constants depending only µ and R).
Replacing the equation xxxxx D[Q−1 ] = D∗ and repeating the earlier arguments we have
2
(62)
< µ2 dℓ∗
dK fKℓ
where ( fµν ) = G−1 .
Since | ( fµν ) |=| G−1 |, 0 there exists a permutation of the indices
n
Q
1, 2, . . . into ℓ1 , ℓ2 . . . , ℓn with
fKℓK , 0 Since | G | fKℓK is an integer,
K=1
its absolute value is at least 1. Hence 1/ fKℓK is bounded by the absolute
41
value of | G | and a fortiori by µ. Consequently, from (62) we have
dK < µ3 dℓ∗K , K = 1 · · · n. Among the (n − K + 1) indices ℓK , ℓK+1 . . . ℓn ,
there should be at least one not exceeding K so that
min(dK , dK+1 , . . . dn ) < µ max(d1∗ , d2∗ , . . . dK∗ )
′
and hence, by means of (59) and the analogous assumption on the d∗ s
We have dK < µ4 dK∗ , K = 1, 2 . . . n. The relation (61) now allows us to
conclude that
(63)
dK g2Kℓ < µ5 dℓ , K, ℓ = 1, 2, . . . n
Let p denote the largest number among 1, 2, . . . n such that the relation dK ≥ µ5 dℓ holds for K = p, p + 1, . . . n and ℓ = 1, 2, . . . p − 1. For
each g among p + 1, p + 2, . . . n there exists then a K = K(g) ≥ q and an
ℓ = ℓg < g with dK < µ5 dℓ ( with the appropriate interpretation for the
border cases p = 1, n). In view of (59) we have then
dg < µ6 dg−1 , g = p + 1, p + 1, . . . n
(64)
(59) and (64) together imply that dℓ | dK is a bounded quotient for
ℓ, K = p, p + 1, . . . n.
i.e.
dℓ < µ7 dλ
and consequently, from (63).
g2Kℓ < µ5 µ4 , ℓ = p, p + 1, . . . , n
By choice of p, dK ≥ µ5 dℓ , ℓ = 1, 2, . . . , p − 1 and K = p, p + 1, . . . n. 38
Also gKℓ is an integer for each K, ℓ. Hence (63) can hold only if gKℓ = 0
for these indices, viz. K = p, p + 1, . . . n; ℓ = 1, 2, . . . p − 1.
Thus G is a a matrix of the type
!
G1 G12
G=
0 G2
(n−p+1)
where the elements of G2 = G2
are bounded by a µ8 . In case p = 1,
the proof of Lemma 6 is already complete. In the alternative case, we
write, analogous to G,
!
!
D1 0
D∗1 0
∗
D=
,
D =
0 D2
0 D∗2
42
B∗ B∗12
B = 1
0 B∗2
!
B1 0
B=
,
0 B12
∗
!
and obtain
D∗1 [B∗1G1 ] = D1 [B1 ]
(65)
by equating
!
!
D∗1 [B∗1G1 ] ∗
D1 [B1 ] ∗
D [B G] =
and D[B] =
∗
∗
∗
∗
∗
∗
By assumption D∗ [B∗G] is reduced so that in particular D∗1 [B∗1G1 ] is
reduced. It is now immediate that our assumptions on D∗ , B∗ and G are
also true of D∗1 , B∗1 and G1 , so that by induction assumption the elements
of G1 are all bound. What remains then to complete the proof is only to
show that the elements of G12 are bounded.
From the matrix relation
′
G′1 D∗1 [B∗1 ]G12 + G′1 B∗1 D∗1 B∗12G2
′
39
′−1
with the help of (65), we get G12 = G1 B∗1 B12 − B∗1 B∗12G1 ,
As the elements of all the matrices occurring on the right side are
bounded, the same is true of the elements of G12 , and in all these cases
the bound depends only on µ and n. The proof is now complete. Lemmas 5 and 6 now yield
Lemma 7. If Y = Y (n) > 0 is a reduced matrix and U, a unimodular
matrix such that Y[U] is also reduced, then the elements of U all lie
between bounds which depend only on n.
As an immediate consequence, we infer that, equivalent to a given
matrix Y > 0 there exists only a finite number of reduced matrices, and
further, this number is bounded by a constant which depends only upon
n.
We now proceed to determine the structure of the space R of the
reduced matrices and its relationship to the space p of all real positive
symmetric matrices. We first observe that p is an open domain in the
space Y of all real symmetric matrices. For if GK ∈ Y , GK < PK =
43
1, 2, . . ., and if GK → G ∈ Y , then for each K, there exists a vector
E which may be supposed to be of length 1 with GK [EK ] ≤ 0. Then a
subsequence EνK clearly converges to a vector E , again of length 1, and
G[E ] = lim GνK [E νK ] ≤ 0
K
whence G < P. This certainly implies that P is an open do main. Let
now G be a boundary point of P so that G < P. Then there exists a
sequence yK ∈ P with yK → G. Since γK [E ] > 0 for every E , 0, we
get G[E ] ≥ 0 for E , 0. Also G[E ] = 0 for at least one E , as otherwise
G will belong to P. Every symmetric matrix G with this property, viz.,
G[E ] ≥ 0 for every E , 0, the equality holding for at least one such E ,
shall be called semi positive. We can now state that any boundary point
of P is semi positive. Conversely too, every semi positive matrix G is a 40
boundary point of P, since G+ ∈ E lies in P for every ∈> 0 but G < P.
In P, we now consider the domain R of all reduce γ > 0. Let R = (Rµν )
be a point of R. This means by Lemma 3 that
R[YK ] ≥ rKK ≡, nKK+2 ≥ 0
(46)′
for any integral YK whose last n − K + 1 elements are coprime, K =
1, 2, . . . n − 1.
1
Interpreting the n(n+1) independent elements of R as the cartesian
2
coordinates in the Euclidean space of the same dimension, the above
inequalities define a cone with the apex at the origin. We shall show that
much more is true.
Let Ro be a boundary point of R. Then either Ro ∈ P in which case
it is positive or R0 ∈ BdP in which case it is semi positive. Consider first
the case when R0 is positive. Then it satisfies (46) and also there exists
a sequence yK > 0 in Y such that yK < K and yK → Ro . We represent
Ro according to (56) in the form Ro = D[B] with a diagonal matrix D =
(dµν dµν ) and a triangular matrix E = (bµν ). The transformation (rµν ) →
(dν , bµν ) defines a topological mapping of a neighbourhood of Ro on to a
neighbourhood of (dν , bµν ) (where, in the indices for b, we assume µ < ν.
We represent yK in the same way, viz. yK = DK [BK ], and in view of the
topological character of the above mapping, we conclude that DK → D
44
41
and EK → B. Since D[P] is reduced, the elements of D, B satisfy (59)
and the same is therefore true of the elements of DK , BK for sufficiently
large K. Let uK be unimodular such that yK [uK ] is reduced. yK < R
by assumption so that u , K. Lemma 6 then shows that for sufficiently
large K, dK belongs to a finite set of matrices and in particular, there exist
an infinity of K′ s for which the UK′ s are the same, say, UK = U , ±E.
As K → ∞ through this sequence of values, we have
Ro [U] = lim γK [U] ∈ R
K
since each yK [U] ∈ R and R is easily seen to be a closed set. Thus
what we have shown is that if Ro is a positive boundary point of R,
there exists a unimodular matrix u , ±E such that
Ro [U] ∈ K.
(66)
Indeed Ro [U] is a boundary point of R and this, we proceed to :
establish. More generally we show that if Ro , R1 ∈ R and K1 = Ko [U]
for some unimodular matrix U , ±E, then both Ro and R1 are boundary
points of R.
Assume first that U is not a diagonal matrix. Let Y1 , Y2 , . . . Yn
be the columns of U and let YK be the first column which is different
form ±nK (If no such YK exists, then U would be diagonal, contrary
to assumption). Then the Kth column YK of U −1 has also the same
property. Let us write Ro = (rµν ), R1 = (Sµν ). Then we have
SK = Ro [Y ] ≥= R1 [YK ] ≥ SK
Hence the equality holds throughout and
Ro [YK ] = rK = SK = R1 [SK ]
If Ro is an interior point of R, then the strict inequality must hold
in (46) for all YK . As we have shown the equality to hold for one YK it
follows that Ro ∈ BdR. The same is of course true of R1 too. We stop
here to make the following remark.
From Lemma 7, it is clear that YK and SK belong to a finite set of
primitive vectors. So (66) allows us to conclude that from the infinite
45
42
set of inequalities ((46)′ ) defining R, it is possible to determine a finite
subset such that at least one of this finite set of inequalities reduces to an
equality in the case of any positive boundary point of R. In other words,
the positive boundary points of R all lie on finite number of planes and
these planes bound a convex pyramid E containing R. Of course we
have still the case when U is a diagonal matrix to settle, to complete
the proof. In this case all diagonal elements are ±1, the sign changing
at least once. Let then the sign change for the first time from the qth to
(q + 1)th element. By changing U to −U if necessary, we shall have
′
Sqq+1 = Yq+1
R0 Yq = x′q+1 R0 xq = −rqq+1 .
But due to one of the reduction in equalities rqq+1 ≥ 0, Sqq+1 ≥ 0.
Hence it follows that
rqq+1 = 0 = Sqq+1
(67)
It is immediate that Ro and R1 are again boundary points of R and
one of the inequalities in (46) reduces to an equality. Now our earlier
remark is unreservedly valid.
We now show the interior of R is non void. Consider a compact
set L ⊂ P with non null interior. Then its interior is of the highest
dimension, viz. n(n + 1)/2. Let y ∈ L . Then by (56) we can represent
y in the form y = D[B] and then, since L is compact the ratios dν | dν+1
and ±bµν are bounded by µ ( say ), µ depending only on L . So now,
if we determine a unimodular U such that y[U] is reduced, then by
Lemma 6, the elements of U are all bounded, the bound depending
only on L and n : in other words, even though the y′ s belonging to L
may be infinite, there are only a finite number of U ′ s such that y[U] is
reduced. If R has no interior points, then the y[U]′ s are all boundary
points of R and hence belong to a finite set of planes. This would then 43
mean that a finite number of planes is mapped by a finite set of U ′ s on
to a set of dimension n(n + 1)/2 which is clearly impossible. Hence we
conclude that R has interior points.
Let now R be an interior point of R. We consider the segment
(1 − λ)T + λR, 0 ≤ λ ≤ 1 where T < E but T ∈ R. Our aim is to
show that such a T is semi positive. Since E is a convex set, all the
46
points on the above segment are points of E and all, except possibly
T , are interior points of E . On this line there exists a boundary point
Ro of R. Ro cannot be positive as otherwise Ro would lie on one of
the planes which bound E so that it is a boundary point of E too, and
consequently Ro = T ∈ R - a contradiction to the choice of T . Hence
R0 is semi positive. The characteristic roots of a semi positive matrix
are all ≥ 0 and one at least among them is zero. Hence | Ro |= o so that
o ) and approaching R through a sequence of matrices
writing Ro = (rµν
o
R = (rµν ) ∈ R we obtain, in view of the reduction conditions, viz.
± 2r1ν ≤ r1 ≤ r2 ≤ . . . ≤ rn ,
r1 r2 . . . rn < C1 | R |→ C1 | R0 |= 0
o = 0, ν = 1, 2, . . . n
that r1ν
But Ro = (1− λo T )+ λo R for some λ0 , 0 ≤ λ0 ≤ 1; we have therefore
in particular
0 = (1.λo )t1ν + λ0 r1ν , ν = 1, 2, . . . n
43
We may consider the above
as linear equations in the variable λ so
t
r1ν
r
1ν
1ν
that the determinant = 0 or b1ν = tµ ν = 2, . . . n.
t11 r11 r11
But Γ is a given point and R is an arbitrary point in the segment. We
therefore conclude that t11 = 0 = t1ν and then from an earlier equation
λ0 = 0. Thus T = R and hence is semi positive. We have therefore
shown that ε − R consists of semi positive boundary points of R. Conversely, any such point lies in ε − R since an arbitrary neighborhood of
this point includes exterior points of R. Now we obtain the main result
of Minkowski’s reduction theory, viz.
Theorem 2. The space R of the reduced matrices y > 0 is a convex
pyramid with the vertex at the origin. If U runs over all unimodular
matrices, wherein −U will not be considered as different from U, then
the images of R under the group of mappings y → y[U] cover the
space P of all symmetric matrices y < 0 without over lapping (except
for boundary points). In other words, R is a fundamental domain in P
for the group of mappings Y → Y[U] acting on it.
Further, R has a non void interior and its boundary lies on a finite
set of planes.
Chapter 4
The Fundamental Domain of
the Modular Group of
Degree n
The tools now at our disposal enable us to construct a fundamental do- 44
main in the generalised upper half plane for the modular group acting
on it. We need a few preliminaries.
We first prove that given any point Z ∈ Y there exists only a finite
number of classes {C, D} of coprime symmetric pairs C, D for which the
absolute value of | CZ + D |, written as || CZ + D ||, has a given upper
bound K, i.e. such that || CZ + D ||≤ K. Let r denote the rank of C.
We employ the parametric representation of (C, D) given by Lemma 1.
Then
!
!
C1 0
01 0
′
C = U1
U , D = U1
U2−1
(68)
0 0 2
0 E
where U2 = (QR), Q = Q(n,r)
We have then
!
!
e1 0 ′
D1 0
CZ + D = U1
U2−1
U ZU2 +
0 0 2
0 E
!
!
Z[Q] Q′ ZR
Q′
′
U2 ZU2 = ′ Z(QR) = ′
R
R ZQ Z[R]
47
and
48
4. The Fundamental Domain of the Modular Group of Degree n
!
C1 Z[Q] + D1 ∗ −1
Thus CZ + D = U1
u
0
E 2
and
| CZ + D | =| U1 || U2 |−1 | C1 Z[Q] + D1 |= ± | C1 Z[Q] + D1 |
= ± | C1 || Z[Q] + P | where P = C1−1 D1
We shall subsequently need this relation
| CZ + D |= ± | C1 | Z[Q] + P |
45
(69)
We may observe that P is a rational symmetric matrix which uniquely determines and is uniquely determined by the class {C1 , D1 }. For
if C0−1 D0 = C1−1 D1 = P, then D0C1′ = C0 D′1 which means that {C0 , D0 }
= {C1 , D1 }, while if A, B is any pair in the class {C1 , D1 }, then A =
UC1 , B = UD1 , for some unimodular U, and A−1 B = C1−1 D1 = P.
What is more, while P = C1−1 D1 is rational symmetric for any coprime
symmetric pair (C1 , D1 ), conversely too, any rational symmetric matrix
P is of the above form. We need to only observe that in this case, we
can choose unimodular
a matrices U3 , U4 such that
ν
U3 PU4 = δµν
, viz. a diagonal matrix with the diagonal elebν
ments aν/bν , ν = 1, 2, . . . n where aν , bν are coprime integers with ℓν > O
and then, defining C1 = (δµν bν )U3−1 , D1 = (δµν aν )U4 , we have P =
C −1 D1 .
We now return to the representation (68) of C, D. Let X, Y denote
the real and imaginary parts of Z. Since Q in the above representation
can always be replaced by QU3 , U3 -unimodular, while preserving the
form of (68), we can assume that Y[Q] is reduced. Let then


S = X[Q] + P

(70)



T = Y[Q]
and lat F (r) a real matrix with T [F] = E, S [F] = H = (δµν bν ). Then
(S + iT )[F] = H + iE and (S + iT ) = (H + iE)[F −1 ]
Now |Z[Q] + P| = |S + iT | = |T |
r
Y
(i + hν )
ν=1
and
49
||Z[Q] + P||2 = |T |2
r
Y
(l + h2ν ).
ν=1
||CZ + D||2 = |C1 |2 |T |2
r
Y
(1 + h2ν )
(71)
ν=1
By assumption ||CZ + D|| ≤ K
; also, C1 being integral, |C1 | > 1,
2
and trivially 1 + hν > 1. (71) now implies that |T | is bounded. But T is
reduced so that one of the inequalities (47-49) will imply that
r
Q
Y[Y ] < Q1 |T | where we denote Q = Y1 Y2 Yn ) Since each of the
ν=1
factor Y[Yν ] has a positive lower bound, vis Y[Y ν] ≥ λYν′ Yν ≥ λ > o 46
where λ denotes the smallest characteristic root of Y, we conclude from
the above that each factor Y[Yν ] is bounded. As a consequence, the
Yν′ S belong to a finite set of primitive columns and there are only a
finite number of possible choices for Q. In particular, therefore, the
elements of T = Y[Q] are bounded and since T = F ′−1 F −1 the same is
true of the elements of F too. Also, the number of distinct T ′ s being
finite, |T | has a positive lower bound and being an integer, |C1 |2 ≥ 1 so
that we infer from (71) that the h′ν s and what is the same, the matrix H
are bounded. But
S + iT = (H + iE)[Γ−1 ] and we already know that F −1 is bounded.
Hence S + iT = Z[Q] + P is bounded and so also P = S − ×[Q] Since
|C1 |P is integral and |C1 | is bounded as is seen from the relation
|C1 |2 |T |2
r
Y
(1 + h2ν ) < K
ν=1
it follows that the number of distinct P′ s occurring here is finite. We
know however that distinct classes {C1 , D1 } correspond to distinct P′ s
so that the number of distinct classes {C1 , D1 } occurring are finite. As
we have shown already that the number of classes {Q} is also finite we
conclude that the number of classes {C, D} in our discussion, viz. those
which satisfy ||CZ + D|| < K for a given K > O is finite and this was
what we were after.
Let Y, Y ∗ be the imaginary parts of two points Z, Z ∗ ∈ Y which
are equivalent with regard to M. We call Z ∗ higher than Z if we have
50
|Y ∗ | > |Y|. If Z ∗ = (AZ + B)(CZ + D)−1 , then we have from (24) that
Y ∗ = (ZC ′ + D′ )−1 Y(C Z̄ + D)−1 so that
47
−2
|Y ∗ | = |Y| CZ + D
(72)
Thus Z ∗ is higher than Z if and only if ||CZ + D|| ≥ 1 where we
assume Z ∗ = (AZ + B)(CZ + D)−1 As a consequence we have that in
each class of equivalent points there exist at least one highest point. For,
in the alternative case, there would exist a sequence of equivalent points
ZK ,
!
A B
ZK = XK + iYK = (AK Z1 + BK )(CK Z1 + DK )−1 K K ∈ M
C K DK
such that |Y1 | < |Y2 | < . . . ad inf. and this will imply in its turn by (72)
that
1 > ||CZ Z1 + D2 || > ||C3 Z1 + D3 || > . . . ad inf. In other words we end
up with the conclusion that an infinite number of classes {CK , DK } have
the property that ||CK Z1 + DK || < 1 a contradiction to an earlier result.
Now we state
Theorem 3. The domain f defined by the following inequalities represents a fundamental domain in Y with regard to M. We denote
Z = x + iy, X = (xµν ), Y = (yµν ) and then Z ∈ F is defined by
(i) ||CZ + D|| ≥ 1 for all coprime symmetric pairs C, D,

Y[yK ] ≥ yKK ,


 (Minkowski’s reduction conditions) where yK
(ii) yKK+1 ≥ 0,

K = 1, 2, . . .
is an arbitrary integral column with its last n − k + 1 elements
coprime.
1
1
≤ xkℓ ≤ , K, ℓ = 1, 2, . . . n
2
2
Also F is a connected and closed set which is bounded by a finite
number of algebraic surfaces.
(iii) −
51
48
Proof. We first show that given any point Z ∈ Y , F contains an equivalent point Z1 = M < Z >. In fact, since we know that any class of
equivalent points contains a highest point, we can assume that Z is a
highest point and then ||CZ +!D|| ≥ 1 for all coprime symmetric pairs C,
U ′ S U −1
where U is unimodular and S an integral
D. If now M =
0
U −1
symmetric matrix to be specified presently, we shall have
Z1 = Z[U] + S , Y1 = Y[U] = U ′ YU and X1 = X[U] + S where
we assume Z1 = X1 + iY1 . We can determine U such that Y1 = Y[U]
is reduced and then S can be chosen such that X1 satisfies the last of the
three conditions in the theorem. The first condition is then automatically
ensured as |Y1 | = |Y| so that Z1 is a highest point as Z is. The Z1 thus
determined clearly serves.
Further, no two distinct interior points of F can
! be equivalent. For
A B
let Z, Z1 ∈ Int.F and Z1 = M < Z >, M =
. Then, in the condiC D
tions stipulated in theorem 3 for Z, we have strict inequality throughout
except in the cases where these reduce to identities in Z, viz. when
YK = ±nK and (C, D) = (0, U −1 ), U being unimodular. Now Z1 =
(AZ + B)(CZ + D)−1 so that
(−C ′ Z1 + A′ )(CZ + D) = E
(73)
Since Z ∈ F and (C, D), (−C ′ , A′ ) are coprime symmetric, we have
||CZ + D|| ≥ 1 and || − C ′ Z1 + A′ || ≥ 1. Since their product is equal to 1
by (73), we conclude that the equality holds in both cases. It therefore
follows that (C, D) is one among the exceptional pairs singled out earlier, 49
viz. C = o, D = U −1 for some unimodular matrix U. Then from (73)
we have

Z1 = Z[U] + S 


(74)
′−1
′

A=D =U
where we write S = BU. In particular, writing Z1 = X1 + iY1 we have
X1 = X[U] + S , Y1 = Y[U]. Since Y and Y1 are both reduced matrices
and Y is an interior point of R, theorem 2 states that U = ±E. Then
X1 = X + S and condition (iii) of theorem 3 will require that S = O.
52
50
Thus Y1 = Y and X1 = X and consequently Z1 = Z. We have therefore
shown that two points of F can be equivalent only if both are boundary
points of F , a property typical of a fundamental domain and this settles
the first part of theorem 3.
We now show that F is a closed set. Let ZK ∈ F , K = 1, 2, . . . , and
ZK → Z. Clearly the conditions (i) and (iii) are satisfied by Z as they
are true of each ZK . Condition (ii) also would be fulfilled by Z except
that we do not know from the fact YK > o for each R, that Y > o.
We shall show that this is so. More generally we show that |Y| has a
positive lower bound for all Y such that Z = X + iY ∈ F . If (C1(r) , D(r)
1 )
(n,r)
is any coprime pair with |C1 | , o and Q , a primitive matrix, then the
pair (C, D) given by (68) is always symmetric and coprime so that for
(69) we have ||CZ + D|| = ||C1 Z[Q]
! + D1 || ≥ 1. Choosing in particular
(r)
E
, we have ||CZ + D|| = ||Zr ||, Zr being
C1 = E (r) , D1 = o and Q =
0
the matrix which results when Z is deprived of its last (n − r) rows and
columns and this is true for each r. If now ZǫF , then ||CZ + D|| ≥ 1
by one of the condition of F so that we can conclude that ||Zr || ≥ 1
for each r and every ZǫF . In particular, setting r = 1, this means that
||Z1 || = |Z11 | ≥ 1 which in its turn implies by reason of |X11 |. being less
1
than or equal to that
2
1√
y11 ≥
3
(75)
2
where x11 + iy11 = Z11
In view of the reduction conditions (47 - 49) we have
n
Q
yn11 ≤
yrr < C1 |Y| and the above inequality now implies the der=1
sired result. Specifically we have
|Y| > C1−1 (
1√ n
3) > o
2
(76)
for every Z = X + iy ∈ F
The proof of the connectedness of F is a bit more involved.
let (C, D) be a given coprime symmetric pair and use the represen-
53
tation (69) for C, D. We recall the following notation.
P = C1−1 D1 , S = X[Q] + P, T = Y[Q], T [F] = E, S [F] = R = (δµν hν )
We further introduce Z1 = X + iλY with λ ≥ 1. Then we have, as in
deriving (71),
Z[Q] + C1−1 D1 = (H + iE)[F −1 ], Z1 [Q] + C1−1 D1 = (H + iλE)[F −1 ],
r
Q
||CZ + D||2 = |C1 |2 |T |2 (1 + hν )2 and
ν=1
r
Y
||CZ1 + D|| = |C1 | |T |
(λ2 + h2ν )
2
2
2
(71)′
ν=1
51
It is immediate that ||CZ1 + D|| ≥ ||CZ + D|| and that Z ∈ F implies
that Z1 ∈ F for λ ≥ 1, Z1 = X + iλY.
We now ask how to choose λ such that Z1 = X + iλY ∈ F for two
matrices X, Y which satisfy the condition (iii) and condition (ii) respectively of theorem 3, and we show that this will be fulfilled if λ is chosen
sufficiently large-specifically if
λ≥
C1
, C1 = C1 (n)
y11
(77)
Let λ satisfy the above condition (77). Then from ((71)′ ) we have
||CZ1 + D||2 ≥ λ2r |T |2 so that
||CZ1 + D|| > λr |T | = λr |Y[Q]|
As in earlier contexts we can assume that T = Y[Q] is reduced
by an appropriate choice of Q. Then, in view of (47 - 49), |T | ≥
r
Q
C1−1
Y[yν ] ≥ C1−1 yr11 ≥ λ−r where we denote Q = (y1 y2 . . . yr ) and
ν=1
assume without loss of generality that C1 > 1. Hence ||CZ1 + D|| ≥
λr |T | ≥ 1 for all symmetric coprime pairs (C, D) and this is precisely
what ensures us that Z1 ∈ F
Let now Zν ∈ F , Zν = Xν + iYν , ν = 1, 2, and let λo ≥ 2C1/ √3 join
Z1 and Z2 by the polygon consisting of
Z = X1 + iλY1 , 1 ≤ λ ≤ λo ,
(78)
54
Z = (1 − λ)(X1 + iλ0 Y1 ) + λ(X2 + iλo Y2 ), o ≤ λ ≤ 1,
Z = X2 + iλY2 , i ≤ λ ≤ λo
(79)
(80)
52
We prove that this polygon lies completely in F . The result we
proved above and (75) together imply that the lines defined by (78) and
(80) both lie in F . It remains then to consider only the line determined
by (79). Since y1 , y2 are reduced, and R is convex, (1 − λ)Y1 + λY2 ∈ R
for 0 ≤ λ ≤ 1. Also since x1 , x2 satisfy the condition (iii) of theorem 3,
so does (1 − λ)x1 + λx2 , o ≤ λ ≤ 1. Hence, in view of our earlier result,
Z = {(1 − λ)x1 + λx2 } + iλo {(1 − λ)y1 + λy2 } belongs to F provided λo
satisfies (77). We shall show that it does. This is in fact immediate since
2C1
C1
λo ≥ √ =
√
√
3
(1 − λ) 23 + λ 23
≥C I /(1 − λ)y′11 + λy′′
11
It follows therefore that the points Z in (79) all belong to F . We can
now conclude that F is a connected set.
Our assertion concerning the boundary of F still remains to be settled. Specifically, we have got to show that the boundary of F consists
of a finite number of algebraic surfaces. First we note that every positive
matrix Y satisfies the inequality
|y| ≤ y11 y22 ..ynn
53
(81)
where we assume y = (yµν )
For we can write y = R′ K with a non-singular real matrix K =
(W1 W2 · · · Wn ) Then it is known that
2
|K| ≤
n
Y
ν=1
Wν′ Wν
=
n
Y
yνν
ν=1
But |K|2 = |Y| and this proves what is desired. We proceed to determine a lower bound for the smallest characteristic root λ of a positive
reduced matrix Y. Let λ1 , λ2 , · · · , λn denote the characteristic roots of
55
Y and let Yν denote the matrix which arises from Y by deleting its νth
row and column. It is known then that the characteristic roots of Y−1
are λ−1
ν , ν = 1, 2, . . . , n. Denoting by σ(A), the trace of a square matrix
A, we have from (49), (81) and (47)
1
1
1
1
1
= max
≤
+
+ ···
= σ(Y −1 )
ν λν
λ
λ1 λ2
λν
n
n
X
X
y11 y22 . . . yν−1 yν−1 · yν+1 yν+1 . . . ynn
|yν |
≤ CI
=
|y|
y11 · y22 . . . ynn
ν=I
ν=1
n
X
1
nC1
= C1
≤
y
y11
ν=1 νν
(82)
Thus if y > o is reduced and y = (yµν ) and if λ be the minimum of
the characteristic roots of y, then
λ ≥ y11 /nC1
(82)
1√
3 from (75) so that in this
If now Z = X + iY ∈ F then y11 ≥
2
case
√
(83)
λ ≥ 3/2nC1 , C1 = C1 (n)
Let Z ∈ Bd · F and let ZK → Z, ZK < F . To each ZK we can 54
determine an equivalent point WK ǫF say, WK = (AK ZK + BK )(CK ZK +
D−1
K ). Then
!
!
E o
AK BK
,±
C K DK
o E
Case. i: Let CK , O for an infinity of indices. By passing to a subsequence, we may assume this to be true for all CK′ s. Also, since their
ranks belong to the finite set 1, 2, . . . n, we can assume all CK , s to be of
the same rank r > o. To each class {CK , DK }, we determine the corre(n,r) } with |C | , 0, given by Lemma
sponding classes {Co(r) , D(r)
o
o } and {Q
1. Both classes {Co , Do } and {Q} depend on K though the notation is not
suggestive. Then from (70) and (71) we have
||CK ZK + DK ||2 = ||Co ZK [Q] + Do ||2
56
= |Co |2 |T |2
r
Y
(1 + h2ν )
(84)
ν=1
where T = yK [Q], S = xK [Q] + Co−1 Do ,
xK + iyK = ZK , T [F] = E
and S [F] = H = (δµν hν )
As usual we assume that T is reduced. Let y1 , y2 , . . . yr be the columns of C. Then T = (Uν′ YK Uν ) so that by (49),
r
Y
ν=1
YK [Uν ] < C1 |⊤|
(85)
As is well known, the smallest characteristic root λ(K) of yK is given
55
by
λ(K) = min
YK [E]
′
E E=1
(E − real column ).
Since λ(K) is a continuous function of YK and YK → Y we have
lim λ(K) = λ the smallest characteristic root of Y, and further λ ≥
√K
3/2nC1 , from (83), so that
√
(86)
λ(K) ≥ 3/4nc1
for sufficiently large k. We shall be concerned only with these k′ s in the
rest of the proof. Then we have
√
YK [Uν ] ≥ λ(K) U ′ ν Uν ≥ λ(K) ≥ 3/4nC1
and therefore from (85),
|⊤| >
√3 r
4n
C1−r−1 > o
Clearly WK satisfies the relation
(−CK′ WK + A′K )(CK ZK + DK ) = E
(87)
57
and since WK ∈ F we have || − CK′ WK + A′K || ≥ 1
We therefore conclude that ||CK ZK + DK || ≤ 1 Hence
1 ≥ ||CK ZK + DK ||2 = |Co |2 |⊤|2 π(1 − h2ν )
which implies that |Co | is bounded as |⊤| and (1 + h2ν ) have positive
lower bounds as a result of (87). By appealing to (84) and (87) we then
conclude that |Co |, |⊤|, h1 , h2 , . . . hr all lie between bounds which do not
depend upon K and Z. Since
√
3 ′
U ν Uν ≤ λ(K) U ′ ν Uν ≤ Y[Uν ]
4nc1
C1 |⊤|
4n
≤Q
≤ ( √ )r−1C1r |⊤|
Y
[U
]
µ
3
µ,ν K
the vector Uν is bounded. This proves that the matrix Q belongs to a 56
finite set of matrices which does not depend on Z. Since the diagonal
elements Uν YR Uν of T are bounded as a consequence of (84) and (87)
and T > 0, it follows that all the elements of T are bounded and this in its
turn, in view of the relations T = E[F −1 ]; S = H[F −1 ] implies that the
1
elements of F −1 and S are bounded. Now we observe that |xR |x(R)
µν | ≤ 2
1
and xR → x = (xµν ) so that |xµν | ≤ and consequently X is bounded.
2
But S = xR [Q]+Co−1 Do i.e.Co−1 = δ− xR [Q]. The above then shows that
P = Co−1 Do is bounded; in their words only a finite number of choices
exist for P. But we know that P determines the class {Co , Do } uniquely.
Hence the number of choices for {Co , Do } is also finite. As we have
already shown that the number of classes {Q} is finite, it follows that the
number of distinct classes {CR , DR } is finite; in other words the classes
{CR , DR } are equal for an infinity of k′ s.We shall denote this common
class by {C, D}. Then, taking the appropriate sequence of k′ s,
||CZ + D|| = lim ||CR ZR + DR || ≤ 1, as ||CR ZR + DR || ≤ 1
R
for each k.
Since Z ∈ f, we have the reverse inequality and the equality results.
Hence in this case Z ∈ Bd.f satisfies the equation
||CZ + D|| = 1
(88)
58
56
for some out of a finite number of paris (C, D) which are not mutually
equivalent.
Case ii: Suppose CK = O for all sufficiently large k′ s. Considering only
these k′ s, we have
WK = ZK [UK ] + S K
with a unimodular matrix UK and a symmetric integral matrix S K as in
(74). If VK be the imaginary part of WK , then YK [UK ] = VK ∈ R. We
wish to show that the number of [uK ]′ s is finite. Since the imaginary part
Y of Z belongs to R, representing Y in the form Y = D[B], D = (δµν dν ),
dν
and B = (bµν ), we have from (59) that
, bµν are all bounded by
dν+1
C11 = C11 (n). But YK → Y so that representing YK analogous to Y
(K)
). we have DK → D
as yK = DK [BK ], DK = (δµν dν(K) , and, BK = (ℓµν
(R)
(K)
).
and(ℓµν
and BK → B. We therefore deduce that the ratios dν(K) /dν+1
are bounded, say, by 2C11 . Lemma 6 now implies that there can be
only a finite number of UR′ s with yR [UR ] = DR [BR UR ] reduced, in
other words, for an infinity of k′ s, the UR′ are the same, say UR = U .
As R → ∞ through the sequence of these values, we have y[U ] =
limR yR [U ] and since each yR [U ] ∈ R and Y > 0, it follows that
y[U ] ∈ R. But we already know that y ∈ R and this, by theorem 2, is
possible only if both Y and Y[U ] belong to the boundary of R. Hence
y[UR ] = YR
57
(89)
where UR is the first column of U such that UR , ±H R provided
such a UR exists. This is therefore true if U is not a diagonal matrix.
Let now U be a diagonal matrix. Two cases arise, viz. either U ,
±E or U = ±E. In the former case it follows, as in (68) that
YRR+1 = 0
(90)
for some k. In the latter case, taking the k’s for which UR = U = ±E
we have WR = ZR + S R with S R , O as WR ∈ f while ZR < f. Then
the condition (i) of theorem 3 for the elements of f implies that
xµν = ±
1
2
(91)
59
for at least one pair (µ, ν).
Thus any point Z ∈ Bd, f satisfies one or the other of the system of
equations (88) - (91), in other words, Z satisfies one of a finite set of
equalities, and each one of them defines an algebraic surface. It therefore follows that a finite number of algebraic surfaces bound f and this
completes the proof of theorem 3.
We insert here the following remarks for future reference. A group
of symplectic matrices shall be called discrete if every infinite sequence
of different elements of the group diverges, or equivalently any compact
subset of it contains only finitely many distinct elements. The modular
group of degree n is then clearly discrete.
We shall call a group of symplectic substitutions discontinuous in 58
Y , if for every z ∈ Y , the set of images of z relative to the given group
has no limit point in Y . Concerning the modular group then, we have
Lemma 8. The modular group of degree n is discontinuous in Yn
The proof is by contradiction. Let z1 , z2 · · · be a sequence of equivalent points, all different, converging
! to a point z ∈ Y . Then ZR = MR <
A
BR
∈ Mn
Z1 > for some MR = R
C R DR
Let zR = xR + YR and Z = X + iY.
′ D−1 y (C z + D )2 by (24). If Y = R′ R this
Then YR = (Z̄1′ CR
R
1
R 1 R 1
gives YR = Ω̄′ Ω with Ω = R(CR Z1 + DR )−1 . Also y′R s are bounded
as YR → Y, and it follows that Ω is bounded. This in its turn implies
that (CR Z1 + DR )−1 is bounded. Also ||CR z1 + DR || is bounded as
||CR Z1 + DR ||2 = |Y1 |/|YR | from (72). We therefore infer that CR Z1 +
DR is bounded. Considering the real and imaginary part separately this
implies that CR and hence also DR are bounded and consequently also
AR and BR as the relation ZR (CR Z1 + DR ) = AR Z1 + BR shows. Thus
′ s are all bounded for K = 1, 2, . . . and therefore for an infinity
the MR
′
′ s are identical, and for these k′ s the Z ′ s are identical,
of k s the MR
R
contradicting our assumption. This proves the Lemma.
Chapter 5
Modular Forms of Degree n
The possible of developing a modular form into a Fourier series rests 59
upon a general theorem in complex function theory. First of all we have
to deal with the following facts.
Let E be the Gaussian plane. A subset K of direct product E n =
E × E × · · · × E (n times) shall be called a Reinhardt domain with zero
centre, when the following conditions are satisfied.
1) R is a domain in the sense of function theory, viz. an open connected
non empty set.
2) R is invariant under the group of transformations (Z1 , Z2 · · · Zn ) →
(Z1 eiϕ1 , Z2 eiϕ2 , . . . Z1 eiϕn )ϕν - real. We now stare
Lemma 9. Let R be a Reinhardt domain with (o, o · · · o) and
f (Z1 , Z2 , . . . , Zn )a functions regular in R. Then we can develop f into a
power series
P
ν1 ν2
νn
f (Z1 , Z2 , . . . Zn ) = ∞
ν1 ,ν2 ,·,νn =−∞ oν1 ν2 ···νn .0Zn Z2 · · · Zn
This representation is valid in the whole domain R and the power
series converges uniformly on the manifold |Zν | = rν , ν = 1, 2, . . . n when
(r1 , r2 , . . . , rn ) ∈ R.
Proof. We first prove the Lemma in the special case of the domain
R :|Zν |<ρν ,1≤ν≤h
61
5. Modular Forms of Degree n
62
o < σν < |Zν | < ρν , h < ν ≤ n.
60
Consider the contracted domain
|Zν | < ρ′ν < ρν (1 ≤ ν ≤ h)
o < σν < σ′ν < |Zν | < ρ′ν < ρν (h < ν ≤ n).
Let L1ν be the circle |Zν | < σ′ν , h < ν ≤ n assigned with the negative
sense of rotation and let L1ν be the circle |Zν | = ρ′ν (1 ≤ ν ≤ n) assigned
with the positive sense of rotation. Then we have by mean of the Cauchy
integral formula, f (Z1 , Z2 , . . . Zn )=
Z
Z Z
Z
f (ζ1 , . . . ζ1 )
1 n
···
···
)
ds1 · d
=(
2πi
(ζ1 − Z1 ) · (ζ1 − Zn )
L21
=
2
X
νh+1 ,...νn
61
L2h L2h+1
1 n
((
) )
2πi
=1
Z Z
L2
n+L2n
νh+1
Z
L21 L2h h+1
·
Zν,h
n
f (ζ1 , . . . ζ1 )
ds1 · dζ
(ζ1 − Z1 ) · (ζ1 − Zn )
By expanding the quotient 1/ρν − Zν as a convergent power series as
in the one variable case of each ν, a simple argument shows that every
integral in the above sum is expressible as a power series, a typical term
of which contains non negative powers of
±1 , Z ±1 , . . . Z ±1 where in Z ±1 (r ≥ h + 1) the upper
Z1 , Z2 · · · Zh , Zh+1
n
r
h+2
sign holds in cases where νr = 2 and the lower sign holds in cases
where νn = 1. Clearly, the coefficients of these power series do not
depend upon the choice of ρ1ν , σ1ν . The uniform convergence as stated in
the Lemma is immediate from a consideration of the derived series as in
the one variable case.
In the general case of any Reinhardt domain with center (0, 0, . . . 0)
by the result shown above, in every subset R′ ⊂ R of the type |Zνi | = ρi ,
1 ≤ i ≤ h,
0 < σν < |Zν | < ρν , ν , ν1 , ν2 · · · νh ,
there exists a representation of f (Z1 , Z2 , . . . Zn ) as a power series with the
desired properties. We need then only prove that two representations
in two subsets R′ 1 , R′ 2 are identical in their intersection and then the
63
representation in any two subsets R′ 1 , R1′′ are identical as we can always
find a sequence of such subsets, viz.
R′ 1 = R1 , R2 , . . . Rm = R′′
with Ri ⊂ Ri+1 non empty for any two indices i, i + 1. But for any two
such subsets Ri′ , R2′ their intersection R′ 2 ∩ R′ 2 is again a subset of the
same type and it is then immediate that the power series representing
f (Z1 , Z2 , . . . Zn ) in the two sets are identical. !Lemma 9 now follows.
A B
Let K be given integer and M =
, a symplectic matrix. We
C D
introduce the notation
f (Z)|M = f (M < Z >); CZ + DK
1
(92)
where f (Z) is an arbitrary function defined on Y . It is easy to see that
( f (Z)|M1 )|M2 = f (Z)|(M1 M2 ). We now fix the conception of a modular
form by the following.
Definition. A modular form of degree n ≥ 1 and weight K, is a function
f (Z) satisfying the following conditions.
1) f (Z) is defined in Y and is a regular function of the
pendent elements Zµν (µ ≤ ν) of Z.
n(n + 1)
inde- 62
2
2) f (Z)|M = f (Z) for M ∈ M
3) In this case n = 1, f (Z) is bounded in the fundamental domain F on
Y relative to M.
We shall show later that the last condition is a consequence of the
earlier conditions in the case n > 1.
!
E S
Considering a matrix M of the form M =
∈ M where S
O E
is an integral symmetric matrix, we infer from condition (2) above that
f (Z + S ) = f (Z) for an integral symmetric matrix S ; in other words a
modular form is a periodic function of period 1 in each of the variables.
Hence g(ζµν ) = f (e2piiZ µν) is a single valued regular function in the
64
domain into which Y is mapped by the mapping ζµν = f (e2piiZ µν viz.
the domain defined by the inequalities ζµν = ζνµ , O, (−ℓoc|ζµν |) > O. It
is easy to see that this is a Reinhard domain and then Lemma 9 assures us
of a power series representation for g(ζµν ) valid throughout this domain.
This power series can be looked upon as a Fourier series of f (Z) and can
be written in the form
X
f (Z) =
a(T )e2πiσ(T Z)
(93)
T
where T = (tµν ) runs over all rational symmetric matrices such that tµµ
and 2bµν (µ , ν) are integral. Such matrices are called semi integral. To
verify the above fact, we need only observe that
σ(T Z) =
n
X
tµρ Zρµ =
µ,ρ=1
63
n
X
tµµ Zµµ + 2
X
tµν Zµν
µ<ν
µ=1
so that
Pn
P
e2πiσ(T Z) = e2πi( ν=1 tµµ +2 µ<ν tµν Zµν )
n
Y
Y zt
t
=
ζµνµν
ζµνµν
µ=1
µ<ν
The condition (2) in the definition of a modular form yields in the
case of the modular substitution Z1 = Z[U], U-unimodular the transformation formula
f (Z1 ) = |U|−K f (Z).
(94)
Observing that the trace of a matrix product is invariant under a
cyclic change in the succession of the factors and in particular σ(AB) =
σ(BA) for any two matrices A, B, we obtain
X
f (Z1 ) =
a(T )e2πiσ(T Z[U])
=
=
T
X
T
X
T
′
a(T )e2πiσ(T U ZU)
′
a(T )e2πiσ(UT U Z)
65
=
X
a(T [U ′ −1 ])e2πiσ(T Z)
T
On the other hand,
|U ′ |−K f (Z) =
X
T
|U ′ |−K α(T )e2πiσ(T Z)
A comparison of coefficients by mean of (94) now yields
|U|K a(T [U ′ −1 ]) = a(T )
where U is an arbitrary unimodular matrix. Replacing U by U ′−1 we 64
get
a(T [U]) = |U|K a(T )
(95)
The special choice U = −E leads to
a(T )
(−1)nK a(T )
=
(96)
This proves that modular forms not vanishing identically can exist
only in the case nK ≡ 0(2). Also we deduce from (95) that for properly
unimodular matrices i.e. U such that |U| = 1,
a(T [U]) = a(T )
(97)
We apply these results to prove.
Lemma 10. A modular form f (Z) which is bounded in the fundamental
domain F of Y relative to M has a representation of the form
f (Z) =
X
a(T )e2πiσ(T Z)
(98)
T ≥O
and conversely any modular form representable by (98) and in fact any
series (98) which converges everywhere in Y is bounded in F .
1
We prove the direct part first. We shall denote by H the cube − ≤
2
Q
1
xµν ≤ (µ ≤ ν)
and put [dx] =
dxµν
2
µ≤ν
66
Since the Fourier series of f (Z) converge uniformly in H, we obtain
Z
Z
a(T ) =
· · · f (Z)e−2πiσ(T Z) [dx]
H
65
and consequently
−2πσ(T y)
a(T )e
=
Z
···
H
Z
f (Z)e−2πiσ(T x) [dx]
where Z = X + iY ∈ Y
Let yo be a reduced matrix such that σ(T yo ) < O for some fixed T .
By (77), if λ is chosen sufficiently large, the largeness depending only
on yo then Z = X + iλYo ∈ F for x ∈ H and y-reduced. With such a
choice of λ, the assumption | f (Z)| < C for Z ∈ F implies by means of
the relation
Z
Z
−2πγσ(T yo )
a(T )e
=
· · · f (Z)e−2πiσ(T yo ) [dx]
X
that
|a(T )|e−2πλσ(T yo ) ≤ C.ϑoℓH = C.
As λ → ∞ our assumption σ(T yo ) < 0 implies that
a(T ) = 0
(99)
Let now Y be an arbitrary positive matrix with σ(T y) < 0 and choose
a unimodular matrix U such that y = yo [U ′ ], yo ∈ R, the space of
reduced matrices. Then we have
0 > σ(T y) = σ(T yo [U ′ ]) = σ(T [U]yo )
and (99) and (95) now imply that
a(T [U]) = ±a(T ) = 0
for those T for which there exist positive matrices Y with σ(T y) < 0.
Hence it is sufficient that the matrix T in the Fourier series (93) runs
only over those matrices with the property σ(T y) ≥ 0 for all y > 0.
67
66
Let R = R(n) be any nonsingular real matrix, R = (µ1 , W2 , . . . Wn )
and let y = RR′ . Then Y > 0 and the above implies for the matrices T oc
P
curing in (93) that σ(T y) = σ(T [R]) = nν=1 T [Wν ] ≥ 0. This is true of
all non-singular real matrices R and σ(T [R]) is a continuous function of
R so that it is true of singular matrices too with real elements. In other
words T [ε] ≥ 0 for all real column E and what is the same, T ≥ 0. This
settles the first part of the Lemma.
Conversely we have to prove that the sum of a Fourier series (98)
which converges everywhere in Y is bounded in F . The matrices T
occurring in (98) are all semi integral matrices which are further positive. Let T = RR′ , R = (W1 , W2 · · · Wn )-real and let Z = X + iY ∈ F .
The
√ characteristic roots of y have by (83) a positive lower bound C =
3
, C1 = C1 (n). Hence
2nc1
σ(T y) = σ(y[R]) =
n
X
ν=1
y[Wν ] ≥ C
n
X
Wν′ Wν
ν=1
′
= Cσ(R R) = Cσ(T ).
iC
E
2
for all T , where C is a suitable
The convergence of the series (93) at the particular point Z =
implies that |a(T )|e−πCσ(T ) < C
constant. If now Z ∈ F then
|a(T )e−πCσ(T Z) | = |a(T )|e−πCσ(T y)
≤ |a(T )|e−πCσ(T )
≤ C e−πCσ(T )
and consequently,
67
| f (Z)| = |
X
a(T )e2πiσ(T Z) |
≤|
X
a(T )e2πσ(T y) |
≤
T ≥0
T ≥0
X
T ≥0
C e−πcσ(T )
68
The last sum is independent of Z and we will be through if only we
show that this series is convergent. The convergence of this series is a
consequence of the fact that the number of semi integral matrices T ≥ 0
whose trace (which is always an integer) is equal to a given integral
value t increases at most as a fixed power of t as t → ∞. For it σ(T ) = t,
T ≥ 0− semi integral, then writing T = (tµν ) we have tνν ≤ t for all
′ s together is
indices ν so that the number of possible choices for all tνν
n
atmost (t + 1) . Since T ≥ 0 we have tνν ≤ tµµ tνν ≤ (t + 1)2 so that
′ s for a given (µ, ν) consistent with
±2tµν ≤ 2(t + 1) and the number of tµν
this inequality is at most 4t + 1. It therefore follows that the number of
T ′ satisfying our requirements can be majorised by
(t + 1)n (4t + 1)n(n−1)/2 ≤ C1 tn+n(n−1)/2 = C1 tn(n+1)/2
C1 being suitable positive constant. We can now estimate
X
C e−πcσ(T ) = C
T ≥0
≤
68
∞
X
X
e−πct
t=0 T semi integral,
∞
X
C C1
e−πct tn(n+1)/2
t=0
and the last sum is clearly convergent. The proof of Lemma 10 is now
complete.
We now apply Lemma 10 to show that every modular form is bounded
in the fundamental domain F of the modular group. In case n = 1 this
is true by the definition of a modular form. Assume n > 1. Since the
Fourier series of a modular form may be considered as a power series,
the convergence of such a series is absolute. Hence every partial series
P
of (93), viz T ∈K a(T )e2πiσ(T >.) where K denotes an arbitrary set of
semi integral matrices, converges absolutely. Let T be a fixed matrix
such that a(T ) , 0 and kT the set of matrices T 1 = T [U] where U denotes an arbitrary proper unimodular matrix. Then a(T 1 ) = a(T ) by (95)
and the series
X
e2πiσ(T 1 Z)
(100)
g(Z, T ) =
T 1 ∈kT
69
converges absolutely. We shall show that this is possible only if T ≥ 0.
Let ν(T, m) denote the number of matrices T 1 ∈ kT with σ(T 1 ) = m. We
shall use the abbreviation t for e2π .
From (101) we get
g(i.e, T ) =
≥
X
T 1 ∈kT
∞
X
m=1
−2piσ(T 1 )
e
=
ν(T, −m)tm ≥
∞
X
ν(T, m)t−m
m=−∞
∞
X
m=1
ν(T, −m),
If T 0 we shall show that ν(T, −m) ≥ 1 for an infinity of m′ s
and this will provide a contradiction as we know that the series (100)
is convergent. If T 0 we can find an integral column Y such that 69
T [Y] < 0. Let U = E + h(h1 Y, h2 Y · · · βn Y) with integers h, hi , i =
12, . . . n. Since U − E has rank 1, we contend
that |U| = 1 + trace {h(h1 , Y, h2 Y, . . . hn Y)}
= 1 + hσ(h1 Y, h2 Y, . . . , hn y).
To verify this we need only observe that while for any matrix A we
have
|tE + A| = tn + σ(A)tn−1 + · · · ,
if A is of rank 1, the coefficient of tn−2 and lower powers of t which
depend on subdeterminants of A all vanish and consequently |tE + A| =
tn + σ(A)tn−1 ,
In particular, choosing t = 1 we obtain |E + A| = 1 + σ(A) which is
precisely what we desired.
Now σ(h1 Y, h2 Y, . . . hn Y) is a linear form in the h′ s and for n > 1,
there exists a non trivial integral solution of the equation
σ(h1 Y, h2 Y, . . . hn Y) = 0
P
and then nν=1 h2ν > 0.
With these h′ν s and the free variable h we compute
σ(T 1 ) = σ(T [U])
70
= σ{T [E + h(h1 Y, h2 Y, . . . hn Y)]}
= {T + hT (h1 Y, h2 Y, . . . hµ Y)
+ h′ ki Yβν Y, . . . hn Y′ π + h2 T [h1 Y, ·hn Y]}
= σ(T ) + 2hσ(λ(h1 Y, h2 Y, . . . hn Y)) + h2 σT [(h1 Y, . . . , hn Y)]
n
X
= σ(T ) + 2hσ(T (hY , . . . hn Y)) + ρ2 T [Y]
h2ν
ν=1
70
Choosing h suitable, this shows clearly that σ(T [U]) = −m, |U| =
1 is solvable for an infinite number of positive integers m and this is
equivalent to saying ν(T, −m) ≥ 1 for an infinity of m′ s. Thus we have
shown that, for the series (101) to converge absolutely, we must have
T ≥ 0 and this in its turn implies that only such T ′ s occur in the series
(93) representing f (Z). Lemma (10) implies that f (Z) is bounded in f.
We have now proved
Theorem 4. Every modular form is bounded in the fundamental domain
F of the modular group acting on Y .
We proceed to show that the modular forms of negative weight (K <
0) must necessarily vanish identically.
Let f(Z) be a modular form of degree n and weight K. Then as a
result of (72) it is easy fo see that h(Z) = |Y|K/2 | f (Z)| is invariant under
all modular substitutions, viz.
h(M < Z >) = h(Z)
(101)
for M ∈ M. If we assume K < 0, then by theorem (4) f(Z) is bounded
in F and by (76) |y| has a positive bound in F . It follows therefore that
h(Z) is bounded in F and hence also throughout Y . Let then h(Z) ≤ C
for Z ∈ Y .
By means of the representation
a(T )e−2πσ(T y) =
Z
···
H
Z
f (Z)e−2πiσ(T x) [dx]
71
we conclude that
|a(T )|e−2πσ(T y) ≤ sup | f (Z)| = C |λ|−K/2 .
x∈H
Z=x+ty
and letting ∈→ 0 the limit process yield that a(T ) = 0. We then have
71
Theorem 5. A modular form of negative weight vanishes identically.
We can therefore assume in the sequel that the weight K of a modular
form F is non negative. Later on we shall show that if K = 0 then F is
necessarily a constant.
We now introduce an operator which maps the modular forms of
degree n > 1 into those of degree n − 1 with the same weight. This
operators will be denoted by φ and the image of F (Z) under will be
denoted by F (Z)|φ. The use of this operator will be particularly felt in
such cases where proofs are based on induction on n
We write, in place like these, where we are concerned with modular
forms of different degrees,
Y = Y n , F = Fn , M = M n .
It is straight forward verification that if Z ∈ Yn the matrix Z1 arising
from Z by cancelling its last row and! column belongs to Yn−t and if
Z 0
Z1 ∈ Yn−1 then the matrix Z = 1
∈ yn provided λ > 0
0 iλ
!
Z 0
We can then form the function f
define for every Zi ∈ yn−1
0 iλ
!
Z C
and f, a modular form of degree n − 1. We shall show that lim f
λ→∞ 0 cλ
exists, denoted by f (Zt ) and this will be the modular form f (z)|φ of
degree n − 1 and weight K.
Let L be a compact subset of yn−1 and Z1 = X1 + iY1 ∈ L.
We show first that for any T 1 ≥ 0,
72
σ(T 1 y1 ) ≥ λσ(T 1 )
(102)
where γ = γ(L) > 0. We need only consider the case σ(T 1 ) > 0 as
otherwise T 1 = (0) and the inequality reduces to a trivial equality. Then
72
by reason of homogeneity of both sides in T 1 , we can assume σ(T 1 ) = 1
The equations:
σ(T 1 ) = 1, T 1 ≥ 0, Z1 ∈ L ⊂ Yn−1
define a compact Z1 , T 1 -set and on this set, σ(T 1 Y1 ) is a function, continuous in T 1 and Y1 -If we show that this function is positive at every
point, then it has a positive minimum γ in this set and we would have
proved (102). Let R = Rn−1 , 0 be determined with T 1 = RR′ and let
R = (W1 , W2 , . . . Wn−1 ).
P
Then σ(T 1 Y1 ) = σ(Y1 [R]) = n−1 ν = 1y1 [Wν ], and the last sum
is positive as Y1 > 0 and at least one of the columns is nonzero. This
settles our contention. Since the Fourier series
X
f (Z) =
a(T )e2πiσ(T Z)
T ≥0
i
γE, we have
2
|a(T )| ≤ C eπγσ(T ) for T !≥ 0 and a certain positive constant C .
Z 0
Writing Z = i
, Z1 ∈ L
and decomposing T analogously
0 tλ
!
T
µ
we get from the above that
as T = (tµν ) = 1
ν tnn
converges everywhere and in particular at the point Z =
|a(T )|e2πiσ(T Z) ≤ C eπνσ(T ) e−2πσ(YT )
= C eπλσ(T ) e2π(σ(T 1 y1 )+λtnn)
= C eπγσ(T 1 )−2πσ(T 1 y1 )−π(2λ−γ)t) nn
≤ C e−piγσ(T 1 )−πγtnn
= C e−piγσ(T )
73
assuming λ ≥ γ. Thus if Z1 ∈ L, λ ≥ γ(L), Z =
|a(T )|e2πiσ(T Z) ! ≤ C −piγσ(T )
!
Z1 0
then
0 tλ
73
It is now immediate that the series
X
f (Z) =
α(T )e2piiσ(T z)
T ≥0
which is majorised by C T ≥0 e−piγσ(T ) independent of Z, converges
uniformly for Z1 in every compact domain L ⊂ Yn−1 and λ ≥ γ(L).
Then
!
Z1 0
lim f(Z) = lim f
0 ℓλ
λ→∞
λ→∞
X
=
a(T ) lim e2πiσ(T Z)
P
T ≥0
=
X
T ≥0
λ→∞
a(T ) lim e2πiσ(T 1 Z2 )−2πλtnn
λ→∞
In the last series, the terms involving T for which tnn > 0 vanish in
the limit and only there terms for which tnn = 0 survive. We than obtain
lim f(Z) = f1 (Z1 )
X
=
a(T )e2πi5(T 1 z1 )
λ→∞
T ≥0tnn =0
X
a(T 1 )e2πiσ(T 1 ,z1 )
(103)
T ν 0≥0
!
T1 0
where by definition a(T 1 ) = a
73
0 0
It is clear that t fi (Zi ) is regular in Yn−1 as the corresponding Fourier
series converges uniformly in every compact subset of Yn−1 . Further
f1 (Z2 ) is bounded in the fundamental domain fn−1 as in the series (103)
only those T 1′ s occur which are semi positive. It remains to show that
f1 (Z1 ) is actually a modular
! form of degree n − 1 and weight R
A B1
∈ Mn−1 We complete M1 to a modular matrix
Let M1 = 1
C 1 D1
M of degree n as follows.
!
!
!
A B
A1 o
B1 0
M=
where A =
,B =
C D
o 1
0 0
74
!
!
C1 0
D1 0
C=
, and D =
0 0
0 1
!
Z1 0
If the Z =
, L1 ∈ Yn−1 we have M < Z >= (AZ + B)(CZ +
0 ℓλ
D)−1 =
(
!
!
!)
AI 0 z1 0
B1 0 n o−1
⋆
=
+
0 1 0 iλ
0 0
!
!−1
A1 Z1 + B1 0 Ci Zi + 01 0
=
0
iλ
0
1
!
!−1
A1 Z1 + B1 0 (Ci Zi + 01 )1 0
=
0
iλ
0
1
!
M1 < Z 1 > 0
=
0
iλ
74
Also |CZ + D1 = |C1 Z1 + D1 |. Hence we obtain from the relation
−R = f, z) that
f (M < Z >)|CZ + D1
!
!
M1 < Z 1 > 0
Z1 0
R
|C Z + D1 | = f
which as λ → ∞ yields
0
λ 1 1
0 λ
f1 (M1 hZ1 i)|C1 Z1 + D1 |−R = f1 (Z1 ),
That is to say f1 (Z1 )|M1 = f1 (Z1 ) for M1 ∈ Mn−1 and it is immediate
that f1 (Z1 ) is a modular form of degree n − 1 and weight R.
Chapter 6
Algebraic dependence of
modular forms
We are interested here in the question: when a modular form of degree n 75
vanishes identically, in other words, when two when two given modular
forms of the same degree are identical. The following theorem provides
a useful criterion in this direction.
Theorem 6. Let sn denote the least upper bound of σ(Y −1 ) for Y −1 such
that X + iY ∈ fn and let
X
f(Z) =
a(T )e2πiσ(T Z)
T ≥0
be a modular form of degree n and weight k ≥ 0. If a(T ) = 0 for T
R
sn (i.e. if a certain finite set of Fourier coefficients
such that σ(T ) ≤
4π
vanish), then f (z) vanishes identically.
Proof. First we note that {sn }, n = 1, 2, . . . , is an increasing sequence.
For, we know from (82) that
σ(y−1 ) ≤
2nC1
na
≤ √ for Z = x + oy ∈ Fn .
yM
3
75
6. Algebraic dependence of modular forms
76
2nc1
In particular then, S n ≤ √ < ∞ Let now Z1 ∈ Fn−1 . We claim
3
!
Z1 0
that Z =
∈ Fn provided λ s sufficiently large. We prove this as
0 M
!
y1 0
= (ym ) say.
follows. Let Z = x + iyu, Z1 = x1 + y1 Then y =
0 λ
We have!to verify the reduction condition of Minkowski for Y. Let
YR∗
be a primitive column with the integers g1 , g2 , · · · gn as
YR =
gn
elements. Then we have
y[YK ] = y1 [YK ] + λg2n
76
If gn = 0, then gK , gk+1 · · · gn−1 are themselves coprime and then
Y1 [yK ] ≥ yK B as y1 is reduced and k is necessarily less than n. It then
follows that Y[YK ] = y1 [YK ] ≥ yKK If yn , 0, then y[YK ] = y[YK ] +
λgK +λ, so that y[YK ] ≥ yKK provided we choose λ ≥ gn−1 , n−1 Trivially
yKK+1 ≥ o for all k as this is true of y1 .
It is now immediate that if
λ ≥ yn−1,n−1
(104)
!
y1 0
satisfies Minkowski’s reduction cond
0 λ
itions. Clearly, since y1 is reduced modulo 1, χ is also reduced modulo l. It only remains therefore to verify that Z is a highest point in the
set of all points which are equivalent to Z relative to Mn provided λ is
sufficiently large; in other words
then y =
kCZ + Dk ≥ 1
As in (71) we have , in the usual notations in (70),
kCZ + Dk2 = |C1 |2 |T |2
r
Y
(1 + h2ν )
ν=1
and to infer that kCZ + Dk2 ≥ 1 it suffices to ensure ourselves that
kT k ≥ 1.
77
77
r
Q
ν=1
As usual, we assume that T is reduced. Then by (49), C1 |T | >
y[yν ], Q = (y1 , y2 , . . . yr ) = (Q∗ y′ ) (say) where Y ′ = (q1 , q2 , . . . qr ) is
the last row of Q and let Q∗ = (y∗1 , y∗2 , . . . y∗r ) where yν = (y∗ν , qν ). Then
Y[yν ] = Y1 [y∗ν ] + λq∗r and by (83),
√ since the smallest characteristic root λ
3
= C, we have y1 = y∗ν | ≥ Cy∗ν y∗ν with
of λ1 is at most equal to
2(n − 1)C1
a positive constant C = 1 which depends only on u. Hence choosing
λ ≥ C, we have
y[U ] = y1 [U ∗ν ] + λq2n ≥ C(y∗ν , y∗ν + y2ν ) = cy′ν yν ≥ C
since Uν′ Uν ≥ being a primitive vector.
Thus



c, in any case
y[ην ] ≥ 

λ, if qν = C
Hence if xxxx , o at least one qν , ∗ ∗ ∗ ∗ ∗ so that one of the factors
of the product ***** is greater than or equal to λ while the others are
at least equal to C. Consequently ***** and a fortiri ***** if U , 0.
This means that |T | ≥ xxxxxxx choosing λ with λ ≥ max ∗ ∗ ∗ ∗ ∗ and
this in its turn implies that ||CZ + 0|| ≥ 1 as desired. If however *****
then ***** is itself primitive and *****. The pair {C1 , C2 } and *****
then determine a class ***** of coprime symmetric pairs of order n − 1.
A simple consideration shows that
||Co Z1 + Do ||2 = |Co |2 |T |2
r
Y
(1 + h2ν ) = ||CZ + D||2 .
ν=1
Since Zi ∈ fn−1 , ||Co Z1 + D2 || ≤ 1- the same is therefore true of ||CZ + D|| 78
too. Thus if λ ≥ max(yn−1,n1 , C, C1C 1−n and Zi ∈, then
!
Zi 0
Z=
∈ fn .
0 λ
(105)
Let Z1 ∈ Fn−1 be chosen such that σ(γ1−1 ) ≥ s−ε
n−1 where Z1 = X1 +iy1
and ε is a given positive number.
78
!
Zi 0
Then, if Z = × + iy is determined by (105) we have y =
and
0 λ
σ(y−1 ) = σ−1
1 ) = 1/λ .
Now Sn ≥ σ(y−1 ) = σ(y−1 ) + 1/λ ≥ Sn−1 + 1/λ − ε. Since this is true
for all λ ≥ max ∗ ∗ ∗ ∗ ∗ letting λ → ∞ we obtain that sn ≥ sn−1 − ε
The arbitrariness of ∈ now implies that zn ≥ sn−1 , in other words,
the sequence ***** is monotone increasing.
We shall need one more fact for proving our theorem, viz. that if
T = T (n) is a semi positive matrix, and 0 < rank T = n < η then there
exists a positive matrix T i = T (n) = T 1 and a unimodular matrix V such
that
!
T 1(n) 0 (r)
T [V] =
(106)
T > 0.
0 0 1
79
Indeed, due to our assumption we have |T | = 0 so that there exists
a rational column W with T [W]. Clearly we can assume W to be
primitive. Then W can be completed to a unimodular matrix Vo with
(o)
) is a matrix all the elements
W as the last column and then T [Vo ] = (tµν
(o)
= 0 since T [W] = 0
of whose last row and column vanish; in fact, tnn
(o)
(o)
and then tnν = 0 = (tνn = 1, 2, . . . n as T ≥ 0. Let then
!
T0 0
T = T o′ = to(n−1) ≥ 0
T [V] =
0 0 o
If r = n−1, then clearly T o > 0 and we are through. In the alternative
case we can repeat the above with T 0 in the place of T and this can be
continued until we arrive at T 1 = T 1(r) satisfying (106).
We now take up the proof of the main theorem. The proof is by
induction on n. We assume that either n = 1 or if n > 1 then the theorem
is true for all modular forms of degree n − 1. With this assumption on n
we shall prove that the theorem is true for modular forms of degree n.
Let
X
f (Z) =
a(T )e2πiσ(T Z)
(107)
T ≥0
In the second case, viz. when n > 1 we have
X
f (Z)|φ =
a(T 1 )e2πiσ(T Z)
T 1 ≥0
79
where
!
T1 0
a(T 1 ) = a
0 0
!
K
T 0
sn−i then σ 1
= σ(T 1 ) < kν S n−1 and our hy- 80
If σ(T i ) <
0 0
4π
!
T1 0
pothesis now implies that a(T 1 ) = a
= 0. In other words,
0 C
f (z)|φ satisfies the conditions of Theorem 6 so that, due to our assumption f (z)|φ ≡ 0 as it is a modular form of degree n − 1. This means
!
T1 0
that a(T ) = 0 for T such that |T | = 0 For, by (106), T =
[u],
0 C
U-unimodular, and by (95), since T 1 ≥ 0,
!
T1 0
= a(T 1 ) = 0
(108)
±a(T ) = a
0 C
For n = 1, the above is true by one of the assumptions in Theorem
6. Thus in any case |T | = 0 implies a(T ) = 0 so that in (107) only those
T ′ s with T > 0 survive; in other words
X
f (Z) =
a(T )e2πiσ(T Z)
T >0
We now wish to prove a preliminary result, viz. if Z = x, iy ∈ Fn
then
Z
K/2
lim |y|
(Z) = 0
(109)
|y|→∞
Let y = (yµν) , T = (tµν ) > 0 and introduce y1 , T 1 by requiring that
√
y = y1 [K], T 1 = T [K] = [K] where K = (δµν yµν ). Then f = y[K −1 ]
!
1 ∗
is a matrix of the type 1
and is positive. Hence y1 is a bounded
∗ 1
matrix with |y2 | = |y|(y11 , y22 , ynn )1 ≥ C1−1 > 0 as a result of (49). These
show that y1 belongs to a compact subset y of the space of all positive
symmetric matrices, which set depends only upon n.
According to (102) we then have
σ(T y) = σ(T 1 [k−1 ]y1 [K]) = σ(T 1 y1 )
80
≥ γσ(T 1 ) = γ
81
γ
X
yνν tνν
(110)
ν=1
with a positive constant γ depending only on n. Since the√ series (107)
converges everywhere
and in particular at the point z = 23 γ2 iE we get
√
3
that a(T )e− 2 πγσ(T ) is bounded, and by multiplying f(Z) by a constant
factor if necessary, we can assume the bound to be 1. We may now
estimate the general term of (107) as follows :
|a(T )e2πiσ(T Z) | ≤ e
√
3
2 πγσ(T )−2πσ(T Y)
= e−
√
3
2
(t11 + t22 + · · · tnn −
Pn
P
πγ( ν=1 tνν )−2πγ( nν=1 tνµ yνµ
≤e
2πσ(T Y)
)
P
−πγ( nν=1 tνν yνν )
e
.
In the above we have made use of (75) and (110).
n
n
pQ
Q
P
yνν by (81), Y being
Since
tνν yνν ≥ n n nν=1 tνµ yνν and |y| ≤
ν=1
ν=1
positive, we have from the above
|a(T )e2πiσ(T Z) | ≤e−πγn
√
n
nπnν=1 tνν yνν
p
√
n
≤ e−πγn |Y| n t11 t22 , . . . tnn
Since K ≥ 0 for a suitable constant C we have
K
K
and then |Y| 2 f(Z) =
γ √
n
|Y| 2 eπn 2
P
|Y|
<C
K
T >0
|Y| 2 a(T )e2πiσ(T Z) is majorised by the series
−πγn √
√
n
|γ| n t11 t22 ...tnn
P
2
C
e
T >0
82
The last series converges by arguments as in pages (66 - 67) since the
number of semi integral positive matrices T = (tµν ) with t11 t22 . . . tnn = t
n(n+1)
for a given t, can be estimated by C t 2 , viz. a fixed power of t and
then, the last series is further majorised by the convergent series
C
∞
X
t=1
C1 tn(n+1)/2 e−T γ/2∗∗∗∗∗
81
It now turns out that for Z ∈ Fn
|Y|K/2 F (x) = 0(e−e∗∗∗∗∗
(111)
with some ∈> 0 as |y| → ∞ and this, in particular, implies (109). If
h(Z) = |y|K/2 |F (Z)|, Theorem (4) and (109) imply that h(Z) is bounded
in the fundamental domain. But we know from (101) that h(Z) is invariant under modular substitutions and the h(Z) is bounded throughout its
domain. Since further h(Z) is continuous in Y it attains its maximum;
in other words there exists a point Zν ∈ such that h(Z) ≤ h(Zo ) for z ∈ Y .
Consider h(Z) in a neighbourhood of Zo . Let z = x − ly be a complex
variable Z = Zo − zE and t = ezπiz .
Let g(t) = F (z)e−λσ(Z) with λ determined by
ηλ
= 1 + [k/4π sn ] where [x]
2π
denotes the integral part of x, viz. the largest integer not exceeding x.
K
Using our assumption that a(T ) = 0 if σ(T ) < 4π
sn we have
X
a(T )e2πiσ(T Z)−iλσ(z)
g(t) =
K
sn
σ(T )> 4π
=
X
a(T )e2πiσ(T zo ) tσ(T ) e−iλσ(z)
X
a(T )e2πiσ(T zo )−iλσ(zo ) tσ(T )− 2π
K
Sn
σ(T ) 4π
=
λn
K
Sn
σ(T ) 4π
λn
K
K
Here the exponent σ(T ) −
>
Sn − [ Sn ] − 1 ≥ −1 which
2π
4π
4π
λn
λn
is an integer, that σ(T ) −
≥ 0.
means, as σ(T ) −
2π
2π
This shows that the function g(t) is regular in a circle |t| ≤ ρ and we
can assume ρ > 1 by choosing y < 0 with its absolute value sufficiently
small.
By the maximum principle, there exists then a point t1 with |t1 | =
ρ > 1 such that |g(t1 )| ≥ |g(1)|. If the z-point corresponding to t1 is
denoted by z1 , we have
K
|g(t)| = |f(Z)|eλσ(y) = h(z)|y| 2 eλσ(y)
83
82
and putting t = 1 the above now implies that
K
K
h(zo )|Yo | 2 eλσ(yo ) ≤ h(z1 )|Yo | 2 eλσ(y1 )
R −2πy
where Z1 = Zo + z1 E, y1 = yo + yE,
= |t1 | = ρ.
y=−
1
log ρ
2π
Let h(zo ) = sup h(z) = M (say). Then the above yields
z∈y
K
K
M ≤ M|y| 2 |y0 | 2 eσ(y1 −yo ) = Meψ(y)
84
(112)
where
ψ(y) = λny −
K
log(|y1 ||yo |−1 )
2
λny −
= ψ(y)
K
log |E|yγo−1
2
Now ψ(0) = 0 and ψ′ (o) = λn K2 σ(γo−1 )
K
sn
2
λn K
sn ) > 0.
= 2π( −
2π 4π
≥ λn −
Hence it follows that ψ(y) is monotone increasing in a neighbourhood of y = 0 so that it is negative for sufficiently small y < 0. But
ψ(y) < 0 implies from (112) that M = 0 which in its turn means that
h(z) and consequently f (z) vanishes identically. The proof of theorem 6
is now complete.
The above theorem has several interesting consequences. In the first
instance it implies that all modular forms of weight 0K = 0) are necP
essarily constants. For if F (z) = a(o) +
a(T )e2πiσ(T Z) be such a
T ,0
form, then F (Z) − a(0) is a form satisfying the conditions of theorem 6.
Hence it follows from theorem 6 that F (Z) − a(o) ≡ 0, in there words
F (Z) ≡ a(o). We therefore assume in the sequel that k > 0.
83
Now consider modular forms F (Z) of degree 1(n = 1) and weight
K
2
K ≤ δ. Then from (75) we deduce that S1 √ . Then σ(T ) ≤
S1
4π
3
2 2
4
implies that σ(T ) ≤ √ = √ < 1 which in its turn means that
π 3
π 3
σ(T ) = 0 as sigma(T ) is an integer, and consequently T = 0. Hence if
X
f(z) = a(0) +
a(T )e2πiδ(T Z)
T ,0
and if a(0) = 0, then the conditions of theorem 6 are satisfied so that by 85
its conclusion, f(z) ≡ 0. In other words, modular forms of degree 1 and
weight K ≤ 8 are uniquely determined by their ‘first’ Fourier coefficient
a(0). It is now immediate that any two modular forms of degree 1 and
weight K ≤ 8 are proportional and what is the same, a modular form
of degree 1 and weight K ≤ 8 is unique, except for multiplication by
a constant. Indeed, the same result is true of modular forms of degree
2 too. For if Z = x + iy ∈ f2 and f (Z) is a modular form of degree 2
and weight K ≤ 8 , then writing Y = (yµν ) we have, by Minkowski’s
reduction conditions,
0 ≤ 2y12 ≤ yy11 ≤ y22
3
and then |Y| = y11 y22 − y212 ≥ y11 y22 .
4
√
3
Since y11 ≥
by (75), this means that
2
σ(Y −1 ) =
so that s2 ≤
1
8 1
16
y11 + y22 4 1
≤ (
+
)≤
≤ √
|Y|
3 y11 y22
3 y11 3 3
16
√ . Also
3 3
K
K
2
32
s2 ≤ ≤ √ < 2 and then , σ(T ) <
s2
4π
π 3 3
4π
implies that σ(T ) < 2 This means that at least one of the two diagonal
elements and consequently one of the elements t12 , t21 where T = (tµν )
84
86
also vanish. Then it is clear that |T | = o. Then assumption a(o) = o will
imply by our earlier result on modular forms of degree 1 that f(Z)|φ ≡ 0
and then as in (106) and (108), we have a(T ) = 0 for |T | = 0. In
other words if we assume a(0) = 0 then the conditions of theorem 6 are
satisfied and we are able to conclude that f(Z) ≡ 0. It now follows as in
the earlier case that every modular form of degree 2 and weight R ≤ 8 is
uniquely determined by the first Fourier coefficient a(0).
The above results are for the moment hypothetical. In other words,
their significance is based on the assumption that modular forms of the
desired degree, not vanishing identically actually exist. Their existence
we prove later, by constructing the so called Eisenstein Series.
An interesting application of theorem 6 is to prove the algebraic
dependence of any set of sufficiently large number of modular forms.
Specifically we state
+ 2 and let fν (Z) be a modular form deTheorem 7. Let h = n(n+1)
2
gree n and weight Kν > 0, ν = 1, 2, . . . h. Then there exists an isobaric
algebraic relation
X
νh
Cν1 ν2 , . . . ν2 f1 fν2
(113)
Z .fh = 0
not all of whose coefficients vanish, the summation extending over all
integers νi ≥ o with the property
h
X
νε Ki = mKi K2 , Kh
(114)
l=1
where m is an integer which depends only upon n
We may remark that the product of two modular forms of weight
k and ℓ is a modular form of weight KK, their degrees being the same
µ
µ
. Consequently all the power products fν11 f2 2 . . . , fh h that occur in (113)
are modular forms of the same weight in view of (114) and hence the
name isobaric for the relation (113).
86
Proof. All modular forms of degree n and weight k form a linear space
mnK . By means of theorem 6 we infer that this space is of finite dimension
85
αn (K). In fact αn (K). is at most equal to the number of solutions of the
relation σ(T ) ≤ (K)
4π Sn with semi positive integral T’s, a rough upper
K)
estimation of which is provided by 4π
Sn∗∗∗∗∗ by arguments as at the end
of page (66); in other words we have
dη (K)(
K)
Sn + i)n(n+1)/2
4π
(115)
We now find a lower estimation of the number of possible power
products F = fν11 fν22 , . . . fνhh subject to the conditions

µi ≥ C, i = 1.2 . . . 2 and integral,





h
X



νi Ki = ηKi K2 , . . . Kh m = m(n)


(116)
i=1
Let us assume for the moment that m is divisible by 2h − 2.
Let (x1 , x2 , ∗ ∗ ∗ ∗ ∗) be a system of integers with
o ≤ xν ≤ ∗ ∗ ∗ ∗ ∗
(117)
where K = K1 K2 · · · Kh . The number of such systems is clearly H ∗∗∗∗∗
There exists then one coset modulo K1 which contains the sums
K1 x1 K2 x2 + . . . Kr−1 xh−1 (mod)Kn for at least H−1
Kh + 1 different systems
*** as otherwise, the total number of different systems (x1 , x2 , . . . xh−1 )
could be at most H − 1/Kh , Kh = H − 1 while actually there are H such 87
systems. Consider then the systems (x1 , x2 , . . . xh−1 ) corresponding to
this coset. Let (ξ1 , ξ2 , . . . ξh−1 ) denote a fixed system among these and
let (η1 , η2 , . . . ηh−1 ) a variable system. Then clearly we have
h−1
X
i=1
ki ηi −
h−1
X
i=1
Ki ξi ≡ o mod Kh
We shall further assume o ≤ ξi < Ki , l = 1, 3, . . . h − 1
We introduce now νi = ηi Kh − ξi , = 1, 2, . . . h − 1 and νh =
1 Ph−1
hk ( i=1 Ki νi
(118)
m−k
ηn
−
86
We shall verify that the system h{ν} satisfies the conditions (116).
Clearly ν1 , ν2 , . . . νh−1 are non negative and integral. Also using (117)
we have
h−1
1 X
mK
−
(Ki ηi + Ki (Kh − ξi ))
νh =
Kh
Kh i=1
≤
h−1
h−1
X
mK
1 X mK
−
−
−
−Ki
Kh
Kh i=1 2h − 2 i=1
= mK/2Kh −
=
87
(119)
h−1
X
−Ki
i=1
h−1
X
(h − 1)K
−
Kh
i=1
−Ki ≥ 0
The last but one of these relations is a consequence of the fact that
m is divisible by 2h − 2 and a fortiori m2 ≥ h − 1 while the last step
is immediate by observing that kKh ≥ Ki 0 = 1, 2, . . . h − 1.νh νh is certainly integral as seen from (119) by means of (118). We have therefore shown that each system (η1 , η2 , . . . ηh−1 ) leads to a permissible system of exponents ν1 , ν2 , . . . νh . Consequently,the number of possible
νh
ν2
power products fν1
1 f2 · · · fh is at least as great as the number of the sysH−1
tems (η1 , η2 , . . . ηh−1 ) which is at least H−1
Kh + 1. Denoting Kh by q we
have proved that there exist at least q + 1 modular forms of the kind
ν2
νh
fν1
1 f2 · · · fh
Therefore, in case
q ≥ dn (mK)
(120)
there exists a non trivial relation
X
νh
Cnu1ν2 · · · νh fν1
1 · · · fh = 0
with constant coefficients Cν .
As a result of (115), (120) will be satisfied provided we have q ≥
mK
( π sn + 1)2−2 which in turn will be true if
K
q ≥ mh−8 ( Sn + 1)h−2
π
(121)
87
Since H =
h
Q
(1 +
mK
(2,p,−2)Kν )
q=
H−1 Y
mK
≥
)
(1 +
Kh
(2,
p,
−2)K
ν
ν=1
ν=1
we have
h
=
mh−1 K h−2
(2h − 2)h−1
Hence (121) is certainly satisfied if
mh−1 K h−z
K
≥ mh−z ( Sn + 1)h−z
h−1
π
(2h − 2)
Sn
+ 1)h−2 ,
i.e.
if m ≥ (2h − 2)h−1 (
π
Obviously, consistent with this requirement, the assumption that m is 88
divisible by 2h − 2 is permissible and theorem 7 is established.
Chapter 7
The symplectic metric
The paragraph deals with the symplectic metric in Y defined by a pos- 89
itive quadratic differential form which is invariant
! under all symplectic
A B
substitutions. Let Z1 , Z2 ∈ Y and M =
∈ S n . Let Zν∗ = M <
C D
Zν >, ν = 1, 2. A simple computation yields by means of the relation
Zν∗ = (AZν + B)(CZν + D)−1 = (ZνC ′ + D′ )−1 (Zν A′ + B′ )
and the typical relations for a symplectic matrix established in §1, the
formulae
Z2∗ − Z1∗ = (Z1 C ′ + D′ )−1 (Z2 − Z1 )(CZz + D)−1
Z2∗ − Z̄1∗ = (Z̄1 C ′ + D′ )−1 (Z2 − Z̄1 )(CZ2 + D)−1
Z̄2∗ − Z̄1∗ = (Z̄1 C ′ + D′ )−1 (Z̄2 − Z̄1 )(C Z̄2 + D)−1
Z̄2∗
−
Z̄1∗
′
′ −1
(122)
−1
= (Z1 C + D ) (Z̄2 − Z1 )(C Z̄2 + D)
Now for any points Z1 , Z2 ∈ Y it is easily seen that Z2 − Z̄1 and
Z̄2 − Z1 also belong to Y and by (25), every Z ∈ Y is nonsingular. Thus
the inverses (Z2∗ − Z̄1∗)−1 (Z̄2∗ − Z1∗)−1 exist. From (122) we now obtain that
(Z2∗ − Z1∗)(Z2∗ − Z̄1∗ )−1 (Z̄2∗ − Z̄1∗)(Z̄2∗ − Z̄1∗)−1
′
′ −1
′
′
= (Z1C + D ) ̺(Z1 , Z2 )(Z1C + D )
89
(123)
7. The symplectic metric
90
where
̺(Z1 , Z2 ) = (Z2 − Z1 )(Z2 − Z̄1 )−1 (Z̄2 − Z̄1 )(Z̄2 − Z1 )−1
90
(124)
It is immediate from (123) and (124) that the characteristic roots
of ̺(Z1 , Z2 ) for any two Z1 , Z2 ∈ Y are invariant under the symplectic mapping Zν → M < Zν >, M ∈ S n . In particular it follows that
σ(̺(Z1 , Z2 )) is an invariant function of Z1 , Z2 meaning
σ(̺(Z1 , Z2 )) = σ(̺(M < Z1 >, M < Z2 >)), M ∈ S n
(125)
For any matrix Z, we define dZ the matrix of the differentials as
dZ = (dZµν ) where Z = (Zµν ). For two matrices Z1 , Z2 the following
relations are easily verified.
1)
d(Z1 + Z2 ) = dZ1 + dZ2
2)
d(Z1 + Z2 ) = dZ1 .Z2 + Z1 dZ2
(126)
As a consequence of (2) above we have, if |Z| , 0,
0 = d(E) = d(ZZ −1 ) = dZZ −1 + Z · dZ −1 so that
dZ −1 = Z −1 dZ · Z −1
(126)′
With these preliminaries about differentials, we once again take up
the main thread. In (124), we specialize Z1 , Z2 as Z1 = Z, Z2 = Z + dZ
and obtain
1
ρ(Z, Z + dZ) = dZY −1 dZ̄y−1 , Z = X + iY.
4
(127)
From (125) it now follows that
¯ −1 )
dS 2 = σ(dZ.Y −1 dZY
91
(128)
is an invariant quadratic defferential form in the elements dXµν , dYµν of
dX, dY respectively. We may also consider d′ s2 as a hermitian form
in the elements dZµν of dZ. Further, if R be a real matrix such that
Y −1 = RR′ and if Ω = (ωµν ) = R′ dZR then Ω = Ω′ and dS 2 =
P
P
σ(R′ dZRR′ d̄ZR) = σ(Ω′ Ω̄) = µ,ν ωµν Ω̄µν ≥ 0. If now µ,ν ωµν ω̄µν =
91
0 then Ω = 0 and this will require that dZ = 0 as R , 0. Thus ds2 > 0
if dZ , 0 and what is the same, ds2 > 0 represents a positive quadratic
form. Now we can consider Y as a Riemannian space with the fundamental metric given by ds2 , called the symplectic metric. The decomposition dZ = dX − dY yields
ds2 = σ{(dX + dY)Y −1 (dX − dY)Y −1 }
= σ{(dXY −1 )dXY −1 + (dY − Y −1 dYY −1 }
taking only the real part, since the imaginary part has got to vanish as
ds2 > 0. Thus we get
ds2 = σ(Y −1 dX)2 + σ(Y −1 dY)2
(129)
as expression which is well known in the case n = 1.
We now carry over these results to the generalized unit circle, which
we may recall is the set K of W ′ s satisfying the condition
W = W ′ , E − W ′ W̄ > 0
(2b)′
we know that we can map Y into K by means of the mapping
W = (Z − iE)(Z + iE)−1 = (Z + iE)−1 (Z − iE),
If W1 , W2 correspond to Z1 , Z2 we have
W2 − W1 = 2i(Z1 + iE)−1 (Z2 − Z1 )(Z2 + iE)−1
E − W̄1 W2 = −2i(Z̄1 − iE)−1 (Z2 − Z̄1 )(Z2 − iE)−1
W̄2 − W̄1 = −2i(Z̄1 − iE)−1 (Z̄2 − Z̄1 )(Z̄2 − iE)−1
−1
(130)
−1
E − W1 W̄2 = 2i(Z1 + iE) (Z̄2 − Z1 )(Z̄2 − iE)
and then
(W2 −W1 )(E − W̄1 W2 )−1 (W̄2 − W̄1 )(E −W1 W̄2 )−1 = (Z1 +iE)−1 ̺(Z1 , Z2 )(Z1 +iE)
In particular therefore, the matrix represented by the left side has
the same characteristic roots as ̺(Z1 , Z2 ) and hence also the same trace.
92
92
Setting Z1 = dZ, Z2 = Z + Z we have W1 = W, W2 = W + dW and then
from the above, and from (127), (128) we have
ds2 = 4σ(̺(Z, Z + dZ))
= 4σ(dW(E − W̄W)−1 dW̄(E − W W̄)−1 )
(131)
This provides another useful expression for ds2
We observe at this stage that given two points Z1 , Z2 ∈ Y we can
always find a symplectic substitution which brings Z1 , Z2 into the special
position
Z1 = iE, Z2 = iD = i(δµνδν ) 1 ≤ d1 ≤ d2 ≤ · · · ≤ dn
93
For, in view of the 1− 1 correspondence between Y and K it suffices
to determine a permissible substitution which takes two assigned points
dν − 1
) and in view of
W1 , W2 into 0 and a special points D1 = (δµν ,
dν + 1
the homogenity of the space K it suffices to determine a mapping which
leaves the origin fixed and takes a given point W into a point of the
type D1 viz, a diagonal matrix with positive diagonal elements in the
increasing order. This we know is possible by Lemma 2 by a mapping
of the kind
W → u′ Wu, u − unitary,
and this settles our claim,
We now introduce a parametric representation for positive matrices
y > 0. We choose matrices F = F (r) , G = G(r) and H = H (r,n−r) where r
is a fixed integer in 1 ≤ r < n to satisfy
!
!
!
F
FH
F 0
E Hnr
= ′
(132)
y=
H E G + F[H]
0 G
0 E
Such a choice is always possible as a comparison of the two extremes shows. Also from (132) it is clear that Y > 0 is equivalent with
F > 0, G > 0. By means of (132) we also have
!
F −1 + G−1 [H ′ ] HG−1
−1
y =
−G−1 H ′
G−2
93
and obtain by a simple computation, using properties (126), ((126)′ ),
that
σ(y−1 dy)2 = σ(F −1 dF)2 + σ(G−1 dG)2 + 2σ(G−1 dH ′ FdH)
(133)
We wish to remark at this stage that all terms in (133) are positive 94
quadratic forms in the appropriate elements; a possible doubt can only
be about the last term but by specialising dF, dG it is easily seen that the
last term represents a positive quadratic form in the elements of dH.
We proved to consider the existence and uniqueness of geodesic
lines in Y and we prove the following
Theorem 8. Given any two points Z1 , Z2 ∈ Y there exists a uniquely
determined geodesic line joining them. The length s(Z1 , Z2 ) of this
geodesic line is given by
v
t n
X
1 + λν 2
)
(134)
(log
s(Z1 , Z2 ) =
1 − λν
ν=1
where λ2ν , ν = 1, 2, . . . , n are the characteristic roots of
ρ(Z1 , Z2 ) = (Z2 − Z1 )(Z2 − Z̄1 )−1 (Z̄2 − Z̄1 )(Z̄2 − Z1 )−1 .
In the special case Z1 = iE, Z2 = iD = l(δµν ) a parametric representation of the geodesic line given by
Z = Z(t) = (δµν dνt ; 0 ≤ ε ≤ 1
Proof. Since we know that there always exists a symplectic substitution which takes Z1 , Z2 into iE, iD and that the characteristic roots λ2ν
of ρ(Z1 , Z2 ) are invariant under such a substitution, it clearly suffices
to prove the theorem for these special values of Z1 , Z2 Assume for a
moment that a geodesic line joining Z1 = iE to Z2 = iD exists with
a parametric representation Z = Z(t), 0 ≤ t ≤ 1 such that the elements
Zµν = Zµν (t) have continuous derivatives with respect to t. We shall show 95
that such a line is unique. Also, in the course of the proof we explicitly
determine what Z(t) is. Then the existence of a geodesic line with the
desired properties is immediate - in fact, the curve represented by Z(t)
which we have explicitly determined is the desired geodesic line.
94
We shall denote the differential coefficient with respect to t as
d
(∗) = (˙∗) By means of (128) and (133) we have
dt
Z Z2
Z 1
2 1/2
s(Z1 , Z2 ) =
(ds ) =
σ(Z)−1 Z̄Y −1 1/2 dt
Z1
1
=
Z
1
σ(Y −1 x)2 + σ(Y −1 y)2
Z
1
σ(Y −1 x)2 + σ(F −1 Ḟ)2 + σ(G−1Ġ)2
σ(F −1 Ḟ)2 + σ(G−1Ġ)2 dt
0
≥
1/2
Z
0
=
σ(Y)−1 ZY −1 Z̄
0
=
0
Z
0
1
dt
1/2
dt
+ 2σ(G−1 )Ḣ 1 F Ḣ) 1/2 dt
We claim that the last inequality
! is actually a equality. For otherwise,
F(t) C
in the equation Z(t) =
we will have a curve joining Z1 and
0 G(t)
Z2 whose length will be actually
Z 1
{σ(F −1 Ḟ)2 + σ(G−1 Ġ)2 }dt < s(Z1 , Z2 )
0
contradicting our assumption. But then, the equality can hold in the
above when and only when
Ẋ = 0 = Ḣ
96
(135)
We therefore conclude that (135) holds identically in t, which means
that X ≡ 0 ≡ H as X(0) = 0 = H(0) then the parametric representation
of our geodesic line is given by
!
F(t)
0
Z(t) = i
(136)
0 G(t)
where F = F (r) , G(n−r) and r is arbitrary with 1 ≤ r < n. According to the permissible cases r = 1, 2, . . . , n − 1 the same geodesic line
95
will admit of (n − 1) formally different representations so that it follows
from (136) that Z(t) is necessarily a diagonal matrix Z(t) = i(δµν yµν (t))
Consequently we obtain
v
v
t k
Z 1u
Z 1 tX
n
X
−2
2
s(Z1 , Z2 ) =
Yνν Yνν dt =
Yνν2 dt
0
0
ν=1
ν=1
putting nµν = log Yνν . In the n dimension Euclidean space with nνν , ν =
1, 2, . . . n as rectangular cartesian coordinates, s(Z1 , Z2 ) then represent
the Euclidean length of our curve. But in this case we know that the
curve of shortest length is the straight line segment joining the points
corresponding to Z1 , Z2 . Hence we conclude that
log Yνν (t) = ηνν (t) = t log d2 (ν = 1, 2, n)0 ≤ t ≤ 1.
In other words we have the parametric representation of the geodesic
line joining Z1 = iE, Z2 = iD = i(δµν dν ) as
Z(t) = i(δµν dνt ), 0 ≤ t ≤)
Now we compute the characteristic roots λ2ν of (iE, iD)
̺(iE, iD) = (D − E)(D + E)−1 (D − E)(D + E)−1
dν − 1 2 )
= δµν (
dν + 1
dν − 1 2
dν − 1
) . Consequently ±λν =
which means that 97
dν + 1
dν + 1
1 + λν 2 ) = (log dν )2 .
log
1 − λν
We know that s(Z1 , Z2 ) is the Euclidean length of the segment joining (0, 0, . . . 0) and (log d1 , . . . log dn ).
so that λ2ν = (
Hence s(Z1 , Z2 ) =
n
X
=
n
X
(log dν )2
ν=1
ν=1
(log
1/2
1 + λν 2 1/2
)
1 − λν
96
and the proof is complete.
Having thus obtained the ’shortest distance’ s(Z1 , Z2 ) between any
two points Z1 , Z2 we wish to remark that the symplectic sphere defined
as the set of points Z with Z ∈ Y , s(Z, Z0 ) ≤ r − Z0 being a fixed point
in Y − is a compact set. Certainly we can assume that Z0 = iE.
Then
n
X
1 + λν 2 1/2
)
≤r
(137)
s(Z, iE) =
(log
1
−
λ
ν
ν=1
where λ2ν are the characteristic roots of ρ(Z, iE). But
̺(Z, iE) = (Z − iE)(Z + iE)−1 (Z̄ + iE)(Z̄ − iE)−1 = W W̄
98
where W is the point the in the Generalised unit circle which corresponds to Z under the usual mapping and we know from (26) that W W̄ <
E. We therefore conclude that λ2ν < 1, ν = 1, . . . r. From (137) it is
clear that λν cannot be arbitrarily close to 1 so that we should have
λ2ν ≤ 1 − δ < 1 for some δ > 0. Then W W̄ ≤ (1 − δ)E and the set
of W ′ s consistent with this inequality is clearly compact. Therefore the
same is true of the corresponding Z set too and hence also of our symplectic sphere.
We add here one more result for future reference, viz.
Lemma 11. Given a point Z0 ∈ Y there are at most a finite number of
modular substitution which have Zo as a fixed point.
Proof. We first observe that the modular group is discrete and hence
also countable. Let M , ±E be any modular substitution. Then the
equation M < Z >= Z define for a given M a complex analytic manifold
n(n + 1)
. Consider now all analytic manifolds
of dimension less than
2
of this kind, viz those formed by the fixed points of a given modular
substitution. In view of the above fact, given any point Zo ∈ Y there
always exist points Z in any neighbourhood of Zo with M < Z >, Z for
any M ∈ Mn , M , ±E. If now Mk < Z0 >= Z0 for an infinity of K′ s, say
K = 1, 2 . . . , andMK , ±Ml K , ℓ then choosing Z with M < Z >, Z
for any M different from ±E we have MK−1 < Z >, Z for ℓ , K i.e.
MK < Z >, Mℓ < Z > for , K. On the other hand, S (MK < Z >
97
, Z0 ) = S (MK < Z >, MK < Zν >) = S (Z, Z0 ). Consequently the
points MK < Z >, K = 1, 2, are all distinct and lie on a compact subset
of Y so that they have at least one limit point contradicting Lemma 8.
We therefore conclude that MK are identical after a certain stage and the
Lemma is proved.
We are now on the look out for the invariant volume element of 99
the symplectic geometry. We make a few preliminary remarks. Let
R = (rµν ) denote a variable n− rowed symmetric matrix and Ω = (ωµν )
∂(S )
denote the
a n × n square matrix. Let S = (sµν ) = R[Ω] and let
∂(R)
functional determinant of the n(n + 1)/2 independent linear functions
Sµν (p ≤ ν) with respect to the independent variables rµν(µ≤ν) . Then
∂(S )
, C if and only if the mapping R → S is 1 − 1. But this mapping
∂(R)
is 1 − 1 when and only when |Ω| , 0. For, while the one way implication
is trivial, viz. when |Ω| , 0 the mapping R → S is actually invertible
as R = S [Ω−1 ], to realize the converse, we argue that if |Ω| = 0 there
exists a row XXXXX with WΩ = 0. The choice of has R = νν leads
to R[Q] = Ω′ RΩ = 0 without R being zero, verifying that the mapping
∂(S )
= 0 when
R → S = R[Ω] is not 1 − 1. It therefore follows that
∂(R)
and only when |Ω| = 0. In other words, considered as polynomials in
∂(S )
the n2 variables Ωµν ,
and |Ω| have the same zeros. Since |Ω| is an
∂(R)
irreducible polynomial in these n2 variable, we conclude by means of a
∂(S )
well known algebraic result that
= C|Ω|n+1 with a constant C , 0.
∂(R)
The index (n + 1) in the right side is suggested by a comparison of the
degrees of both sides in ωµν . The special choice Ω = E yields to the
determination C = 1. Thus we have
∂(S )
= |Ω|n+1 for S = R[Ω].
∂(R)
(138)
!
A B
Let now M =
∈ ZY and consider the symplectic substituC D
98
tion Z → Z ∗ = M < Z >. One of the relations 122 gives
Z2∗ − Z1∗ = (Z1C ′ + D′ )−1 (Z2 − Z1 )(CZ2 + D)−1
100
and it then follows that
dZ ∗ = (ZC ′ + D)−1 dZ(CZ + D)−1 )
∂(Z ∗)
requires only
∂(Z)
the knowledge of the linear relation between dZ and dZ ∗ given by the
last formula. By means of (138) we then get
The computation of the functional determinant
∂(Z ∗ )
= (C + D)−n−1
∂(Z)
(139)
Decomposing Z, Z ∗, Z̄, Z̄ ∗ into their real and imaginary parts we get
Z = X + iY, Z ∗ = X ∗ + iY ∗
Z̄ = X + iY, Z̄ ∗ = X ∗ + iY ∗
and it is immediate that
∂(Z, Z̄)
∂(Z ∗, Z̄ ∗ )
=
∂(X, Y) ∂(X ∗ , Y ∗)
From Z ∗ = M < Z > and Z̄ ∗ = M < Z̄ > we then obtain using (139)
and (72) that
∂(X ∗ , Y ∗) ∂(X ∗ , Y ∗ ) ∂(Z ∗, Z̄ ∗ )
=
∂(X, Y)
∂(Z ∗, Z ∗)
∂(Z, Z̄)
∗
∗
∂(Z , Z̄) ∂(Z, Z¯∗)
=
=
∂(Z)
∂(Z, Z̄)
Z∗
= ||CZ + ν||2r2 = ( )
Z
101
∂(Z, Z̄)
∂(X, Y)
It now follows that the volume element
dv = |y|n−1 [dx][dy]
(140)
99



[dx] = πµ≤ν drµν
with 

[dy] = πµ≤ν dyµν
is invariant relative to symplectic substitutions.
We supplement our above results with the following two lemmas.
Lemma 12. If Z, Z ∗ ∈ Yn and Z1 , Z1∗ ∈ Yr denote the matrices which
arise from Z, Z ∗ by deleting their last (n − 2) rows and columns, where
r ≤ n then
s(Z1 , Z1∗) ≤ s(Z, Z ∗)
(141)
Proof. Since we know that the geodesic lines are uniquely determined,
it clearly suffices to prove the corresponding inequality for ds, viz. ds1 ≤
ds in an obvious notation. In other words we need only show that
2
2
2
−1
2
−1
2
σ(y−1
1 dy1 ) + σ(x1 dx1 ) ≤ σ(y dy) + σ(y dx)
where Z = Z (n) = x + iy and Z1 = Z1(r) = x1 + iy1 . We use the parametric
!"
!#
y∗1 E H
representation we had for positive matrices to write y = ∗
x
0 E
and appeal to (133) to infer that σ(y−1 dy)2 ≥ σ(y−11 dx1 )2
!
x∗
Since X = 1∗ we also have, as in the above case, σ(y−1 dx)2 ≥
x
−1
2
σ(y1 dx1 )
The desired result is now immediate.
101
Lemma 13. If Z, Z ∗ ∈ yn and Z1 = X1 + y1 , Z1∗ = X1∗ + LY1∗ be as in the
previous Lemma, and if S (Z, Z ∗) ≤ ̺ then there exist positive constants
σ(Y1∗ )
|Y ∗ |
1
1
Mν = Mν (̺ν)ν = 1, 2 with
≤
≤ M1 ,
≤ 1 ≤ M2 .
M1 σ(Y1 )
M2 |Y1 |
Proof. Since the condition s(Z, Z ∗) ≤ ̺ implies by Lemma 12 that
s(Z1 , Z1∗) ≤ ̺ we need only consider the case r = n. In view of the
interchangeability of Y and Y ∗ it clearly suffices to prove one part of
each of the two sets of inequalities. Let then W be an orthogonal matrix (WW ′ = E) so determined that y∗ [W] = D = (δµν , dν ), dν > cν =
1, 2, . . . n. Let Z̃ = (Z − X ∗ )[W D−1 ] = X̃+ and Z̃ ∗ = (Z ∗ − X ∗ )[W D−1 ] =
100
iy∗ [W D−1 ] = iE. Since s is invariant under symplectic substitutions, we
have s(Z̃, iE) = s(Z̃, Z̃ ∗) = s(Z, Z ∗) ≤ ̺
In other words Z̃ belongs to a symplectic sphere which we know is
a compact subset of Yn and consequently
σ(Ỹ) ≤ m1 (n, ̺),
1
≤ (ỹ) ≤ m2 (n, ̺)
m2
(142)
with suitable constants Mν ν = 1, 2. Also |Ỹ| = |y|Wd−1 | = |y|D−1,2 =
|y|Y ∗ . Then (142) implies that
1
≤ |y||y∗ |−1 ≤ M2
M2
Further, if y[W] = H = (hµν ) then
−1
σ(Ỹ) = σ(H[D ]) =
Pn
≥ Pν=1
n
hνν
2
ν=1 dν
102
=
n
X
h
νν
d2
ν=1 ν
σ(y)
σ(H)
=
.
∗
σ(Y ) σ(y∗ )
(142) again implies now that σ(γ)σ(y∗ )−1 ≤ m1 (n, ̺) and as remarked
earlier, the interchange of y and y∗ leads to the other half of our inequality. The lemma is thus established.
Chapter 8
Lemmas concerning special
integrals
We need to settle some special integrals to be applied subsequently in 103
our consideration of the Poincare′ -theta series. First we generalize Euler’s well known Γ-integral
Z∞
e−ty y−s−1 = Γ(s)t−s , t > o, Re.s > o
o
to the case of a matrix variable.
Lemma 14. If T = T (n) > 0 and Re.s >
Z
e−σ(T γ) |y| s−
n+1
2
[dy] = π
n(n−1)
4
n−1
, we have
2
n−1
Y
ν=o
γ=γ(n) >o
Γ(s − ν/2 )|T |−s
(143)
The proof is by induction on n. We first observe that it suffices to
consider only the case T = E, as in the general case, we can write
T = RR′ and changing the variable y in the above integral to y∗ = y[R]
101
8. Lemmas concerning special integrals
102
we shall have
|y∗ |
n+1
2
∂(y∗ )
= |R|n+ℓ by (138) so that
∂(y)
[dy∗ ] = |[R]|−n+ℓ 2[dy] = |y|−n+ℓ 2|R|−n−1
∂(y∗ )
[dy]
∂(y)
= |y|−n+ℓ 2[dy]
and |y| s = |y∗ | s |T |−s as y = y∗ [R−1 ].
Then the integral we had, transforms into
Z
n+ℓ
∗
−s
e−σ(y ) |y∗ | s− 2 [dy∗ ]
|T |
y∗ >o
and what we claimed is now immediate. We shall assume then that
T = E. Let
!#
!"
!
y1 o E w
y1 Y
.
−
Y=
o y o 1
Y ′ ynn
104
The above is a special case of our representation (132) where we
have set r = n − 1. The following relations are immediate.


ynn = y + y1 [w]

(144)



= y1 [w]
Clearly then, the functional determinant
E
∂(y)
∂(y1 , ynn )
=
is of the type *
∂(y1 , y, w) ∂(y1 , y, w)
*
∂(y)
= |y1 |
∂(y1 , Y, W )
0
1
0
0
* so that
y1
(145)
Also |y| = |y1 |Y, σ(y) = σ(y1 ) + Y + y1 [W ], and y > o when
and only when y1 > o and Y > o all these being immediate consequences of (144). Hence we have, writing d[W ] = πn−1
ν=1 dϑν there
W ′ = (ϑ1 .ϑ2 , . . . ϑn−1 ),
Z Z Z
Z
−σ(y) s− n+1
e−σ(y1 )−y−y1 [W ]
e
|y| 2 [dy] =
y>0
y1 >o y>o W
103
|y1 | s−
n+1
= y(s −
)
2
Z
e−σ(y1 ) |y1 | s−
y1 >o
n−1
2
=π
y(s −
n−1
)
2
Z
y1 >o
n−1
2
n−1
2
Y
W
e−σ(y1 ) |y1 | s−
s− n+1
2
[dW ]dY [ny1 ]
e−y1 [W ] [dW ] [dy1 ]
(n−2)+1
[dy1 ]
2
observing
that, if we write y = R′ R on Y = (w1 , w2 , . . . wn−1 ) = (RW ) then the 105
integral within the parenthesis, upto a factor, reduces to
Z
w21 −w2n−1
e
dw1 . . . dwn−1 =
Zα
2
e−w dw
σ
n−1
n−l
= π1/2 .
By the induction assumption, the last integral, viz.
Z
n−2
Y
n−1 n−2
y(S0ν/2).
e−σ(y1 ) |dy1 | = π 2 , 2
ν=0
y1 >o
Substituting this in the above we get
Z
n−1
Y
n n−1
−σ(y) S− n+1
. 2
2
2
e
|y|
[dy] = Π
y(S − ν/2)
ν=0
y>0
and the proof is complete.
With Rn denoting space of all reduced matrices y = yn > 0 we state
Lemma 15. Let f (y) be a real and continuous functions on the ray y ≥ o
n−2
> 0 . Then
and let s +
2
Z
y∈Kn
|y|≥t
n+1
f (|y|)|y| [dy] =
ϑn ,
2
s
Z1
f (y)ys+
n−1
z
dy
0
where ϑn is a certain positive constant, depending only on n.
(146)
104
Proof. We first prove the lemma in the special case when f (y) ≡ 1 and
s = o. Then (146) reduces to
Z
n+1
(146)′
[dy] = ϑn t
2
y∈Kn
||y|≤t|
If only we know that the integral on the left of ((146)′ ), exists then in
view ofR its homogenity in the variables y, ((146)′ ) certainly holds with
ϑn =
[dy].
y∈Kn
|y|≤L
106
We shall now show that the integral
Z
[dy] exists.
yn (t) =
y∈K
|y|≤∈
Towards
R this end, we approximate yn (t) by the following integral
|y ≤∈ [dy] where δ denotes a small positive number. The
yn (δ, t) =
y∈Kn
yl l≥δ
above requirements in y will imply, in view of the reduction conditions
that all y′νν s are bounded as
δ ≤ yi1 ≤ . . . ≤ ynn .Πnν=1 yνν ≤ c, |λ| ≤ Ci t
and as a consequence, all the y′Nn s are bounded. In other words the domain of integration for Jn (s, ∈) is finite (i.e. Jn (δ, ∈)) is a proper integral
) so that Jn (δ, ∈)
R certainly exists. We need then only show that Jn (δ, ∈)
has a limit as → o and then this limit is obviously Jn (δ, ∈). Specifically we show that if 0 < δ < δ′ , then the difference Jn (δ, ∈), Jn (δ, ∈)
(which is positive ) can be made arbitrarily small by taking δ sufficiently
small. This difference is representable as the integral.
Z
[dy]
(147)
y∈Kn ,|y|≤∈,δ≤y11 ≤δ
105
If we write λ = yx11 yxi , yi = y(n−1)
, then the domain of integration
i
on (147) is contained in the following domain
′
δ ≤ yii ≤ δ
1
|y2v | ≤ yi1 ν = 2, 3, . . . , , .
2
|yi > 0|
as the inequalities yi1 |yi | ≤ yi1 y22 · · · ≤ ci |λ| ≤ t which are consequences 107
of (81) and (49), show. Denoting the last domain as L , we have, in
view of the induction hypothesis ,
Z
Z
Z
Z
[dy, ]dy, z..dyi ndyi i
[dy] =
L
δ≤y11 ≤δ |yi ν≤1 \2|y11 y1 ∈Kn−1
ν=2.3,n |ri |≤C,t\y11
= ϑn−1
Z
Z
y<y11 δ |y1 1|≤1 /2 y11 |
= ϑn−1
Z
δ≤y11 ≤δ
= (ci )n\2 ϑn−1
C t n/2 1
dy12 . . . dy1n dyii
y1 1
C t n/2
1
dy12 . . . , dy1n dy11
y1 1
Z
yn/2−1
dy11 <∈′
11
δ′ ≤y11 ≤δ
if δ is sufficiently small, ((146)′ ) now follows.
R
It is also clear that the existence of
[dy] which we proved above
y∈Kn
|y|≤t
already implies the existence of the integral on the left side of (146), taking into consideration the continuity on f (y) in y ≥ 0, so that it remains
only to establish the equality of the two side of (146). Let us introduce
the following truncated integral
Z
f (|y|)|y| s [dy]
I(a, t) =
y∈Kn
a≤|y|≤t
106
and observe that
Z
I(a, t + h) − I(a, t) = I(t, t + h) =
f (|y|)|y|A [dy]
y∈Kn
t≤|y|≤t+ℓ
108
By a mean theorem, in view of the continuity of the function f (y).y2
in the closed interval [t, t, K], the last integral is equal to
Z
[dy]
f (t + ϑh)(t + ϑ + h)s
y∈Kn,t≤|y|≤t,h
with a certain ϑ in the interval [0,1]
Thus we have from ((146)′ ),
n
o
I(a, t + h) − I(a, t) = f (t + ϑ|)(t + ϑ)s J(t + h) − 7(t) ϑ
= f(t + ϑh)(t − ϑh)s {(t − h)
n+1
2
−t
n+1
}ϑn
2
and therefore
n+1
(t + h n+1
I(a, t + R) − I(a, t)
2 )t 2
= f (t)t s ϑn lim → o
h→0
h
h
h
n
−
1
n+1
ϑn f (t)t s +
=
2
2
lim
The limit in the left side is clearly equal to
that
I(a, t) =
Zt
a
∂I(a, t)
n+1
dt =
ϑn
∂t
2
Z
o
∂(t(0.t))
. It now follows
∂t
t
f (t)t s +
n+1
dt
2
n+1
> 0 by assumption, letting a → 0, and since s +
and since s +
2
n+1
> 0 by assumption, letting a → 0, what results is precisely what
2
we wanted. We single out important special cases of (146), viz.
107
1) Choosing f (t) ≡ 1 in the above, (146) yields that
Z t
Z
n−1
n+1
n+1
ϑn
ys+ 2 xy =
|y| s [dy] =
2
2s + n + 1
o
y∈Kn
|y|≤t
ϑn τn t s+
n+1
s
provided s +
n+1
> 0.
2
(148)
d
2) Setting f (y) = e−ay (a, 0, ∂ > 0) in the above, and letting t → ∞
in (146), that the limit exists is seen by considering the right side and 109
we get
Z
Zy
n−1
d
∂
2
+
1
e−a|y| |y| s [dy] =
2n e−ay ys+ 2 dy
2
y∈Kn
0
n+1
s n + 1 −s∂ n+1
(149)
ϑn |( +
)d −2d
2d
2
2d
n+1
> 0C.
provided a > 0, ∂ > 0 and s +
2
It is remarkable that all integrals which we use and need for a consideration of the Poincare’ series are easily computed as seen from the
above.
=
Chapter 9
The Poincare’ Series
This section is devoted to the construction of modular forms in the shap 110
of the Poincare’ series. Let T ≥ 0 be a semigroup of modular matrices
of the form
!
u S o u′−1
(150)
o u′−1
where u is a unit of T meaning T [u] = T , and S o is symmetric and
integral. Let V(T ) denote a set of modular matrices which constitutes a
complete system of representatives of the left cosets of M modulo A(T ).
Then
X
M=
A(T )S
(151)
S ∈V(T )
With k standing for an even integer, we introduce the Poincare’ series as
X
g(Z, T ) =
e2πσ(T S <z>) |cz + D|−K
(152)
S ∈V(T )
!
A B
where we assume S =
C D
First we note that the Poincare’ series does not depend on the choice
of V(T ). For, this only requires that (152) is left invariant if we replace
the matrices S occurring
in (152) by MS where
!
u S o u′−1
is a matrix of the type occurring in (150). Due to 111
M=
0 u′−1
109
9. The Poincare’ Series
110
this replacement σ(T S < Z >) will go over into σ(T MS < Z >) and
σ(T MS < Z >) = σ{T (S < Z > [u′ ] + S o )}
= T (S < Z > [u′ ] + S o )
= σ(T [u]S < Z >) + σ(T S o )
= σ(T S < Z >) + σ(T S o )
since u is a unit of T . Also
X
X
X
σ(T S o ) =
tµν soνµ =
tµµ soµµ −
tµν 2soµν
µ,ν
µ<ν
so that σ(T S o ) is an integer as S o = (soµν ) is semintegral. It therefore
follows that the exponential factors occurring in (152) are left unaltered
by such a replacement. The same is trivially true of the other factor
|cZ + D|K as it goes over into |u−1 cZ +u−1 D| = |u′ |K|cZ + D|K = |cZ + D|K ,
u being unimodular and k being an even integer. It is immediate that
g(Z, T ) is independent of the choice of V(T ). We now contend that
g(Z, T )|M = g(Z, T )|
(153)
!
Ao Bo
∈ M where
for any M =
C o Do
g(Z, T )|M = g(M < Z >, T )|C0 Z + D0 |−K
For, g(Z, T )|M =
2πi(σ(T S )<Z>) |C Z
0
S ∈V(T ) e
P
=
X
S ∈V(T )M
+ D0 |−K
e2π(σ(T S )<Z>) |cZ + D|−K
= g(Z, T )
112
since V(T )M represents a set of representatives of the left cosets of
A(T ) in M in view of V(T ) doing so. What is more, g(Z, T ) depends
only on the equivalence class of T . In other words a replacement of T
by T [y] where V is unimodular but arbitrary, does not affect g(Z, T ). We
may remark are this stage that all our considerations are formal for the
111
present. V being any unimodular matrix, the group A(T [V]) consists of
all modular matrices
!
u1 S 1 u′−1
1
with (T [v])[u1 ] = T [V]
0
u−1
1
and S 1 = S 1′ (integral). This means that if we introduce
v C
u = vu1 v , S = vS 1 v and R =
0 v−1
−1
′
!
then R ∈ M and, with T [u] = T ,
!
!
′
u1 S − 1u−1
u S u′ −1
−1
1
.
R =
R
0
u−1
0 u−1′
1
An argument in the reverse direction is also valid and thus we obtain
RA(T [v])R−1 = A(T )
From (151) we have
X
A(T )S =
M=
S ∈V(T )
X
(154)
A(T [v]S )
S ∈V(T [v])
so that using (154)
113
M = RMR−1 =
X
RA(T [v])R−1 RS R−1
S ∈V(T [v])
=
X
.
S ∈RV(T [v])R−1 A(T )S
Thus RV(T [v])R−1 is a set of the kind V(T ) and multiplication on
the right by a modular substitution will again lead to a set of the same
kind so that RV(R[v]) is itself a set of the type V(T ). Now
X
g(Z, T [V]) =
e2πσ(T [v]S <Z>) |cZ + D|K .
S ∈V(T [v])
112
Since
σ(T [v]S < Z >) = σ(T S < Z > [v]) = σ(T RS < Z >)
we conclude by arguments as in earlier contexts that
g(Z, T [v]) = g(Z, T )
(155)
where V is an arbitrary unimodular matrix.
We are heading towards the proof of the absolute and uniform convergence of the Poincare’ series (152) in every compact subset of Y
provided
K > min(2n, n + 1 + rank T )
(156)
114
and then for such R′ s, g(Z, T ) is a regular function in Y . The absolute
convergence will in particular justify all our earlier considerations which
were hitherto formal and then by means of (153), we would have proved
that g(Z, T ) represents a modular form of degree n > 1 and weight k.
The case n = 1 requires some further consideration.
In view of (155) and (106), it suffices to consider the convergence
of the Poincare’ series in the special case when
!
T1 0
T=
, T 1 = T 1(r) > 0.
0 0
(157)
!
u1 u2
If u is a unit of T , we decompose u analogous to T as u =
u3 u4
and use the relation T [u] = T to infer that u2 = 0 and T 1 [u1 ] = T 1 . Let
us denote by Ar (r ≥ 0) the group of modular matrices
!
u S 0 u′−1
(158)
M=
0 u′−1
!
E (r) o
with u =
u- unimodular and S o symmetric integral. If T be
u3 u4
as in (157) which it is throughout our subsequent discussion, then Ar is
actually a sub-group of A(T ) of finite index (A(T ) : Ar ). In fact, if ε(T 1 )
113
denotes the number of units of T 1 = T 1(r) for r > 0 and denotes unity if
r = 0, then
(A(T ) : Ar ) = ε(T 1 ).
Let Vr be a set of representatives of the left cosets of Ar in M so that
X
M=
Ar S , r ≥ 0.
(159)
S ∈Vr
It is clear that to each S ∈ V(T ) there corresponds ε(T 1 ) elements
of Vr say S 1 , S 2 , . . . S ε(T 1 ) all of which belong to the same left coset of 115
A(T ) in M and consequently,
!
1 X 2πσ(T S <Z>)
A B
−K
e
|cZ + D| , S =
g(Z, T ) =
C D
ε(T 1 )
S εVr
P
By (159), M = S ∈Vo A0 S so that if we assume
P
P
Ao = R Ar R, then M = RS Ar RS . Comparing this with (159)
we conclude that Vr is obtained as all possible products RS where R
runs through a set of representatives of left cosets of Ar in Ao and S run
through all the elements of Vo . In view of an earlier result (pp. 11) we
can take for Vo the set of all matrices whose second matrix rows constitute a class of coprime symmetric mutually non-associated !pairs. We
u ∗
, ν = 1, 2
now determine a representative system for R. If Rν = ν
0 ∗
are two matrices which belong to the
! same coset of A in Ao then there
u
S o u−1
of the type occurring in (158) with
exists a matrix M =
ou−1
MR1 = R2 . This is easily seen !to mean that uu1 = u2 and consequently
!
E (r) u3
E (r) u3
′
′
′
′
the above gives u1
u1 u = u2 . Since u =
= u2 ′ . If
o u4
o u4
u′ν = (p(r,r) w), ν = 1.2, such an equation can hold only when p1 = p2 .
It is now easy to infer that if P = P(r,r) runs through all primitive matrices and to each P we make correspond a unique matrix u∗ obtained by
completing P to a unimodular matrix in an arbitrary way, then the class 116
of all matrices
!
u∗
o
(161)
R=
′−1
o u∗
114
provides a compete representative system of the left cosets of Ar in Ao .
We now have
1 X X 2πiσ(T RS <Z>)
g(Z, T ) =
e
|CZ + D|−K
ε(T 1 ) R ∗∗V
o
where (C, D) which should strictly denote the second matrix row of the
product RS can be assumed to be the second matrix row of S in view
of the special form (161) of R. Since to each primitive P = P(n,r) we
have corresponded a unique R, the summation in the last series can be
regarded as one over the set of P′ s instead of the R′ s. In fact the general
term of the last series depends only on P and not on the rest of the
columns of R as the equations
σ(T RS < Z >) = σ{T (S < Z >)[u′∗ ]}
= σ{T [u∗ ]S < Z >}
= σ(T 1 [P′ ]S < Z >)
show. We have thus shown that
1 X X 2πiσ(T 1 [P′ ]S <Z>)
e
|CZ + S |−K
g(Z, T ) =
∈ (T 1 ) P S ∈V
(162)
o
!
A B
where S =
C D
We proceed to construct a suitable fundamental domain for Ar in Y .
We use the parametric representation (132) for y > o as
!"
!#
y1 o
E y
y=
(132)′
o y2 o E
117
(n−r)
and v = V (r,n−n) .
where y1 = y(r)
1 , y2 = y2
The substitution (158) takes Z = X − iy ∈ Y
!"
y1 o E
′
′
Z[u ] + S o = X[u ] + S o + i
o y2 o
!"
y1 o E
′
= X[u ] + S o + i
o y2 o
into
!
!#
o E u3 ′
y2 o u4 ′
!#
u3 ′ + vu4 ′
u4 ′
115
y
o
= X[u ] + S o + i 1
o y2 [u1 ′ ]
′
!"
E u3 ′ + vu4
o
u4 ′
!#
Comparing this with
y
o
Z = X + iy = × + i 1
o y2
!"
E V
o E
we deduce that a transformation of the type (158)
changes.
× → ×[u′ ] + S o y1 → y1
Y2 → y2 [u4 ′ ].
!#
effects the following




.
′

V → u3 + Vu4 
(163)
We need a few notations. We shall denote by
1) KK,n−n the set of all matrices V = V (K,n−n) = (ϑm ν) satisfying the
1
1
conditions: − ≤ ϑµν ≤ + for all µ, ν,
2
2
2) Kn the set of all matrices × = x(n) with
× = ×′ , −
1
1
≤ ×µν ≤ + , µ, ν = 1, 2, . . . r
2
2
3) Kn′ the set of all reduced positive matrices y − y(n) (in the sense of
Minkowski)
By an appropriate choice of u4 in (158), we can obtain y2 [u4 ′ ] in 118
(163) as a reduced matrix, in other words Y2 [u4 ′ ] ∈ Kn−n and in general,
uΛ is uniquely determined. We can then choose an integral matrix u3
such that in (163), u3 vu4 ′ ∈ KK,n−n and u3 is also uniquely determined
in general. The choice of u3 and u4 fixes u and then we can choose S o
such that [u′ ] + S o ∈ Kn . In general S o is also uniquely determined.
These considerations prompt us to define a fundamental domain yu for
A2 on K as the set of Z.
!"
!#
y1 o E V
Z = × + iy = × + i
where
o y2 o E
× ∈ yWn , y2 ∈ Kn−n , V ∈ yn−n .
(164)
116
For purposes of proving the convergence of the Poincare’ series, we
in fact consider a majorant of it, viZ. the series
h(Z, T ) =
1 X −2πσ(T yS ) h/2
e
|y2 |
ε(T Z ) S ∈V
(165)
n
119
which arises from (??) by replacing each term of (??) by its absolute
value multiplied by a factor |y|K = |y|K/2 ||CZ + D||K where we denote
S < Z >= ×S + y2 . The absolute or uniform convergence of the above
series implies the corresponding fact for the Poincare’ series. In the
sequel therefore, we shall concern ourselves with this new series.
We first prove the uniform convergence of h(Z, T ) on special compact subsets of Y , viz. the symplectic spheres and the general case will
immediately follow. Let Z0 ∈ Y be fixed and let Ko denote the symplec1
tic sphere s(Z, Z) ≤ ρ Assume ρ to be so chosen that Ko does not have
2
a non empty intersection with any of the S (Ko )Y without being identical with it, S denoting an arbitrary modular substitution. The validity
of this assumption is an immediate consequence of the modular group
being discontinuous on Y (cf. Lemmas 8). Let Z, Z ∗ be arbitrary points
of Kν . Then we have s(Z, Z ∗) ≤ ρ and s(s < Z >), S < Z ∗ >≤ S , S ∈ M.
With a view to obtaining a convergent majorant for h(Z ∗ T ) which does
not depend on Z ∗ ∈ Ko we work
! out the following estimations.
rn r
If Z = × + iy and y =
with y1 = y(r)
1 then in view of the special
∗y
form (157) for T we have
σ(T y) = σ(T 1 y1 ).
Let R = R(n) be a real matrix such that y = RR′ and let wν , u =
1, 2, . . . r denote the columns of R. E denoting any real column, let µ1 =
minν=1 r1 [∈] and µ2 = µε=1 T 1 [ε], Then we have σ(T 1 Y2 ) = σ(T 1 (ε)) =
ν
P
σ(Γ1 [Wν ]). Also
ν=1
µ1 σ(y1 ) = µ1
r
X
ν=1
4′ν Wν ≤
r
X
ν=1
T 1 [Wν ]
117
≤ Γ2
r
X
Wν′ Wν = µ2 σy1
ν=1
so that
Γ1 σ(y1 ) ≤ (T y) ≤ µν σ(yλ )
ν(µ) Replacing Z by Z ν = xr + 0yν , yν = zν yr .
we have from the above
µ1 σ(yν1 ) ≤ σ(T j2 ) ≤ µ2 σ : yν1
Since S (Z, Z ∗ = ̺), by means of Lemma 13, we have
σ(y∗1 )
1
≤ m1
≤
m1 σ(y1 )
Combining all these it results that
σ(T y∗ ) ≥ µ1 σ(y∗1 ) ≥
µ1
σ(y1 ) ≥ mσ(T y′ , mi0
m1
The interchange of y and y∗ leads to an inequality in the opposite
direction and we have
m≤
σ(T y∗ )
1
≤
σ(T y)
m
The assumption on Z, Z ∗ used to prove the above inequalities are
also true of S hZi, S hZ ∗i, S ∈ M so that, writing
S hZi = xs + tY s , S hZ ∗ i = λ∗s + iy∗s
we also have
m≤
σ(T y∗s )
1
≤
σ(T ys ) m
From Lemma 13 we have an estimate for |y∗s |/|ys | viz.
|y∗ |
1
≤ s ≤ ms
mZ |ys |
and consequently |y∗s |k/Z ≤ M |xs |k/Z with M = mk/Z
s .
120
118
The general term of (165) allows then the following estimation for
Z ∈ Eo , viz
∗
e−2πσ(T ys ) |y∗s |k/2 ≤ M e−2πmσ(T ys ,) |ys |k/2
On integration, the above yields
Z
M
−2πσ(T y∗s ) ∗ k/2
e−2πmσ(T ys ,) |ys |k/2 |ys |n−1 dxT o
e
|ys | ≤
ko
y∈Ro
where ko is the volume of Eo and is given by
Z
Z
−n−1
|y|
[dx][dy] −
|yZ |n [dx][dy]
ko =
Z∈Eo
ko
121
Z
M 1 X
∗
h(Z , T ) ≤
∈−2πmσ(T ys ) |ys |k/Z−n−1 [dxs ][dys ]
ko ε(T 1 ) ε∈V
rZ∈E
Z2
X
M 1
=
∈−2πmσ(T y) |y|k/Z−n−1 [dx][dy]01∈s
ko ε(T 1 ) S ∈V
rZ∈∈hE
2i
(166)
We now break the integral
Z
e−2πmσ(T y) |y|k/Z−n−1 [dx][dy]
shE,oi
into a sum of integrals the domain for each of which can be brought into
a subset of the fundamental domain ′ o′n of An by means of an appropriate substitution in An . Noting that the integrated above is invariant
under the substitutions of An we would have then expressed the above
integral as a sum of integrals, all taken over appropriate subsets of yon .
This is done as follows. Since yon is a fundamental domain for An in ξ
S
M hyO
we have ξ =
n i. Hence
M ∈An
S hKo i = ̺hKo i ∩ Y
[
M ∈An
̺hKo i ∩ Mhyon i
119
Let shKo i ∩ M s∼ hyon i be non empty for ν = 1, 2, . . . , as , the suffix S in
denoting that these depend on S . (It is not difficult to show that
µ s is finite in each case but we do not need this). Let further
M s∞, as
Ls(ν) = S hK̃i ∩ M∈(ν) hyon i, ν = 1, R, . . . , R s .
−I
Then S hRi = ∪aν=1 LS(ν) . If Ls(ν) = M (ν) hLy(ν) i, then Ls(ν) ⊂ L
for each λ and we have clearly
Z
K
e−2πmσ(T y) |y| 2 −n−1 [dx][dy]
S hKo i
=
as Z
X
ν=I
K
e−2πmσ(T y) |y| 2 −n−1 [dx][dy]
LS (ν)
and consequently, from (166)
h(Z ∗ , T ) ≤
122
as Z
XX
K
µ
e−2πmσ(T y) |y| 2 −n−1 [dx][dy]
Jo ε(T 1 ) S εV ν=1
r
(167)
(ν)
LS
We certainly know that the union
∪
S ∈Vr
ν=1,2...as
LS(ν) is contained in yor .
But something more is true, viz. this union is actually contained in the
set of Z = × + iy defined by:
Z ∈ yr , |y| ≤ t
if t is chosen sufficiently large. For |y| is invariant under the substitutions
in Ar and then we need only prove that |y| ≤ t for
s
Z = × + τy ∈ ∪aν=1
LS(ν) = S .
Consider now the set of points {S < Z0 > /S ∈ Vr }. Being a set
of equivalent points it has a highest point in it, say Si < Z0 >. Then if
Si < Z0 >= ×os + iyo s, we have ||yosi || ≥ ||yos || for everyS ∈ vr Let Z
120
123
1
be an arbitrary point is S < K0 >. Then (Z, s < Z0 >) ≤ P so that by
2
Lemma 13
1
|y|
≤ m2 , m2 = n2(ρ, n).
≤
m2 |yos |
Thus |y| ≤ m2 |yos | ≤ m2 |yosi | = t (say). Then with this t (which
S
is contained in the set of Z ′ s
is a fixed number) we have that
s∈Vr
u=1,2...0S
defined by: Z ∈ rr , |y| ≤ t If now ∈0 denotes the number of modular
matrices s having Z0 as a fixed point, viz. s < Z0 >= Z0 , we know
from Lemma 11 that ∈0 is finite and we want to show that every point
S ν
L sas In other words, we
of yr appears at most e20 times in the union
s∈Vr
u=1,2,
have got to show that if ∃ denotes the set of all pairs (ν, s) with Z ∈ L(ν)
s
where >, is a fied point in fr , then this set has at most e20 elements. If
−1
−1
(ν)
shK0 i and
(ν, s), (ν0 , s0 ) be two pairs in f, then Z ∈ M sν hL(ν)
s i ⊂ ms
(ν)−1
−1
(ν)
0)
similarly Z ∈ M s0 0 hL(ν
s0 i ⊂ M s0 s0 hK0 i. This means that the two
−1
−1
spheres M s(ν) S hK0 i, M s(ν0 0) hK0 i have one point in common and hence
the spheres themselves are identical. In particular, their centres are equal
so that
−1
−1
−1
M s(ν0 0 ) S 0 hZ0 i = MS(ν) S hZ0 i.i.e.Z0 = S 0−1 M s(ν0 0 ) M s(ν) S hZ0 i.
124
Keeping (ν0 , S 0 ) fixed, we have now produced for each pair (ν, S ) ∈
f a substitution S 0−1 M s(ν0 0 ) M s( ν)−1
s S which has Z0 as a fixed point and
which consequently belongs to a finite set of e0 elements. It easily follows then that the possible choices for (ν, S ) are at most e20 in number.
Turning to our series h(Z ∗ , T ) and its majorant in (167), we can now
state that
as Z
XX
R
M
m
e−2πmσ(T y) |y| 2 − n − 1[dx][dy]
h(Z , T ) ≤
J0 ∈ (T 1 ) s∈v ν=1
r
Me20
≤
J0 ∈ (T 1 )
Z
Z∈yr
|y|≤t
Z∈L(ν)
s
|y|≤t
R
e−2πmσ(T y) |y| 2 − n − 1[dx][dy]
121
=
Me20
F(mT 1 , t)(say)
J0 ∈ (T 1 )
It only remains therefore to investigate under what conditions the
last integral F(mT 1 , T ) exists. First we treat the case 0 < r < n. In this
case we have the parametric representation for y > 0 from (132) as
!"
!#
y1 o E v
y=
o y2 o E
where y1 = y1 (r), y2 = y(n−r)
, ν = ν(r,n−r) and if Z = × + iy ∈ yr then
2
× ∈ XXXXXρn , ν ∈ Kρr,n−r and y2 ∈ Kn−r As with (145), we can show
α(y)
= |y1 |n−r and consequently
that
α(y1 , y2 , ν)
[dy] = |y1 |n−r [dy1 ][dy2 ][dv]
(We may remark that in [dν] we have the product of the differentials of
all the r(n − r) elements of V as ν is not in general symmetric). Also
|y| = |y1 ||y2 |andσ(T y) = σ(T 1 y1 ). Hence
Z
ϕ−Zamσ(T y) |y|k/Z−n−1 [d×][dy]
F(mT 1 , t) ≡
Z∈Lr
|y|≤t
Z
=
e−2πmσ(T i yi ) |yi |K/2,n−1 [dx][dyi ][dyi ][dy]
×∈Kϕn .ν∈Kϕn ,n.r,|y1 ||y2 |≤t
y1 >0,y2 ∈Kn−r |y1 ||y2 |≤t
=
Z
e−2πmσ(T i yi ) |yi |K/2−r−1 |y2 |K/2−n−1 [dyi ][dy2 ]
yi >0,y2 ∈Kn−r
|yi ||y2 |≤t
since the above integrand is independent of ×, ν and the volumes of Kϕn ,
Kϕr , n − r which are the domains for ×, ν are both unity. Thus, using
(146) and (143) we can now state that
Z
Z
−2πme(T,yi )
K/2−m−1
|y2 |K/2−r−e [iy2 ] [dyi ]
F(mT i , t) =
e
|yi |
yi >0
y2 ∈Kn−r
|y2 |≤t/|yi |
122
Z
n−r−1
[dyi ]
2
r−1
Y
K−n−r−1 n(r−1)
−rt
n−r+1
n−ν
=
ϑr−r t 2 π 4 (2πm) 2 |r|−n/2
)
r(
K−n−r−1
2
ν=a
K−n−r−1
n−r+1
ϑn−r t 2
=
K−n−r−1
125
yi > 0e−2πmσ(T i yi ) |yi |
provided k > n + r + 1.
In particular we have shown the existence of F(mT i t) under the assumption k > n + r + 1 0 < r < n. We now discuss the border cases
r = 0, n.
Let r = 0. Then using (146)
Z
F(mT i , t) =
|y|K/2−n−1 [d×][dy]
×∈nϕn ,y∈Kn,|y|≤t
Z
=
|y|H/2−n−1 [dy]
y∈Kn ,|y|≤t
=
K−n−r−1
n+1
ϑn t 2
K−n−1
provided k > n + 1 Let now r = n. Then
Z
F(mT i , t) =
e−2πmσ(T y) |y|K/2−n−1 [d×][dy]
≤
Z
=π
y>0
×∈Kϕn,y>0,|y|≤t
e−2πmσ(T y) |y|K/2−n−1 [dy]
n(n−1)
4
(2πm)
n(n+1−K
2
|T |
n+1−K
2
πnν=i ⌈(
K−n−ν
)
2
provided k > 2n.
It follows that in any case provided k > min(2n, n + r + 1), r =
rank T the Poincare’ series g(Z, T ) converges absolutely and uniformly
in a neighbourhood of every point Zey and hence also in every compact
subset of y.
Since by our assumption k is an even integer, the above condition on
k actually simplifies into
k > n + 1 + rankT.
123
In the case n = 1, we can say something more. In fact, the Poincare’
series in this case has a convergent majorant which does not depend
on Z ∈∈n so that g(Z, T ) is actually bounded in ⌈n . This result is a
consequence of the following
Lemma 16. Let µ, ϑ, ×, y be real numbers, and let |x| ≤ b and y ≥ δ > 0 126
for suitable constants C , ξ. If Z = x + iy, then
|µZ + ϑ| ≥∈ |νi + ϑ|
(168)
with a certain positive constant ∈=∈ (δ, C .
If µ2 +ν2 = 0, (168) is trivially true so that we can assume µ2 +ν2 > 0.
Then, due to the homogenity of (168) in µ, ν we can in fact assume that
µ2 + ν2 = i. The proof is by contradiction. If the Lemma were not true,
there exists sequences Zn = µn + iyn , un , ϑn , n = 1, 2 . . . , with µ2n + ν2n =
|xn | ≤ E , yn ≥ δ and finally |un 2n + ϑn |2 = (νn yn )2 + (νn xn + ϑn )2 < 1/n.
So, in particular u2n < 1/ny2n ≤ 1/nδ2 so that un → 0XXXXXn → ∞
This implies, since |xn | ≤ ζ that un xn → o and consequently ϑn → o.
But this is impossible as u2n + ϑ2n = 1 1 for every n. This proves our
Lemma.
Reverting to the Poincare’ series in the case n = 1, we have
g(Z, T ) =
1 X Z,∗∗∗∗∗
e
|eZ + D1
ε(T 1 ) o
1Vn
If r = 1, the above series is majorised by the series
1 X 2πtoy
e
|c2 − D1−R
ε(t) S oV
1
obtained by taking absolute values termwise and in the alternative case,
P
|cL + D1
vie, r = 0, it is injorised by
S gVo
√
For Z ∈ δ1 , whence |x| ≤ ζ = 1/2 and |y| ≥ δ = 3/2, Z = x + y, 127
P
either of these has the convergent majorant ζ |ei + d1−K independent
of Z by means of Lemma 15.
We one summarize our results into the following
124
Theorem 9. The Poincare, series (152), Viz.
g(Z, T ) =
X
S ∈V(T )
2πσ(T S <Z>)
e
A B
|CZ|D1 , S =
C D
K
!
converges absolutely and uniformly in every compact subset of Y and
represent a modular form of weight k for K ≡ o(2), k > n + 1 + rankT ,
where T stands for an arbitrary semi integral semi positive matrix.
Later, we shall prove that Poincare’ series actually generates all
modular forms of degree n and weight k provided k ≡ 0(2) and k > 2n.
This proof requires a new tool, viz, the metrization of modular forms
and the next section is devoted this topic.
Chapter 10
The metrization of Modular
forms of degree
In §.5 we defined modular forms of degree n ≥ 1 and weight k and we 128
set up an operator φ which maps the linear space MK(n) of such forms
when n > 1. With a view to facilitating the statement of our
into M(n−1)
K
subsequent results we agree to call constants modular forms of degree
0 and a given weight k and extend the domain of φ to include MK(1) by
specifying that for modular forms f (Z) ∈ MK1 , f (Z)/φ = a, (o), viz, its
‘first’ Fourier coefficient . We can now state that
(n) MK φ ⊂ MK(n−1) , n ≥ 1.
It is of interest to investigate when the reverse inclusion is also true,
viz, when the inclusion relation above is actually a relation of equality. This turns out to be true in all the cases where the Poincare’ series
g(Z, T ) which depend on k and n have been proved to exist (as modular forms of degree n and weight k), as we shall see subsequently . A
modular form of degree n > 0 shall be called a cusp form if
f (Z)|φ ≡ o
Any modular form of degree 0 shall by definition be a cusp form
too. If f (Z), g(Z) be two modular forms of degree n ≥ 0 and weight k,
125
126
129
10. The metrization of Modular forms of degree
of which at least one is a cusp form, we define a scalar product ( f, g) by
R


f (Z)g(Z) |y|K−n−1 [d×][dy], f nn > o, Z − x + y


f
( f, g) = 
x (169)
n



 f ḡ,
if n = 0
We prove the existence of the right side of (169) to make our definition meaningful. Let, of the two, g(Z) be a cusp form.
It follows from (111) that
|g(Z)| ≤ ζe−2∈Y |y| for Z = χ+ ∈ y ∈ Fn
√
with certain constants ∈, ζ0. Clearly f (Z) and ∈−∈ |y| |y|K−n−1 are bounded in fn . We have therefore only to prove the existence of the integral
Z
√
e∈u |y| [dy][dx].
fn
Since fn is contained in the set of zXXXXX defined by:
Z = x + iy, χ ∈ Mn , y ∈ Kn
√
R
the above integral is majorised by
ε−∈u |y| [dy] and this integral, we
y∈Kn
130
know, exists by (149).
We also observe that the integrand f (Z, g, Z̄)|y|K−n−1 [dx][dy] is invariant with respect to the modular substitutions, as the product g(Z)
f (Z)|y|K and dy = |y|−n−1 [dχ][dy] are so, and consequently it is permissible to replace the domain of integration fn occurring in the right side
of (169) by any other fundamental domain for Mn
The cusp forms of degree n and weight k constitute a linear space
yK(n) , a sub space of MK(n) . We define the orthogonal space y(n)
of y(K)
in
K
K
()
(n)
MK in the usual way, is the set of forms in MK , which are orthogonal
to all the forms in y(n)
K in the sense of our inner product. It is well known
(n)
(n)
that MK is the direct sum of y(n)
K and K i.e.
+ NK(n) .
MK(n) = y(n)
K
A much finer decomposition of MK(n) is given in
127
Theorem 10. There exists a uniquely determined representation of MK(n)
as a direct sum
(170)
+ · · · + y(n)
+ y(n)
MK(n) = y(n)
Kn
K1
R0
with
y(n)
⊂ y(n)
Kn
K
(n−1)
(n) (n)
for ν < n and y(n)
Kν ⊂ NK , yKo & ⊂ yR ′′
Proof. We first remark that the mapping φ is 1 − 1 on NK(n) .
(171)
Indeed, due to linearity of φ, this is ensured if f (Z) ∈ NK(n) , f (Z)φ ≡
0 together imply that f (Z) ≡ 0, but this is immediate as, in this case,
f (Z) ∈ NK(n) oy(n)
K = (0).
Also the theorem is innocuous in the case n = 0. We now base
our proof on induction on n. We therefore assume given y(n−1)
Kν , ν = 1,
2 · · · (n − 1) satisfying (170)- (171) with n replaced by n − 1. Let y(n)
(ν <
Kν
n) denote the linear space all f (Z) ∈ NK(n) such that f (Z) φy(n−1)
.
Kν
can
be
characterized
by
the relaSince φ is invertible on NK(n) , ȳ(n)
K
tions:
(n−1)
(n)
(n) (n)
ȳ(n)
Kν ⊂ NK , ȳKν |φ = NK |φ ∩ yKν
By assumption
131
MK(n−1) =
n−1
X
yn−1
Kν
ν=o
so that
NK(n) |& = NK(n) |& ∩ MK(n−1) =
=
n−1
X
ν=o
(n−1)
X
ν=o
|φ
y(n)
Kν
where all the sums occurring are direct.
Since φ is 1 − 1 on NK(n) this means that
NK(n) =
n−1
X
ν=o
¯
y(n)
Kν
NK(n) |& ∩ y(n−1)
Kν
128
and then
MK(n) = NK(n) + y(n)
=
K
n−1
X
ȳ(n)
+ y(n)
Kν
K
ν=o
again all the sums direct. The last decomposition has clearly all the
desired properties.
It remains to show that decomposition is unique .
P
Let MK(n) = nν=o y(n)
Kν be another decomposition satisfying (171).
Then we have for ν < n,
(n−1)
(n)
= ȳ(n)
y(n)
K |ϕ
Kν |φ ⊂ NK |φ ∩ NKν
132
are contained in NK(n) on which φ is 1 − 1. Hence
, ȳ(n)
and both y(n)
Kν
Kν
P
(n)
(n)
=
yKν ⊂ ȳKν for ν < n, and by assumption, y(n)
⊂ y(n)
. Since nν=o y(n)
Kν
K
Kν
Pn (n)
(n)
(n)
n
ν=o ȳKν + yK and both sums are direct, it easily follows that yKν = ȳKν
(n)
for ν < n and y(n)
Kν = yKν .
The proof of theorem 10 is now complete.
As a consequence of (171) we deduce that
(n)
n−ν−1
n−ν−1
n−ν
· · · ⊂ y(n)
⊂ y(n−1)
⊂ y(n−1)
y(n)
Kν = yK .
Kν |ϕ
Kν |ϕ
Kν |ϕ
This means in words that y(n)
|ϕn−ν consists of cusp forms of degree
Kν
(n)
ν. Further φn−ν is 1 − 1 on y(n)
Kµ for µ ≤ ν ≤ n and in particular on yKν .
n−ν = o,
For, we need only show that the conditions f (Z) ⊂ y(n)
Kµ , f (Z)|φ
µ ≤ ν ≤ n, together imply that f (Z) ≡ o. Since in the case ν = n, the
assertion is trivial, we assume ν < n. We resort to induction on n.
Let g(Z) = f (Z)|φ. Then our assumption implies in view of (171)
that
n−1−ν
=o
g(Z) ∈ y(n−1)
Kµ , g(Z)|φ
The induction assumption then implies that g(Z) = 0 which in its
turn means that f (Z) = 0 as we know that φ is 1 − 1 on NK(n) and in
(n)
particular on y(n)
Kµ ⊂ NK (µ < n).
We shall now introduce a scalar product for arbitrary pairs of modular forms. In view of theorem 10, we have a decomposition for any
129
modular form f (Z) ∈ MK(n) as
f (Z) =
n
X
ν=o
fν (Z), f (Z), fν ∈ y(n)
, ν = 0.1.2 · · · n,
Kν
(n)
and
and this decomposition in unique. Let g(Z) be another form in MKν
Pn
(n)
let g(Z) = ν=o gν (2), gν (2) ∈ yKν .
Then we define the scalar product ( f (Z), g(Z)) as
( f (Z), g(Z)) =
n
X
( fν (Z)|ϕn−ν gν (2)|ϕn−ν )
(172)
ν=o
where the scalar product occurring in the right side have the meaning 133
given earlier. If g(Z) is a cusp form, then g(Z) = gn (Z) and all but the
last term vanish in the summation on the right side of (172). Thus (172)
reduces in this case to: ( f (Z), g(Z)) = ( fn (Z), gn (Z)), the right side being
interpreted in the sense of (169). This proves the consistency of our
present definition of the scalar product ( f (Z), g(Z)) with the earlier ones
given by (169) whenever applicable. Also it is clear from (172) that
(p, g) = (g, p) and that the assumption ( f, f ) = 0 implies pν (Z)|φn−ν = 0
P
for o ≤ ν ≤ n and consequently fν , ν ≤ n and f = nν=o fν vanish
identically.
We thus conclude that the metric which (172) gives rise, to, is a
positive hermitian metric.
Let us compute the scalar product (( f, Z), g(Z)) where g(Z) is represented by the Poincare’ series g(Z, T ) and f (Z) is a cusp form, i.e.
f (Z) ∈ y(n)
K . We assume of course that the Poincare’ series converges.
Since f (Z) is a cusp form, we have from (111) that
√
n
f (Z) = 0(r−2m |y|) for Z = x + y ∈ fn
where m is a suitable positive constant. Hence
√
n
|y|R/2 | f (Z)| ≤ Me−m |y| for Z ∈ fn
with a sufficiently large M. Since |y|R/2 | f (Z)| = h(Z) is invariant by
(101) and |y| does not increase when we replace Z ∈ fn by an equivalent
point with respect to Mn , it follows that
134
130
|y|k/2 | f (Z)| ≤ M ∈n
√
n
iy′
for >∈ Yr .
We require these facts to prove that the function!
T (n) o
ρ(Z) ∈2πσ(T ε) |y|k−r−1
with T = 1
, T 1(n) > o is absoo o
lutely integrable over the fundamental domain yr of the groups An . This
is certainly true as a result of our above assertions if the integral.
Z
√
n
G(T ℓ , m) =
e−Zπσ(χy)−m |y| |y|k/2 − n − 1, k×][dy]
(173)
yn
exists. We shall show that this integral does exist.
First we consider the case 0 < r < n. In this case we can use the
usual parametric representation for y > o as
!"
!#
y1 o
E v
Y=
o yw o E
Transforming (173) in terms of the new variables and carrying out
the integrations with suspect to x and v as in earlier contexts, we have
Z
√
√
n
n
k/2−r−1
G(T, m) =
∈−Zπσ(T 1 y1 −m |y1 | |y2 ||y1 |
|y2 |k/2−r−1 [dy1 ][dy2 ]
y1 >o
y2 ∈Kn−r
=
Z
y1 >o

 Z

2πσ<T 1 y1 >
k/2−r−1 
e
|y1 |

y2 oKn−r
em
√
n
|y1 |
√
n



|y2 |
k/2−n−1
|y2 |
[dy2 ] [dy]

−n(k − n − n − 1
!
n(n − r + 1)
n(k − r − r − 1
2
=
ϑn−r m
χ
2
2
Z
×
∈2πσ<T 1 y1 > |y2 |k−n−r 2[dy1 ]
y1 >o
!
−n(k−n−r−1
n(n − r + 1
n(k − n − r − 1 r(r−o/4
2
=
y
ϑn−r m
Π
2
2
1
n−u
k−2
y
(2Π)r0/2 |T | 2 Vν−o
2
131
135
The last two relations which are consequences of (149) and (143)
are true provided k > n + r + 1.
We shall now turn to the border cases r = o, n. In the first case, viZ.
r=o
Z
√
n
G(T 1 , m) =
e−m |y| |y|k/2−r−1 [dy]
y∈Kn
=
n(n + 1)
n(k − n − o − n(k−n−1
ϑn y
m 2
2
2
provided k > n + 1 and in the alternative case when r = n,
Z
√
n
G(T 1 m) =
e2πσ(T y)−m |y| |y|k/2−n−1 [dy]
y>o
≤
Z
y>o
e−2πσ(T y) |y|k/2−n−1 [dy]
n(n − 1
n(n + 1 − k n + 1 − k
k−n−ν
2
2
= Π 4 (2π)
|T |
∗ ∗ ∗ ∗nν=1 χ(
2
provided k > 2n.
The conditions in all these cases under which we have proved (173)
to exist are precisely those under which we proved the Poincare’ series
to exist and in particular we have shown that the integral
Z
H(T 1 , ρ) =
ρ(Z)e2πiσ(T Z̄) |y|k/2−n−1 [dx][dy]
(174)
Z∈yr
exists provided k ≡ 0(2) and k > n + 1 + r, where r = rank T . We
observe that the integrand in H(T 1 , f is invariant under the substitutions
in Ar so that we can choose for the domain of integration any convenient fundamental domain of Ar in the place of yr . In fact we compute 136
H(T 2 , K) by choosing “yr ” in two different ways and this will lead us to
an interesting result.
From the definition of Ar (r ≥ 0) it is clear that, while A0 contains
with every M, - M too, Ar for r > o contains at most one of the matrices
132
M, - M. So we may assume that Vr in the case n > o contains −S with
S . Then US ∈Vr S < S n > is a fundamental domain for Ar which is
covered but once if r = o, while it is covered twice if r > 0.
Let us introduce



1. if r = 0
δr = 

2. if r > 0
and obtain from (174), in view of the invariance of the integrand appearing in it under the substitutions in Ar ,
Z
1 X
f(Z) ∈−2πiσ(T Z̄) |y|k−n−1 [dx][dy]
H(T 1 , f) =
δr S ∈V
rS ∈S <f >
n
Z
X
1
=
f(S < Z >)e−2πσ(T S <Z>) |ys |k−n−1 [dxs ][dys ]
S r S ∈V
rZ∈f
n
where we assume S < Z >= X s + iys
Since dv = |y|−n−1 [dx][dy] is the invariant volume element, using
(92) and (??) we get
Z
1 X
f(S < Z > e−2πiσ(T S <Z>) |cZ
H(T 1 , f) =
δr S ∈V
rZ∈δ
n
+ D|−2k |y|k−n−1 [dx][dy]
Z
1 X
=
f(S < Z >)|cZ + d|−k e−2πiσ(T S <Z>) |cZ̄
δr S ∈V
rZ∈δ
r
+ D||y|K−n−1 [dx][dy]
Z
1 X
ρ(Z)e−2πiσ(T S <Z>) |cZ̄ + D|−K |y|K−r−1 [dx][dy]
=
δr S ∈V
rZ∈S
n
Z
X
1
=
f(Z)|y|K−n−1 (
e−2πiσ(T S <Z>) |cz̄
δr
S ∈V
r
Z∈fn
+ D|−K [dx][dy]
=
ε(T 1 )
δr
Z
Z∈fn
f(Z)g(Z, T |y|K−n−1 [dx][dy]
133
=
ε(T 1 )
(f(Z), g(z, T ))
δr
(175)
137
On the other hand we can computer H(T 1 , f) directly as follows :
The part of the integral involving × in (174) is clearly
Z
Z
−2πiσ(T Z̄)
−4πe(T y)
f(Z)e
[ax] = e
f(z)e−2πiσ(T Z)
×∈δS r
−△πσ(T y)
×∈n0n
=e
a(T )
where a(T ) is the Fourier coefficient of f (z) with respect to the exponent
- matrix T .
But a(T ) = 0 by (108) if |T | = 0 since by assumption f (z) is a cusp
form so that, in particular if K < K, H(T 1 , f and consequently the scalar
product (f(Z)g(z, T )) vanish. Assume then that r = n. Then
Z
f(Z)e−2πσ(T z) [dx]e−△πσ(T y) |y|K−n−1 [dy]
H(T 1 , f) =
×∈xyw
y>o
= a(T )
Z
e−4πσ(T y) |y|K−n−1 [dy]
y>o
n(n + 1
n(n − 1)
n
−nK Y
n + ν n+1
4
(4π) 2
y(K −
|T | 2
= a(T )π
2
ν=s
(176)
Comparing with (175) we get
Theorem 11. If f(Z) ∈ y(n)
and K ≡ 0(2), K > n + 1+ rank T , then
K
( f (z)g(z, T )) is equal to

n(n+1
n(n−1
Qn
n+1
n+ν
2


 ε(T ) a(T )π 4 (4π) 2 −nK ν=1 y(K − 2 )|T | 2 −K for T > o


o
for T = o
where a(T ) represents the Fourier coefficient of f (z) with respect to the
exponent matrix T .
We generalize the above result to the case of f (z) ∈ yK (ν ≤ n) in the
next section.
138
Chapter 11
The representation theorem
Our aim in this section is to establish a result we promised earlier at the 139
end of §9, viz. that the Poincare series actually generate all modular
form of degree n under suitable assumptions. As a first step we
! compute
Z 0
Y (Z, T )|Y . Let z∗ ∈ Yn−1 and x ∈ Y so that Z =
cYn . For
0 r
fixed Z∗ let ϕ(z) = g(Z, T ). Then ϕ(z)
! is a modular form of degree 1 and
a b
weight k. For, if the matrix
is a modular substitution of degree 1
c d
and we set
!
E 0
A=
= A(n) , B =
0 a
!
0 0
C=
= C (n) , D =
0 c
!
0 0
= B(n)
0 b
!
0 0
= D(n)
c d
!
A B
then M =
∈ Mn Also
C D
!
Z
0
M(Z) = (AZ + B)(CZ + o) =
where
0 N<z>
N(z) = (az + b)(az + d)−1
135
(177)
136
11. The representation theorem
so that the mapping z → M < z > is equivalent with the pair of mappings:
Z ∗ → Z ∗ , z → (az + b)(cz + d)−1
Besides, we have also |CZ + D| = cz + d so that from (177)
140
ϕ(z) = g(Z, T ) = g(z, T )|M = g(M < z > T )|CZ + D|−K
!
az + b
=ϕ
(cz + d)−K
cz + d
!
a b
= ϕ(z)
c d
(178)
Clearly ϕ(Z) is regular in the fundamental domain f1 of M1 It only
remains then to verify that ϕ(Z) is bounded in f1 . This is obvious at least
in the case of Z ∗ ∈ fn−1 since in this case Z ∈ fn when the imaginary part
Y of z is sufficiently large, and g(z, T ), being a modular form of degree
n, is bounded in fn .
The general case does not present any difficulty either, as in this case
there exists a substitution
!
A1 B1
∈ Mn−1 so that M1 < Z∗ >∈ fn−1 and if we set
M1 =
C 1 D1
!
!
!
!
A1 o
B1 o
C1 o
D1 o
A =
,B =
,C =
and D =
then
o 1
o o
o o
o 1
!
!
M1 < z > o
A B
∈ M1 , M < z >=
M=
o
z
C D
and |CZ + D| = |c, z∗ + D1 |.
Finally ϕ(Z) = g(z, T ) = g(z, T )|M
= g(M < z > T )|cz + D|−K
! !
MI < Z∗ > o
=g
, T |C1 z ∗ +D1 |−K .
o
z
141
The term to the extreme right is bounded as M1 < Z ∗ >∈ fn−1 and
then ϕ(z) is clearly bounded. It is now immediate that ϕ(z) is a modular
form of degree 1 and! weight k.
A B
Let S =
be an arbitrary modular substitution of degree n
C D
137
(we shall stick to this notation throughout this section). It is easy to see
that the matrix AZ + B depends linearly in z and further z appears only
in its last column. The same is true of (CZ + D) too so that |CZ + D|
is a linear function of z, say |cz + D| = cz + D. It follows then that
|CZ + D|(CZ + D)−1 too is a matrix whose elements are linear in z and
which has its last row independent of z; consequently
|CZ + D|S < Z >= |CZ + D|(AZ + B)(cz + D)−1
(179)
is also linear in z. Let then
az + b
with |CZ + D| = cz + d where a, b, c, d are
σ(T S < Z >=
cz + d
all complex numbers
not depending on z. We shall denote by LS the
!
a b
matrix
defined as above. Since Z ∈ Yn we have |CZ + D| , o
c d
ax + b
and hence cz + d , 0. Since ImLS < Z >= Im
≥ 0 for Z ∈ Y1 it
cz + d
a
a
follows that either o = o in which case is real and ≥ 0 or if c , 0
d
d
d
then Im − ≥ 0. The Poincare’ series (152) now takes the form
c
ϕ(z) = g(Z, T ) =
X
2πi
e
az + b
cz + d (cz + d)−K .
(180)
S ∈V(T )
We construct a suitable V(T ) as follows. Let △ be the cyclic group
generated by the modular substitution
!

 (n) o o 

E
o 1 
G = 


o
E (n)
The mapping Z → G < Z > leaves Z ∗ fixed and takes z into z + 1. 142
With the notation
ωS (Z) = e2πiσ(T S <Z>) |CZ + D|−K
!
A B
2πiσ(T Z)
=e
|S , S =
C D
138
we have ωS G ∈ (Z) = e2πiσ(T Z) |S G
az + b
!
−K 1 o
cz
+
d
(cz + d) |
=e
o 1
a(z + d) + d
2πi
= e c(z + c) + d (c(z + b) + d)−K
2πi
!
a b
where we recall
= LS .
c d
Let
M=
X
A(T )S ∗ .
S ∗∈V∗(T )
If nS denotes the set of the
∗
(S GS ∗−1 )t = S ∗ Gt S ∗−1 ∈ A(T ),
exponents t with the property that
then it is easily seen that the products
S ∗Gt with S ∗ ∈ V ∗ (T )t = t mod nS constitute a set of the type V(T ).
So we obtain from (180) that
X
X
X
ϕ(Z) =
ωS G t(z) =
ϕ(z)
(181)
S ∈V∗(T ) t
mod nS
X
ωS G t(Z)
S ∈V∗(T )
where
ϕK (z) =
t
mod nS
e
t
X
mod nS
e2πcLS <z+b> (c(z + t) + d)−K
t
X
=
=
142
2πi
a(z + t) + b
a(z + t) + d (c(z + b) + d)−K
(182)
mod nS
It is obvious that ϕS (z + 1) = ϕS (z) in other words, the functions
ϕS (z)(S ∈ V ∗ (T ) and consequently ϕ(z) are periodic with period 1.
Hence we can introduce the new variable S =∈2πiz and then ϕS (Z) = λ(S )
and ϕ(z) = λ(S ) are single valued functions of S . We shall see that they
are also regular in |S | < 1.
139
We first prove that c , 0 when and only when nS = 0. Indeed if
a
c = 0 then is real and > 0 so that all the terms occurring in the right
d
side of (182) have the same absolute value, independent of t, and this in
its turn means that the series in the right of (182) is actually a finite sum
as we know the series to converge and consequently nS , 0. Conversely,
if nS , 0 then ωS (z) = ωS Sn (z) or
e2πiLS <z> (cz + d)−K = e2πLS <+nS (c(z + nS ) + d)−K
so that
!K
c(z + bS + d
e
=
.
cz + a
The right side of the last equality is meromorphic in z and this can
be said of the left side only when 0 = 0. This proves our assertion.
We shall now appeal to Lemma 16 to show that
X
lim ϕS (z) =
lim e2πiLS <Z+b> (c(z − t) + d)−K
(183)
2πiLS <Z+nS −π∗∗LS <z>
y→r
t
mod nS
y→∞
provided the limits in the right side all exist. This is trivially true if 143
nS > o as in this case we face only a finite sum in (183). In the case
nS = o the right side of (183) is actually an infinite series, t ranging
from −∞ to +∞, but in this case we show that the right side of (182)
has a convergent majorant not depending on z in a domain of the type
|x| ≤ ϕ, Y ≥ δ > o where x + iy = z and this will validate our taking the
limit in (183) under the summation sign.
Since ImLS < z + t >≥ o we have
|e2πiLS <z+t> | ≤ I.
Also |c(z+ t)+ d| ≥∈ ic(c+ t)+ d| for |x| ≤ E , y ≥ δ > o with a certain
positive ∈=∈ (E , δ). For, since by assumption nS = o, we have c , 0
d
d
and then the above inequality is equivalent with |z + t + | ≥∈ |c + c + |
c
c
or equivalent with |x + t + p + c(y + q)| ≥∈ |c + p + c(i + q) where we
write d/c = p + ν, i.e. with
x + i(y + q) p + t p + t ≥ εc +
+
1+q
1+q
1+q
140
and the last inequality is clearly true by Lemma 15. Thus we obtain
2πiL <z+t>
e S
(c(z + c) + d)−K ≤∈−K |c(+() + d|−K
and consequently
∞
P
t=−∞
∈−K |c(i + t) + d| provides a convergent majorant
we were after.
Reverting to our functions ϕS (ζ) = ϕS (z) and x(S ) = ϕ(z) we observe that they are regular in |ζ| < 1. The only point of doubt is the
origin but all these functions are bounded in a neighbourhood of the origin −ϕ(z) is so because it is a modular form and the ϕS (z)r are so in
view of their having a convergent majorant not depending on z as shown
earlier. Hence the origin is also free from being a singularity for any of
these functions. We can therefore conclude that
χ(o) = lim ϕ(z), χS (o) − lim ϕS (z)
y→∞
y→∞
(184)
Since the Poincare’ series converges uniformly in every compact
subset of Y , in particular
X
χ(ζ) =
χ∞ (ζ)
S V∗(T )
145
converges uniformly on |ζ| = ρ(o < ρ < 1).
Cauchy’s integration formula then yields that
Z
X 1 Z ψ (ζ)
X
ψ(ζ)
1
S
oζ =
αζ =
χS (o)
χ(o) = πi
2
ζ
2πi
ζ
S ∈V∗(T )
S ∈V∗(T )
|ζ|=S
|ζ|=ρ
The results (184) and (185) together imply that
X
lim ϕ(z) =
lim ϕS (z)
y→∞
S ∈V∗(T )
and coupled with (183) this gives
g(Z, T )|φ = lim g(Z, T )
y→∞
y→∞
(185)
(186)
144
141
=
X
S ∈V(T )
lim e2πiσ(T S <Z>) |cz + D|−K
y→∞
(187)
!
A B
provided the limits occurring in the right all exist and
C D
the series converges absolutely. To compute g(Z, T )|φ therefore, and this
was the purpose with which we set out in the beginning of this section,
we need only compute
with S =
lim e2πiσ(T S <2>) |CZ + D|−K , S ∈ V(T ).
y→∞
For this purpose we use the representation (162) which can be rewritten as
1 X 2πiσ(T 1 S <z>[p])
g(z, T ) = P
e
|cz + D|−K
(162′ )
(T, ) p
!
T1 o
where T =
, T 1 = T 1K > o, r being the rank of T and S = 146
o o
!
A B
and the summation for P runs over all primitive matrices ρ =
C D
ρ(n,r) while that for S runs over all the elements of Vo , viz. a complete
set of modular matrices whose second rows are non associated.
A representation for Vo is given by Lemma 1 as follows.
Let {co , Do } run through a complete set of non associated coprime
symmetric pairs of square matrices of order s(1 ≤ s ≤ n) with |co | , o
and let {Q} run through a complete set of non right associated primitive
matrices !Q = Q(n,r) . Let (Co Do ) be completed to a modular matrix
Ao Bo
∈ Mr and Q to an unimodular matrix u = (QR) in any one
C o Do
arbitrary way and set
!
!
A o ′
B o −1
A= o
u,
B= o
u ,
o E
o o
!
!
Co o ′
Do o −1
C=
u,
D=
u
o o
o E
with E = E n−r .
142
!
A B
The resulting matrices S =
as s runs through all integers
C D
| ≤ r ≤ n together with the unit matrix E = E (2n) corresponding to the
case s = rank C = o form a class of the desired type. We can now write
1 P 2πiσ(T,Z[ρ])
((162′ )) using (69) as g(z, T ) = ε(T,)
ρe
n
1 X X X 2πiσ(T,S <z>[ρ])
P
e
|c, z[Q] + Do |−K
(T, ) r=1 ρ ρC ,D
o
(188)
o
{Q}
147
!
Z∗ o
We introduce again Z =
with Z = X + iy, z = x + iy, Z ∗ =
o z
X ∗ + iy∗ and denote Q as Q = (Ymν = (Y1 Y2 · · · Yr . We assume without
loss of generality that y[Q] is reduced. Let Q∗ denote the matrix arising
from Q by deleting its last row and let Q∗ = (Y1∗ · · · Yr∗ .
Then using (47 - 49) we have
1 r
1 r
π Y[Yν ] = Yν=1
(Y ∗ LYo∗ ) + yqznν )
c1 ν=1
c1
(189)
with a certain constant c1 > o.
It is clear from (189) that if qn1 qn2 · · · qnr ) , (oo · · · o) then ||Co Z
[Q] + Do || → ∞ as y → x and consequently
||Co Z[Q] + Do || ≥ |Y[Q]| ≥
lim e2πiσ(T 1 S <z>[P] |Co Z[Q] + Do |−K = o
y→∞
148
The alternative case, viz. qn1 qn2 · · · qnr ) = (oo · · · o) can occur only
if s < n. For, if s = n, then Q is a primitive square matrix and hence is
itself unimodular and its last row cannot therefore consist of all! zeros.
Q∗
. Then
Let then s < n and qn1 qn2 · · · qnr ) = (oo · · · o) so that Q =
o
∗
Q is itself primitive and can be completed
to a unimodular matrix u∗!=
!
R∗ o
u∗ o
(Q∗ R∗ ). The choice R =
is permissible and then u =
o 1
o 1
A simple computation yields that
shZi = (AZ + B)(CZ + D)−1
143
!
!
)
B0 0
Q′
−1
=
Z+
(QR)
R′
0 0
(
!
!
!
)−1
00 0 Q′
D0 0
−1
×
Z
+
(QR)
0 0 R′
0 E
(
!)
! ′!
B 0
A0 0 Q
Z(QR) + 0
=
′
0 0
0 E R
(
!
!
!)−1
C 0 0 Q′
D0 0
×
Z(QR)
+
0 0 R′
0 E
!
!−1
A0 Z[Q] + B0 A0 Q′ ZR C0 Z[Q] + D0 C0 Q′ ZA
=
R′ ZQ
Z[A]
R′ ZQ
E


1
A0 Z ∗ [Q∗ ] + B0 A0 Q∗ Z ∗R∗ 0

1

=  R∗ Z ∗ Q∗
Z ∗[R∗ ]
0


0
0
0
2

C0 Z ∗[Q∗ ] + D0 C0 Q∗1 Z ∗ R∗ 0



0
E
0


0
0
1
!
S ∗ hZ ∗i 0
=
0
Z
(
A0 0
0 E
!
where S ∗ represents a modular substitution of degree n − 1 which has
the same relationship to the classes {C0 , D0 }, {Q∗ } as S has, to {C0 , D0 }
and {Q}. This means
that
!
∗
∗
A B
with
S∗ = ∗
C D∗
!
!
A0 0 ∗′
B0 0 ∗−1
∗
∗
A =
B =
u ,
u
0 E
0 0
!
!
C0 0 ∗′
D0 0 ∗−1
∗
∗
C =
D =
u ,
u ,
0 0
0 0
where u∗ = (Q∗ R∗ ) and E = E (n−1−s) .
149
Writing S hZi = X3 + S YS and S ∗ hZ ∗ i = XS∗ ∗ + S YS∗ ∗ we have
!
YS∗ ∗ 0
YS =
and |C0 Z[Q] ∗ D0 | = |C0 Z ∗ [Q∗ ] + D0 | Let P = (Pµν ) =
0 y
(yn , y2 , . . . , yn ) and P∗ = (y∗n , y∗2 , . . . , y∗n ) be the matrix arising from P by
144
deleting its last row. From (102) we have
σ(τ, Y s [P]) ≥ yσ(Y s [P]) = y
=y
r
X
YS [Yν ]
ν=1
r
X
(YS∗ ∗ [Yν∗ ] + Y P2µν )
ν=1
with a positive constant y = y(τ1 ). Hence σ(τ, Y s [P]) → ∞ as y → ∞
and consequently limy→∞ ∈2πs∈(T 1 S hZi[P]) = 0 provided (Pn1 , Pn2 , .Pnr ) ,
(00, .0).
Thus in this case too, the general term
e2πsσ(τ1 S hZi[P]) |C0 Z[Q] + D0 |−k of ((162′ ))
150
tends to zero as y → ∞. The only case which remains to be settled
is when both (qn1 , qn2 ., qns ) and (pn1 , pn2 ., pnr ) are the respective zero
vectors simultaneously. In this case we have
! s < n and by
! seen that
Q∗
P∗
so that P∗ and
,Q =
an analogous reasoning r < n and P =
0
0
Q∗ are themselves primitive. Further the general term of the Poincare
series ((162!′ )) does
! not involve Z in this case as the relations S hZi[P] =
s∗ hZ ∗ i 0 P∗
= S ∗ hZ ∗, [P∗ ] and |C0 Z[Q] + D0 | = |C0 Z ∗ [Z ∗] + D0 |
0
Z 0
show. Trivially then does the limit lim e2πiσ(τ1 S hZi[P]) |C0 Z[Q] + D0 |−k
y→∞
!
Q∗
exist in this case too. We have therefore shown that when Q ,
0
!
P∗
and in particular when either r = n or s = n, the general
or P ,
0
term of the Poincare’ series ((162′ ))!tends to zero
! as y → ∞ while in the
P∗
Q∗
simultaneously, whence
,P =
alternative case viz. when Q =
0
0
r < n and s < n, it assumes as the limit
∗
∗
∗
e2πiσ(τ1 S hZ i[P ]) |C◦ Z ∗ [Q∗ ] + D◦ |−k , being in fact independent of z in this
case. Since g(Z, T )|φ = limy→∞ g(Z, T )
145
we can now state that g(Z, T )|φ = 0 in case r = n > 1 and
∗
p n−1
1 X X X
1 X 2πiσ(T 1 Z ∗ [P∗ ])
e
+
g(Z, T )|φ =
ε(T 1 )
ε(T 1 )
p∗ {C ,0 }
S =1
e2πiσ(T 1 S
∗ hZ ∗ i[P∗ ])
= g(Z ∗ , T ∗ )
|C0 Z ∗ [Q∗ ] + D0 |−k
ω
0
{Q∗ }
!
T1 0
, viz. the matrix which
in case r < n where
=
=
0 0
arises from T on depriving it of its last row and column. g(Z ∗ , T ∗ ) is
then just the Poincare’ series obtained from ((162′ )) on replacing n by
n − 1. We may define formally g(Z, T ) = 1 for n = 0 so as to validate
the above results for all n ≥ 1 and state
!
T 0
Theorem 12. Let T = 1
, T 1 = T 2(r) > 0, T = T (r) n > 0 and
0 0
k ≡ 0(2), k > n + r + 1. Let g(Z, T ) be a Poincare series of degree n and
weight k and let Z ∗ · T ∗ the matrix which arises from Z, T by depriving
them of their last row and column.
151
T∗
Then
T ∗(n−1)



g(Z ∗ , T ∗ ) , for
g(Z, T )|φ = 

0
, for
r<n
r=n
(190)
The theorems 11 and 12 lead to interesting consequences. An immediate consequence is that the Poincare’ series g(Z, T ) with T > 0
generate the space PK(n) of all cusp forms. For, from (190) we know
that such a g(Z, T ) belongs to PK(n) If f(n)
k , denotes the space generated
(n)
(n)
by g(Z, T ), T > 0 and Kk , the orthogonal space of f(n)
K in PK , then
PK(n) = f(n)
+ R(n)
, the sum being direct. If f(Z)εR(n)
then f(Z)εPK(n) and
K
K
K
is therefore a cusp form. This means that a(T ) = 0 for |T | = 0 in the
standard notation by (108), while if T > 0, (f(Z), g(Z, T )) = 0, as then
(n)
f(Z) ∈ K(n)
K and g(Z, T ) ∈ fK so that from Theorem 11 we have again
a(T ) =
1
(f(Z), g(Z, T )) = 0.
C
146
Thus all the Fourier coefficients a(T ) of f(Z) vanish and f(Z) ≡ 0.
(n)
(n)
This means that R(n)
K = 0 and consequently PK = fK . We now go a
step further and prove
Theorem 13. The Poincare’ series g(Z, T ) with rank T = r (fixed) gen(n)
erate PKr
(r ≤ n) provided k ≡ 0(2) and k > n + r + 1.
152
Proof. The case r = n was just now settled. Assume than that r <
n. Let f(n)
km denote the space generated by g(Z, T ) with rank T = r <
n and apply induction on n. From theorem 11 it easily follows that
(n−1)
(n)
(n)
f(n)
Kr ⊂ MK and due to induction assumption fKr |φ = PKr . Since by
(n)
(n−1)
construction, PK
|φ ⊂ PKr
and φ is 1 − t on MK(n) it follows that
r
(n)
(n)
(n−1)
(n)
= f(n)
|φ ⊂ PKr
Pkn
Kn |φ and consequently PKr ⊂ fKn . The reverse
inclusion is also immediate as is seen from the relation
(n)
(n−1)
(n)
|φ
= PKr
f(n)
Kr |φ ⊂ MK |φ ∩ PKr
establishing thereby theorem 13. In conjunction with Theorem 11 this
yields
Theorem 14. The Poincare’ series g(Z, T ) generate Mk(n) , the space of
all modular form of degree n and weight k provided k ≡ 0(Z), k > 2n
(the usual condition for the convergence of g(Z, T )).
This is precisely the representation theorem we were looking for.
(n)
Further since f(n)
|φ = PK(n) and we have shown that f(n)
= f(n)
= PKr
K
Kn
Kr
we have
(n)
|n−1|
(K ≡ 0(Z), K > n + r + 1)
(191)
PKr
|P = PKr
Repeated application of (191) yields that
(n)
(n) n−r
+ PK(n) (K ≡ (2)K = n + r + 1)
|φ = PKr
PKr
153
(n)
and we therefore infer that
We know that φn·r is 1 − 1 on PK,r
(n)
= rank PK(r)
rank PKr
In particular taking r = 0, PK(0) is the space of all constants so
(n)
(n)
is thus
= rank PK(0) = 1(k ≡ 0(2), k > n + 1). PK,u
that rank PKu
147
generated by a single element g(Z, 0), the so called Einstein Series which
converges for K > n + 1 ≡ 0(2), and represents a modular form not
vanishing identically under these conditions. Setting now r = 1, PK(1) is
the space of all cusp forms of the (classical) elliptic modular forms and
from a well known result
h i
K


h 12 i − 1, if K ≡ Z(12), K > n + 2,
(1)
rank PK = 

 K , if K , 2(12), K ≡ 0(Z), K > n + 2
12
(n)
.
and the same is therefore true of rank PKl
We proceed to generalize the fundamental metric formula given in
Theorem 11.
(n)
(n)
Let f(Z) ∈ PKr
, 0 ≤ r ≤ n. If rank T = s, then g(τ, T ) ∈ PK,s
and
we state that P(Z), g(Z, T ) = 0 yr , s
while if r = s we have in conformity with (172),
( f (Z), g(Z, T )) = ( f (Z)|φr−r .g(Z)|φn−r )
We can apply theorem 11 to the scalar product in the right side
above,!as follows. Assume r > 0 and assume for a moment that T =
f, 0
, y = y(r)
1 > 0.
0 0 1
Then, as a result of theorem 12, g(Z, T )|ϕn−r = g(Z1 , T 1 ) where Zi 154
denotes the matrix arising from Z by deleting its last (n − r) rows and
columns and T 1 is as defined above. Hence, by means of theorem 11,
(f(Z), g(Z, T )) = (f(Z)|ϕn−r , g(Z, T )|ϕn−r )
= (f(Z)|ϕn−r , g(Z1 , T 1 ))
n(n − 1) r(r + 1)
2
Q(T 1 )π
− rK
=
∈ (T 1 )
4(4π)
K
n+1
r+ν
)|T 1 |
−K
× πKn=1 y(K −
2
2
n−r is identical with
where the
! Fourier coefficient a(T ) of f(Z)|ϕ
T 0
a 1
= a(T ), the Fourier coefficient of f(Z) In the case of a gen0 0
!
T1 0
eral T we can always assume that T =
[U] with an appropriate
0 0
148
155
unimodular matrix U. Then △(T ) = |T 2 | and ∈ (T 1 ) are uniquely determined by T and we have as above
K
r(n − 1)
(f(Z), g(Z, T )) =
n(r +1)/K−rK x ×πrν=1 y(K−
a(T )π
∈ (T )
4 (4π)
r+ν
n+1
)(△(T ))
,
K
K−K
our specific assumptions being K ≡ 0(2), K > n + r + 1, rank T = r ≥
(n)
(n)
. The case when r = s = 0 still remains.
, g(Z, T ) ∈ PKn
1, f(Z) ∈ PKr
In this case T = (0) and (f(Z), g(Z, 0)) = (f(Z)|ϕ(n) , g(Z, 0)|ϕn ) as an
immediate consequence of our definition. Now f(Z)|ϕn is a modular
form of degree 0 and hence is equal to C : (0) while with regard to
g(Z, 0)|ϕn we can be a bit more specific and state that g(Z, 0)|ϕn = 1 as
is easily verified by a consideration of the Eisenstein series of degree 1
Hence (f(Z), g(Z, 0)) = a(0). We thus have
Theorem 15. Assume K ≡ 0(2), K > n + r + 1 rank T = r and f(Z) ∈
(r)
. Then
PKr
 2
r(r+1)
Q y(K −

a(T )π r(n−1)


4 (4π) 2 nK

 ∈(T )
ν=1
(f(Z), g(Z, T )) = 
for r>0



a(0)
for
r=0
r+1
r+ν
2
K )(△(T ))
where a(T ) denotes the Fourier coefficient of f(Z) corresponding to the
matrix T .
Chapter 12
The field of modular
Functions
We shall need the following generalization of Lemma 16, viz.
Lemma 17. Let Z = X − iy ∈ Yn
, σ(xx′ )
≤ m1
, σ(y−1 )
156
≤ m2 . Then
||CZ + D|| ≥∈0 ||Ci + D||
(193)
with a certain positive constant ∈n =∈0 (n, m, m2 ), where (C D) is the
second matrix row of an arbitrary symplectic is matrix
Proof. We prove the lemma in stages.
First we show that if λ1 , λ2 , . . . , λn is a set of real numbers with
0 < λ1 ≤ λ2 ≤ · · · ≤ λn and R(n,m) = (rµ,ν = (W1 W2 · · · Wm )), S (n,m) =
(Sµν ) = (σ1 σ2 · · · σm ) with m ≤ n be real matrices such that sµν = λµ rµν
then
(λn−m+1 λn−m+2 · · · λn )−2 |Pµ′ Pν | ≤ |Wµ′ Wν |
≤ (λ1 , λ2 , . . . , λn )−2 |Pµ′ Pν |, µ, ν = 1, 2, . . . , m (194)
By a well known development of a determinant, we have
2
r
·
·
·
r
s,i
s
,m
X
2
· · · · · · · · · |λµ Wν | =
1≤ρ1 <ρ2 <···<ρm <n r s,r · · · r sm ,n 149
12. The field of modular Functions
150
2
S
·
·
·
S
s,i
s
,m
X
2
=
(λρ1 λρ2 , . . . λρn ) · · · · · · · · · S · · · S 1≤ρ1 <ρ2 <···<ρm <r
s,r
sm ,n
Also
2
S s,i · · · S s2 ,m · · · · · · · · · |Pµ′ Pν | =
1≤ρ1 <ρ2 <···<ρm <n S s,r · · · S sm ,n X
157
(194) is now immediate.
We next prove that if y = y(n) > 0, T = T (n) = T ′ real, and σ(y−1 ) ≤
m2 , then
||T + iy|| ≥ ε1 ||T + LE||, ∈1 (n, mν ) > 0
(195)
Determine an orthogonal matrix W such that |y[W] = D2 with
D = (δµν λν ), 0 < λ1 ≤ λ2 ≤ λn and set
T [W] = S = (Sµ,ν ), Q = S [D−1 ] = (qµ,ν ) Then
|Y| = |D|2 = (λ1 λ2 · · · λn )2 Also Sµ,ν = λµ λν qµν
and
||T + iy||2 = ||T [W] + iy[W]||2
= ||Q + iE||2 |D|4
= ||S + iD2 ||2
= ||Q + iE||2 |Y|2
= |Q + iE||Q − iE||Y|2 = |Q′ Q + E||Y|2
(196)
Let Q = (U1 U2 · · · Un )S = (∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗) and introduce
1
R = (W1 W2 · · · Wn ) = (rµν ) where U2 =
Wν , ν = 1, 2, . . . n Then we
λν
have Sµ,ν = λµ λν qµν = λµ rµν so that the result (194) is now applicable
(with the obvious changes). Then
1Q′ Q + C1 = 1 +
n
X
X
ν=1 µ1 <µ2 <µ2
=1+
n
X
X
ν=1 µ1 <···<µ2
|Mµ′ S nµK |
(λµ1 λµ2 λµ2 )−2 1Wµ′2 Wµ′K
151
≥1+
n
X
(λn−ν+1 · · · λn )−2
≥1+
n
X
(λn−ν+1 · · · λn )−4
ν=1
ν=1
X
µ<···µ[ ν]
X
µ<···µ2
|1µµg WµK |
|Pµg PµK |
(197)
Since by assumption σ(y−1 ) ≤ m2 all characteristic roots λn u2 of
y have positive lower bound which depends only on m2 so that for a 158
suitable constant ε1 = ε1 (n1 m2 ) we have
(λ1 λ2 . . . λm µ)4 > ε21 , µ = 0 = 1, 2, , n
(198)
The in view of (196) - (198) we have
kT + iyk2 = |Q′ Q + iE||Y|2
= |Q′ Q + E|(λ1 λ2 . . . λn )r
n
X
X
≥ ε21 (1 +
|ymy , ymK |)
µ=2 m1 >,<mλ
=
=
ε21 |s′ s + E| = ε21 |s + iE||s − iE|
ε21 |T +; E|| T − iE|ε21 || = T + iE||2
Hence ||T + iY|| > ε4 iE|| as desired. We now contend that under the
assumption σ(xx′ ) ≤ m1 , where x = x(n) = x′ and s = s( n) = s′ real, we
have
X
||x + s + iE|| ≥
||s + iE||
(199)
2
P
with a certain positive constant 2 = 2 (n1 m2 ). This only means that
the quotient ||x + s + iE||/||s + iE|| has a positive lower bound and this is
certainly ensured if we show that L = log || × +s + CE|| - log ||S + ℓE||
is bounded. By the mean value theorem of the differential calculus we
have
X
∂ log ||S ∗ + iE|| L=
ξµν
∗
∂Sµν
S ∗ =S +µx
1≤µ≤ν≤n
P
∗ ) and 0 < ϑ < 1. Also the assumption 159
with × = (xµν ), S ∗ = (Sµν
152
σ(xx′ ) ≤ m2 entitles us to conclude that ±χµν < C , µ, ν = 1, 2 · · · n.
Hence
X ∂ log kS ∗ + iEk = |L| ≤ C
∗
S ∗ =S +ϑχ
∂s
µν
1≤ν≤ν≤n
X ∂ log |S ∗ + iE| =C
∗
S ∗ =S +ϑχ
∂s
µν
1≤ν≤ν≤n
∗
X Re |S ∗ + iE −1 ∂|S + iE| =C
S ∗ =S +ϑχ
∂s∗µν
1≤ν≤ν≤n
X ∂|S ∗ + iE| ∗
−1
|S + iE1
(200)
=≤ C
∂s∗µν S ∗ =S +ϑχ
1≤ν≤ν≤n
Let W = (ωµν be an orthogonal matrix such that
W ′ S ∗ w = D = (δµν dν), dν ) ≥ o.
Then W ′ (S ∗ + iE)w = D + iE, or equivalently
(S ∗ iE)−1 = W(D + iE)−1 W ′
(201)
Let (S ∗ + iE)−1 = (ωµν ) and observe that the diagonal elements of
D + iE have the lower bound 1 as they are of the form 1/dν+i , dν ≥ o.
P
Then ωµν = ωµS d3 ¯+ iω̺ν from (201) so that, considering absolute
̺
P
values, (ωµν ω̄µν )1/2 ≤ nρ=1 |ωµρ ωµρ | and consequently ±ωµν ≤ 1. From
(200) we have then
|L| ≤ C
160
n
X
(ωµν ω̄µν )1/2 ≤ C n3
µν=1
In other words L is bounded and this is precisely what we wanted to
show.
We are now in position to face the main Lemma. We first prove it in
the case when |c| , o. What we want to show is then that kCZ + Dk ≥
εo kCi+Dk or equivalently kZ+C −1 Dk ≥ εo kiE+C −1 Dk. With Z = X+iY,
this is the same as kχ + C −1 D + iYk ≥ εo kC −1 D + iEk. In other words
153
kT + iYk ≥ εo kS + iEk with T = χ + C −1 D and S = C −1 D. An appeal to
(195) which is by way legitimate, yields kT + iYk ≥ εi kT + iEk and then
in view of (199), kT + iYkε1 kT + iEk ≥ ε1 ε2 kS + iEk. Setting ε1 ε2 = εo
we have the desired result.
The general case can be reduced to the above case as follows. If
(CD) is the second matrix row λ the symplectic
matrix M, then (D, C) is
!
o −E
the second matrix row of M1 = M
and M1 is again symplectic
E o
matrix. Hence in particular |DZ − C1 , o an consequently |DS + C|
does not vanish identically in S = S ′ . We can therefore determine a
sequence of real symmetric matrices S K such that lim S K = (o) while
K→∞
|DS K+C|,o . Then, in view of the truth of our Lemma for a special case
we established above,
k(DS K + C)Z + Dk ≥ εo k(DS K + C)i + Dk.
Proceeding to the limit as K → ∞ we get the desired result. This
completes the proof of Lemma 17.
With these preliminaries we proved to define the field of modular
functions. It looks quite plausible and natural for one to define a modular
function as a meromorphic function in Yn which is invariant under the
group of modular substitutions and which behave a specified manner as
we approach the boundary of ρn .
It is also true that with this definition, the modular functions con- 161
stitute a field. But unfortunately, it has not been found possible to determine the structure of this function field expect in the classical case
n = 1. With a view to getting the appropriate theorem on their algebraic
dependence we are forced to give a (possibly more restrictive) definition as follows:- A modular function of degree n is a quotient of two
modular forms of degree n both of which have the same weight. It is obvious that such a function is meromorphic in Yn and is invariant under
modular substitutions. However, it is not known whether the converse
is also true. We shall be subsequently concerned with determining the
structure of the field of all modular function of degree n. Specifically
n(n + 1)
modular functions of degree n
we shall prove the existence of
2
which are algebraically independent.
Let fK be a modular form degree n and weight K not vanishing iden-
154
tically. Such a form exists for K ≡ 0( mod 2), K > n + 1, as for instance
fµ (z) = g(z.o) = gK (Z, o) is one such. We introduce the series
M(λ) =
X
(λ − fK o)|CZ + D|K )−1
(202)
{C,D}
162
where λ denotes a complex variable, and (C, D) runs over a complete
set of non-associated coprime symmetric pairs. Since a Poincare’ series
converges uniformly in every compact subset of Y , M(λ) represents a
meromorphic function of λ with all poles, simple. Assume fK ρK , 0 in
a neighbourhood of 0.
Then in such that neighbourhood we have
−M(λ) =
=
X
{C,D}
f(z)−1 |cz + D|−K (1 −
∞
XX
λm−1 (fK (z)|cz + D|K )−n
{C,D} m=1
∞
X
=−
=
m=1
∞
X
λ
)−1
fK (z)|2 + D|K
(fK (2))−m {
1
}λm−1
K
|cz
+
D|
{C,D}
X
ϕm (z)λm−1
m=1
with ϕm (z) = gmK (z, e)/f(2)m .
ϕm (z) is clearly invariant under the modular substitutions so that two
points Z and Z1 which are equivalent with respect to Mn define the same
function M(λ). A partial converse is also true, viz :Lemma 18. Provided Z does not lie on certain algebraic surfaces which
have the property that any compact subset of Y is intersected by only
a finite number of them, the equations ϕm (z) = ϕm (z1 ) for m = 1, 2, . . .
imply the equivalence of Z and Z1 with respect to M.
Proof. Due to the invariance property of ϕm (z), we may assume that
z, z, ε5, and in view of the proviso in our lemma, we can further assume
155
that z ∈ interior F as the boundary of F is composed of a finite number
of algebraic surfaces. We then have
|CZ + D| > L,
|CZ + D| ≥ 1
(203)
for {C, D} , {0, E} . The poles of M(λ) are clearly at the points f(π)|CZ + 163
D|K and then the first of the inequalities (203) implies that among these
is a point nearest to the origin, viz. fK (Z) corresponding to the choice
{C, D} = {0, E} and the residue of M(λ) at this point is
1. Due to one of our assumptions, M(λ) is an invariant function of λ
for change of Z to Z1 so that the poles of the same M(λ) (corresponding to Z) are also given by. f(I1 )|CZ1 + D|K . A consideration
of the existence of a pole of M(λ) nearest to the origin at fK (Z)
with residue 1 implies now by means of (203) that
|CZ1 + D| > 1 for {C, D} , {0, E} and that fK (Z1 ) = fK (z).
As the poles permute among themselves by a change of Z to Z1 it
follows that
fK (Z)|CZ + D|K = fK (Z1 )|C1 Z1 + D|K
and consequently
|CZ + D| =∈ |C1 Z1 + D1 |.
(204)
for a certain permutation {C1 , D1 } of all classes {C, D} , {0, E} where ∈
denotes a Kth root of unity which may depend on C, D, Z, Z1 .
If L is any compact subset of F we can show that, Z being arbitrary
in L and Z1 arbitrary in F , there exists only a finite number of classes
{C1 , D1 } consistent with (204) for any given class {C, D}.
The proof is as follows:
Let K be so determined that |CZ + D| ≤ K for Z ∈ XXXX fixed.
With appropriate choices of the constants m1 , m2 , a domain of the type
σ(xx′ ) ≤ πm1 F (y−1 ) ≤ 12 contains (in an obvious notation) so that Z1 164
can be assumed to lie in one of these domains. Then by Lemma 17,
K ≥ ||CZ + D|| = ε||c, zi + D1 || ≥ εεo ||C1 i + D1 ||
156
where ε = εo (n, m1 m2 ) > o. In other words, the determinant |C1 i + D1 |
is bounded (by K/εo ) so that, by an earlier argument (p. 44), the number
of possible choices for {C1 , D1 } is finite. This settles our claim.
Assume now that rank C = 1. Then |CZ + D| = CZ[g] + d where g
is a primitive column and (c, d) is a pair of coprime integers. Also from
Lemma 1, we shall have |C, Z1 + D1 | = |C2 Z2 + D2 | with Zz = Z1 [θz ]θz =
θzn,m , primitive and C2 = Czn |C2 | , 0, r denoting the rank of C1 . From
(204) we then have
CZ[η] + d =∈ |C2 Zz + D2 |
(205)
where to repeat our assumptions, η denotes an arbitrary primitive column, (c, d) denotes a pair of coprime integers, (C2r , Dr2 ) denotes a coprime symmetric pair of matrices with |C2 | , 0 and finally Z2 stands for
Z1 transformed with a certain primitive matrix θz(n,m) . Keeping η fixed,
we choose (c, d) = (1, 1) and (0, 1) successively and obtain from (205),
the relations
Z[η] + 1 =∈2 |c2 z2 + D2 |, z2 = z1 [θ2(n,m) ]
Z[η] + 1 =∈3 |c3 z3 + D3 |, z3 = z1 [θ3(n,m) ]
(205)′
which by subtraction yields
∈2 |C2 Z2 + D2 |− ∈3 C3 Z3 + D3 = 1
165
(206)
where C3 = C3(s) , D3 = D(s)
3 form a pair of coprime symmetric matrices
(n,s)
with |C3 | , 0, Q3 denotes a primitive matrix and ∈2 , ∈3 stand for kth
roots of unity.
Let nµ,ν be the column having 1 in the µth and νth places and 0 elsewhere. Then



if µ = ν
Zµµ
Z[ηµ,ν] = 
(207)

Zµ,µ + Zνν + 2Zµµ if µ , ν
Choosing ηµ,ν in the place of η in (205)′ , we find from the relation
Z[η] = ε3 |C3 Z3 + D3 | that all the elements of Z can be represented as
157
polynomials in the elements of Z1 and the coefficients of these polynomials belong to a finite set of numbers. To realize the last part, we need
only observe that the columns ηµν are finite in number and so are the
pairs (C3 , D3 ) as the pairs (C1 , D1 ) are so for a given (C, D). Consider
now the equation (206), viz.
∈2 |C2 Z2 + D2 |− ∈3 |C3 Z3 + D3 | = 1
This is an equation in Z1 and all the coefficients belong to a finite set
of numbers which means that the number of possible such that equations
is finite. We can then assume that above holds identically in Z1 as in the
alternative case, the finite number of (non identical) relations in Z im′ s which are representable
ply a finite number of relations among the Zµν
n(n + 1)
independent elements of Z1 which in its
as polynomials in the
2
turn means that Z lies on a certain finite set of algebraic surfaces- a proviso already assumed in the statement of Lemma. Rewriting the above 166
equation then, in the form
|Z2 + P2 | − α|Z3 + P3 | = β
(208)
with αβ , 0, P2 = C2−1 D2 and P3 = C3−1 D3 , and comparing the terms
of the highest degree in this identity in Z1 we get r = s and |Z2 | = α|Z3 |.
The last is again an identity in Z1 . Replacing Z1 by Z1 [µ1 ] where µ1 is
unimodular, and then transforming Z2 . Z3 with a unimodular matrix µ2
it results that
|Z1 [µ1 Q2 µ2 ]| = α|Z1 [µ1 Q3 µ2 ]|
(209)
!
(r)
E
We can assume U1 , U2 to be so chosen that U1 Q2 U2 =
0
!
R
Writing U1 Q3 U2 analogously as U1 Q3 U2 =
with a square maS
!
E (r) 0
trix R = R(r) , and choosing Z1 =
we obtain from (208) that
00
α|R|2 = 1. In particular this means that |R| , 0 and then we can determine a non-singular ν such that
R′−1 S ′ S R−1 = ν′ Hν, ν = ν(r) , H = (δµ,ν hν ).
158
!
λν′ ν0
The choice Z1 =
with a variable λ leads by means of (209)
0E
to the relations
λr |ν|2 = |Z1 |u1 Q2 u2 | = λ|Z1 [u1 Q3 u2 ]|
= L|λR′ ν′ νR + S S | = |λν′ ν + R−1 S S R−1 |
r
Y
= |λν′ ν + ν′ Hν| = |λE + H||ν|2 =
(λ + hν )|ν|2
ν=1
167
Comparing the extremes which provide an identity in λ, we deduce
that H = 0 and therefore that S = 0.
With Z ∗ = Z1 [U1 Q2 U2 ], (208) yields
|Z ′ + P2 [u2 ]| − |Z ′ + P3 [u2 R−1 ]| = β|u2 |2 = β1 , 0
identically in Z ∗ . Replacing Z ∗ by Z ∗ +P2 [u2 ] we obtain
|Z ∗ | − |Z ∗ −P1 | = β1
(210)
identically in Z ∗ with a certain
! symmetric matrix P.
(t) 0
E
Setting P = W ′
K and Z ∗ = λW ′ W with |W| , 0 and a
0 0
variable λ, (210) yields
λr |w|2 − (λ + 1)t λr−t |w|2 = β1
identically λ. 0f necessity then, r = t = 1 and consequently R(r) = ±1.
Since we know that α|R2 | = L, this means that α = 1. Also C2 = C2(r)
and D2 = D(r)
2 reduce to pure numbers and from (206).
∈2 |C2 Z2 + D2 | ∈3 |C3 Z3 + D3 | = 1.
Considering the terms of the highest degree in the elements of Z1 we
get that
∈2 C2 Z2 |θ2 |− ∈3 C3 Z1 [θ3 ] = 0
Also
!
!
!
!
R
R
E
E
u1 θ3 u3 =
=
=
R=±
= ±u1 Q2 u2
S
0
0
0
159
so that Q3 = ±Q2 . The last relation now gives
ε2 C 2 − ε3 C 3
(211)
Choosing Z1 = 0 in (206), (210) implies that
ε2 D 2 − ε3 D 3 = 1
168
(212)
Since Cν , Dν are integers (U = 2, 3) it follows from (211) - (212)
that ∈2 , ∈3 are rational numbers and being roots of unity are therefore
equal to ±1. We can assume without loss of generality that ∈2 , ∈3 = 1.
It is now to be inferred from (206)- (207) that the elements Zµν of Z are
linear functions of the elements S µν of Z1 with coefficients all belonging
to a finite set of numbers, say
X
Zµν = aµν +
aµν,̺σ ζ̺σ
(213)
̺,σ
Since Z = Z ′ and Z1 = Z1′ we may assume in the above that aµν =
aνµ and aµνσ̺ = aµν,̺σ . Choosing in (205), c = 1, d = 0 and η =
(q1 q2 · · · qn )′ from among a given finite set of primitive columns, and
setting (C2 , D2 ) = (C1 , d1 ) a pair of coprime integers (we have already
shown that C2 , D2 are pure numbers) and Z2 = Z3 [µ2 ] with Q2 =
(p1 p2 , pn ), the p′i s being coprime we conclude from (213) that
X
X
X
ζ̺,σ p̺ pσ + d1
(214)
aµν,̺σ ζζσ qµ qν +
aµ,ν qµ qν = C1
µν,̺σ
µ,ν
̺σ
The possible choices for c1 , c2 , pν (ν = 1, 2, . . . n) are finite in number so that, by an earlier argument (p. 156) we can assume, in view of
the proviso in our Lemma that (214) is an identity in the ζ ′ s. Then (214)
implies that
XX
aµν,̺σ qµ qν = C1 p̺ pσ
(215)
µ
µ,ν
and, C1 , 0 as we know that rank C2 = 1. This shows that the symmet- 169
P
ric matrix (Q̺σ ) = ( µ,ν aµν,̺σ qµ qν ) has the rank 1 for each primitive
column g. Taking a sufficiently large number of g, s, the truth of our
160
above statement for all these will imply that (Q̺σ ) has rank 1 identically in g and consequently all sub-determinants of two rows and two
columns vanish. In particular, for all ̺, σ
Q̺̺ Qσσ = 0
(216)
and all these are algebraic conditions.
We shall say polynomials F, G to be equivalent, written F ∼ G, if
they differ only by a constant factor. Let us assume that Q11 , 0 and
QKλ , 0. Then from (216), QKk aλλ , 0, Q1K , 0 and Q1λ , 0. Thus all
QH K , aλλ , a1K , a1λ differ from 0.
Two cases arise now.
Case. i: Let Q11 have no square divisor. Then from (216) Q11 QHx −
̺21K = 0 and it immediate that
Q11 ∼ Q1K ∼ QHX ∼ Qλλ ∼ QHλ .
P
This means that Q̺σ = C̺Cσ Q with Q = µ,ν qµ qν and then, from
P
(213), Qµν,̺σ = C̺Cσ and Zµν = aµν + Cµ,νξ with ξ = ̺σ C̺Cσ ζ̺σ
Since aµν , Cµν belong to a finite set of numbers, these equations represent a finite number of algebraic surfaces, and the assumptions of our
theorem permit us to exclude this case
170
Case. ii: Let us assume then that Q11 assume then that Q11 is not square
free. Then Q11 ∼ L21 , where L, denotes a linear from. Then using (216)
Qnn ∼ L2K , Qλλ ∼ L2λ and Qnλ ∼ Ln Lλ where the L′ s are again linear
forms. By suitably normalizing these forms we can assume that
anλ = Ln Lλ .
Let LK =
Then
P
µ
ℓµn qµ (The b′ s may be complex constants).
Qnλ =
X
µν
ℓµn ℓνλ qµ qν =
X
aµν,nλ qµ qν
µν
and being an identity in the q′ s, this gives
aµν,nλ = ℓµν ℓνλ
(217)
161
In other words
Z = Z1 [B] + A
(218)
with certain matrices A = (Qµν ), B = (ℓµν ) which belong to a finite set.
It remains only to show that B is unimodular and A is integral and then
the Lemma would have been established.
Let us choose y = yµν in (215) and then we obtain
aµν,̺σ + aµν,̺σ + Zaµν,̺σ = Cµν pµν,̺ pµν,σ (µ , ν)
and aµν,̺σ = Cµ pµ pµσ with integers Cµν , pµν̺ , Cµ , pµσ where CµνCµ ,
0. For σ = ̺ these yield in particular, by means of (217),
(ℓµσ + ℓνσ )2 = Cµν p2µνσ (µ , ν)
2 = C p2 .
and ℓµσ
µ µσ
p
p
p
Cµν pµνσ = Cµ pµσ + Cν pνσ µ , ν
171
(219)
and squaring,
p
Cµν p2µνσ = Cµ p2µσ + Cν p2νσ + 2 CµCν pνσ , µ , ν.
(220)
If at least one product pµσ pνσ , 0(µ , ν) then CµCν is clearly the
square of a rational number. This is also true if pµσ pνσ = 0 for all µ,
ν(µ , ν) as then CµνC[ µ] and CµνC[ ν] are square of rational numbers as
is seen from (220). In particular the numbers c1 , c2 , . . . cn have all the
same sign. Since Z ∈ Y and A is a real symmetric matrix, |Z − A| , 0
and consequently from (218), |B| , 0. It turns by means of (218) that
Z1 also belongs to a compact subset of f as Z does by assumption. This
means that all that we have proved above for Z hold also for Z1 Since
we can assume Z1 = Z[B′] + A+ where B∗ = B−1 in view of the proviso
in our lemma, this in particular means that a representation
p of′ the kind
−1
−1
B = (πµ pµν ) can be found for B too, say B = Cµ pµν . Since
BB−1 = E, from the above representation for B, B−1 we have
q
C1C2CnC1′ C2′ Cn |pµν ||p′µν | = 1
162
and as all the c′ s and p′ s are integers this gives
C1C2 · · · CnC”1C2′ · · · Cn′ = 1, |pµν | = ±1.
172
We know that all c′ s are of the same sign and therefore c1 = c2 =
· · · · · · = cn = 1.
It is now immediate that B is unimodular.
We now specialize (C, D) in (204) to be E, 0 respectively and obtain
from (218) that
ε|C1 Z1 + D1 | = |Z| = |Z1 [B] + A| = |Z1 + A[B−1 ]|
This again can be assumed to be an identity in Z1 and we conclude
that ε|C1 | = 1, so that |C1 | = ±1. Then
|Z1 + A[B−1 ]| = ε|C1 Z1 + D1 | = Z1 + C1−1 D1 |
Replacing Z1 + C1−1 D1 in the above by Z1 we get
|Z1 | = |Z1 + C1−1 D1 − A[B−1 ]|
identically in Z1 and as a consequence, A[B−1 ] = C1−1 D1 or
A = C1−1 D1 [B1 ]C1 is a unimodular matrix as we have shown earlier
and this means that C1−1 D1 and consequently A are integral. Thus, from
(218), Z = Z1 [u] + A with a unimodular u and integral A, in other words
Z and Z1 , are equivalent with regard to M. By assumption Z is an interior point of f and Z1 ∈ ξ and therefore Z and Z1 coincide. This proves
Lemma 18.
173
We now prove that any maximal set of algebraically independent
n(n + 1)
functions in the sequence {ϕµ (Z)}, ν = 1, 2, . . ., consists of
ele2
ments. Let such that a set comprise of the functions λq (z) = ϕνa (z)a, =
1, 2, . . .. We first show that the λ′a s are at least n(n + 1)/2 in number. For
if their number is a q < n(n + 1)/2, then λλ · · · λq ϕm are algebraically
dependent for each m. In other words, there exists a polynomial Fm (t)
- which we may assume to be irreducible of one variable over the integrityed main generated by λa (z), a = 1, 2, . . . q, such that Fm (ϕm ) = 0,
163
m = 1, 2, . . . identically in Z. These polynomials being separable, their
discriminants Dm (m = 1, 2, . . .) are all polynomials in λ1 , λ2 , . . . λq not
vanishing identically. By Lemma 18, it is possible to find a bounded
domain G ⊂ f viz. an open connected non-empty subset of f such that
the condition ϕm (z) = ϕm (z1 ), m ≥ 1, Z ∈ G , z, ∈ f will automatically
imply that Z = Z1 . We now construct a sequence of sub-domains of G
as follows. Let G1 ⊂ G be so determined that D1 , 0 for any Z ∈ G1 .
Then let G2 ⊂ G1 be such that D2 , 0 for any z ∈ G2 . This may be
continued and there exists at least one point Zo which belongs to every
Gm so that DK (zo ) , 0 for any K.
Consider now the equations
λo (z) = λo (Zo ), a = 1, 2, . . . q,
(221)
These define an analytic manifold of complex dimension at least
n(n + 1)/2 − q > 0. Thus in G there exists an analytic curve through
Zo on which (221) is satisfied at every point. It is then immediate that
functions λa (z) are all constants on this curve. This means as a consequence that the functions ϕm (z) are also constants on this curve, For
the polynomials Zm (t) have all their coefficients on this curve and being
separable their zeros are all distinct. In a sufficiently small neighbourhood of Zo on this curve we should then have ϕm (z) = ϕm (zo ) for every
m and by the choice of G which contains this curve we have Z = Z1 .In 174
other words the curve reduces to a point- an obvious contradiction to
the assumption that the dimension of the manifold defined by (221) is
positive. Consequently the supposition q < n(n + 1)/2 is not tenable.
We now show that the number of λ′ s cannot exceed n(n + 1)/2. For if it
does, let q > n(n + 1)/2 and consider the modular forms fk , gνa K = fνKa λa ,
a = 1, 2, . . . q. These are at least n(n + 1)/2 + 2 in number so that, by
theorem 7, these satisfy a non trivial isobaric algebraic relation
X
µ
µ
µ
µ
Cµ1 µ2 ···µq+1 gν11K gν22K · · · gνqqK fKq+1 = 0
P
with µi ≥ 0 and µ1 ν1 +µ2 ν2 +µq+1 = λ (λ being a fixed number). From
P
µ
µ µ
the above relation it is immediate that Cµ1 µ2 ···µq+1 λ1 1 λ2 2 · · · λq q =
g µq
g µ1 g µ2
X
∋q K
∋2 K
∋ K
·
·
·
=
Cµ1 µ2 ···µq+1 ν11
ν
fK
fνK2
fKq
164
1 X
µ1 µ2 µq µq+1
= λ
=0
Cµ1 µ2 ···µq+1 gν1 K gν2 K gνq fK
fK
In other words we have established a non trivial relation among the
that cannot be independent - thus providing a contradiction. Thus
n(n + 1)
we conclude that
q6h
2
λ′ s so
q=
n(n + 1)
2
We are now equipped with the relevant preliminaries needed to establish the main result of this section viz.
175
Theorem 16. The field of modular functions of degree n is isomorn(n + 1)
phic to an algebraic function field of degree of transcendence
.
2
That is to say, every modular function of degree n is a rational function
n(n + 1)
of
+ 1 special modular functions. These functions are alge2
n(n + 1)
of them are independent.
braically dependent but every
2
Proof. Let q =
n(n + 2)
and let
2
Ka (z) = ϕνa (z) = gνa K (z)(fK (z))−νa , a = 1, 2, . . . , q
with fK (z) = gK (z, o) be a set of algebraically independent modular functions. The existence of such a set has already been shown. We first prove
that the (q + 1) Eisenstein series gK (z, o), gνa K (z, o) are algebraically independent. For, let
m
X
Lµ (z) = 0
(222)
µ=0
with
Lµ (z) =
X
µo ···µq
(µ)
µ
µ
µ
Cµo µ1 ···µq gKo gνq1K · · · gνqqK , µc ≥ 0,
µ + µ1 ν1 + · · · µq νq = µ, be any algebraic relation among the g′ s. We
wish to show that such a relation cannot be non trivial. First we shall
165
show that Lµ (z) = 0 for each µ. Applying the modular substitution
Z → (AZ + B)(CZ + D)−1 to (222) we obtain
m
X
µ=0
|CZ + D|µ|Z Lµ (z) = 0
(223)
identically in Z for all symmetric coprime pairs {C, D}. We can choose
Cν , Dν1 ν = 1, 2, . . . m such that |Cν Z + Dn u| are all different.
Then we shall have from (223) a system of linear equations in Lµ (Z) 176
whose determinant is nonzero. This requires that Lµ (Z) = 0 for µ =
0, 1, 2, . . . m. Thus
X
µ
µ µ
Cµo ,µ1 ,...µq gKo gν11K · · · gνqqK = 0, µ = 1, 2, . . . m
µo ,µ1 ,...µq
where
µo +
q
X
µ c νc = µ
q=1
Pq
µ
µ
Dividing Lµ (Z) by gK = gKo + i=1 µ1 ν1 the above yields that
P (K)
µ
(µ)
µ µ
Cµo ,µ1 ,...µq λ1 1 λ2 2 · · · λq q = 0 and consequently Cµo ,µ1 ,...µq = 0 for all
indices. Thus we have shown that gK , gνa K , a = 1 · · · q are all independent.
f(Z)
a
Let now λ(Z) be an arbitrary modular function and λ(Z) =
g(Z)
representation of λ as the quotient two modular forms f , g of the same
n(n + 1)
+2 modular forms are algebraically depenweight ℓ. Since any
2
dent by theorem 7, this is true in particular of the two sets
f, gK , gν1 K · · · gνq K and g, gK , gν1 K , . . . gνq K and there exist nontrivial isobaric relations
X
µ µ
ϕ(f) =
Cµν̺1 ···̺q fµ gK gνq1K = 0
X
µ
ϕ(g) =
dµν̺1 ···̺q gµ gνK gνqqK = 0
with all exponents non negative satisfying the condition
µℓ + (ν + ρ1 ν1 + · · · + ρν νν )K = mℓν1 ν2 · · · νν Kq+l
166
a certain integer m = m(n). It is immediate that
µ ≤ mν1 ν2 · · · νν Kq+1
so that the polynomials ϕ(t) and ψ(t) are of bounded degree. We can
assume without loss of generality that ψ(0) , 0, ψ(0) , 0. In ϕ(t) we
replace t by tg and let ψ1 (8) = ϕ(tg). The resultant of ϕ(g) and ψ1 (g)
(as polynomials in 8) is a polynomial in t denoted by σ(t) say. Since
ϕ(0) , 0, ψ(ϕ , 0) it follows that σ(0) , 0, in other words, σ does
not vanish identically. Further ϕ(g) = 0 and {ψ1 (g)}t=λ = ϕ( f ) = 0 so
that ψ and ψ1 have a common zero at the point λ. Hence σ has a zero
too at this point. The degree of σ is clearly bounded (for varying λ ) as
the degrees of ϕ and ψ are bounded. Also, the coefficients of σ are all
isobaric polynomials of gK , gνq rk , in other words, all of them are modular
forms of the same weight. It is clear then as in earlier contexts that, for
a suitable λ, g−λ
K σ(t) = σ1 has coefficients all of which can be written as
polynomials in χ1 , λq and then σ1 satisfies the following conditions:
i) the degree of σ1 is bounded with respect to λ
ii) σ1 . 0, and specifically, σ1 (0) , 0
iii) σ1 (λ) = 0.
178
Let now λ = λo be so chosen that it is a zero of an irreducible
polynomial F(t) which has all the above properties of σ11 and whose
degree in t is maximal (with regard to λ). The existence of λo is easily
proved. Then the q + 1 modular functions λo , λ1 , λq generate the field
of all modular functions. For if K = R[λ1 , . . . , λq ] is the field generated
by λ1 , . . ., λq and K[λo ] that generated by λo , λ1 , . . . λq and if λ is an
arbitrary modular function then
K ⊂ K[λo ] ⊂ K[λo , λ].
All these are finite extensions over K and λ being algebraic over K,
the last is clearly a simple extension over K, say, K[λ1 ]. Then, unless
K[λo ] = K[λ∗1 ], the irreducible equation over K satisfied by λ1 will have
a degree higher than that of the irreducible equation that λo satisfies-a
177
167
contradiction to the choice of λo . Hence K[λo ] = K[λo , λ∗ ] and consequently λ∗ is a rational function of λo , λ, . . . , λq . The proof of Theorem
16 is now complete.
Chapter 13
Definite quadratic forms and
Eisenstein series
A first access to the theory of modular forms of degree n was provided 179
by Siegel’s work on quadratic forms. The main result of this theory can
be expressed in the shape of an analytic identity between Theta series
and Eisenstein series. A brief account of their relationship is given in
this section confining ourselves to positive quadratic forms. We may
remark that we shall prefer to speak of symmetric matrices rather than
of quadratic forms.
We first formulate without proof the main result of Siegel’s theory.
This requires a few preliminaries. Let S = S (m) and T = T (n) denote
positive integral matrices, and let m ≥ n. Denote by α(S , T ) the number
of integral X m,n such that S [X] = T and let α(ε, S ) be denoted by ε(s).
By αq (S , T ) we shall mean the number of integral matrices X m,n distinct
modulo q such that S [X] ≡ T ( mod q). We shall say that S and T
belong to the same class if S [u] = y for some unimodular matrix U and
we shall denote by (S ) a special matrix in the class of S . If m = n and if
to every integer q > 0, there exists a pair of integral matrices χ, y such
that S (χ) ≡ T and T (y) ≡ S ( mod q), then we say that S is related to
T , in symbols: S nT . The set of all matrices related to S shall be called
the genus of S . It is well known that the genus of every positive integral
matrix decomposes into a finite number of distinct classes (If a given
169
13. Definite quadratic forms and Eisenstein series
170
genus contains S , it contains the class of S also). We now introduce the
measure µ(S ) of the genus containing S as
X
µ(S ) =
1/ε(S K )
(224)
(S K )n(S )
Finally let
180
α∞ =
π(2m−n+1)
−1
m−n+2
m
y( m−n+1
2 )y( 2 ) · · · y( 2 )
and
αo (S , T ) =
|S | 2 |T |
m−n−1
2
X α(S K , T )
1
µ(S ) (S )r(S ) ε(S K )
(225)
(226)
K
The last expression is called the mean representation number of T
by the genus of S .
We now quote Siegel’s main result in
Theorem 17. Let S = S (n) , T = T (n) with m > n + 1 be positive integral
matrices. Then
αo (S , T ) = α∞ (S , T ) lim
αq (S , T )
K→∞ mnn(n+1)
2
, q = K!
(227)
We proceed to express thee q-adic representation number αq (S , T )
by means of the Gauss sums.
Let W (n) be a rational symmetric matrix and d the smallest positive
integer such that the quadratic form d, W[ε] has all integral coefficients.
We shall call such a′′ d′′ , the denominator of W[ε]. We now introduce
the generalized Gauss sum g(S , W) for any integral symmetric matrix
S (m) and rational symmetric matrix W (n) as follows.
X
g(S , W) =
e2πiσ(S [A]W)
(228)
A
181
where in the summation, A = A(m,n) runs through a complete set of integral matrices which are all distinct mod d, d denoting the denominator
171
of W[ε]. We have get to show that the sum in (228) does not depend
on the choice of the representatives of the cosets mod d. This will be
fulfilled if we prove the invariance of this sum for a replacement of A by
A + dB where B is an arbitrary integral matrix. In this case we have
σ(S [A + dB]W) − σ(S [A]W) = d2 σ(S [B]W) + dσ(B′ S AW)
+ dσ(A′ S BW)
= d2 σ(S [B]W) + 2dσ(B′ S AW).
Since S [B] is integral and symmetric and dW is semi integral, it
follows that d, σ(S [B]W) ≡ 0( mod 1) and similarly, since B′ S A and
2dW are integral, 2dσ(B′ S AW) ≡ 0(1). Hence
e2πiσ(S [A+dB]W)
= e2πiσ(S [A]W)
and the desired result is immediate.
A useful estimate for g(S , W) when |S | , 0 is given by
Lemma 19. Let S = S (m) be a nonsingular integral symmetric matrix,
W = W (n) a rational symmetric matrix and d the denominator of the
quadratic form W[ε]. Then
|g(S , W| ≤ 2m/2 ||S ||n/2 dmn−m/2
(229)
Proof.
=
|g(S , W)|2 = g(S , W)g(S , W)
X
e2πiσ(S [A1 ]W)−2πiσ(S [A2 ]W)
A1 ,A2
mod d
182
Keeping A2 fixed we first sum over A1 and this we can do after
replacing A1 by A1 + A2 We then get
X
|g(S , W)2 =
e2πiσ(S [A1 +A2 ]W)−2πiσ(S [A2 ]W)
A1 ,A2
=
mod d
X
A1 ,A2
mod d
′
e2πiσ{(A1 S 2 +A2 S A1 +S [A1 ]W)}
172
e2πiσ(S [A1 ]W)
A1
X
2πiσ(S [A1 ]W)
A1
X
=
=
mod d
e2πiσ(2A2 S A1 W)
A2
X
e2πiσ(A2 W1 )
A2
X
e
mod d
1
mod d
1
(230)
mod d
where W1 = 2S A1 W
If W1 = (ωµν ) and A2 = (aµν ) then
X
σ(A′2 W1 ) =
aµν ωµν
µν
and
A2
X
mod d
1
e2πiσ(A2 W1 ) =
aµν
X
e2πiΣµ,ν aµν ωµν
mod d
Y
=
(
µ,ν aµν
X
e2πiΣµ,ν aµν ωµν )
mod d
By a standard result, the sum within the parenthesis of d or 0 according as ωµν is or is not an integer. Hence, the inner sum in the right side
of (230) is 0 if at least one ωµν is not integral and dmn in the alternative
case. Thus
X
e2πiσ(S [A1 ]W) ≤ dmn L
(231)
|g(S , W)|2 = dmn
A mod d
2S A1 W−integral
183
where L denotes the number of cosets mod A1 such that 2S A1 W is
integral.
We now wish to estimate L. For this we can assume that S and W
are both diagonal matrices. For, in the alternative case, we can find
unimodular matrices Ul , Vl , i = 12, with U1 S U2 = D = (dµν dν ),
V1 WV2 = H = (δµν Rν ). Then, if A3 = U2−1 A1 V1−1 , A3 runs over a complete representative system of matrices mod d as A1 does and 2DA3 H
is integral when and only when 2S A1 W is integral. Consequently the
replacement of S , W by D, H, viz. diagonal matrices, will not interfere
with our estimation for L. We assume then that S and W are diagonal matrices, S = (δµνsν ), W = (δµν ων ) and that 2S A1 W is integral. If
173
A1 = (aµν ) this means that
aµν 2dων sν = 2sν aµν dν ≡ 0( mod 1),
1 ≤ µ ≤ m, 1 ≤ ν ≤ n
Since 2dων is an integer, it is obvious that the number of distinct
cosets mod d is just (d, 2dsµ ων ) viz. the greatest common divisor of d
and 2dsµ ων . Hence
L=
Y
Y
(d, 2dsµ ων ) ≤
{(d, 2dων )|sµ |}
µν
µ,ν
n
Y
≤ ||S ||
(d, 2dων )m ≤ ||S ||n {(d, 2dω1 , 2dω2 , 2dω2 )dn−1 }n
n
ν=1
≤ ||S ||n (2dn−1 )m
(232)
In stating the above inequalities we have used the fact that
184
Qq
q−1
(g, g1 , . . . , gq )g
≥ ν=1 (g, gn ) for any q + 1 integers g, gi (l − 1, q) and
this is easily proved by induction on q. Combining (231) and (232) we
get
|G(S , W)|2 ≤ ||S ||n dmn (2dn−1 )m and (229) is immediate. We obtain
now a representation of the right side of (227) by an infinite series.
Lemma 20. Let W = W (n) run through a complete set of rational matrices for which the quadratic forms W[∈] are distinct modulo 1 and let
d denote the denominator of W[∈]. Then
lim
K→∞q
αq (S , T )
mn −
n(n+1)
2
=
X
d−mn g(S , W)e−2πiσ(WT ) , q = K!
(233)
W
where S = S (m) > 0, T = T (n) > o, m > n2 + n + 2 and the series on the
right side of (233) converges absolutely.
174
Proof. We first show that
q
n(n+1)
2
αq (S , T ) =
XX
W
e2πiσ{(S [A]−T )W}
(234)
A
where A runs through all integral matrices distinct mod q and W runs
over all rational matrices such that the quadratic forms W[∈] are distinct
mod 1 and qW[∈] has all its coefficients integral. Let S [A] − T = R =
1
(rµν ) and W = ( (1 + δµν )ωµν )
2
Then
X
X
e2πiσ(RW) =
W
e2π
P
µ≤νr µνω µν
ωµν mod 1
qωµν − integral, µ≤ν
=
Y
(
X
e2πirµν ωµν )
µν ωµν mod 1
qωµν ≡0(1)
185
The sum inside the parenthesis is well known and is equal to q or 0
according as q is or is not a divisor of qµν . It is now immediate that

nm+1

X

q 2 , i f S [A] ≡ T ( mod q)
e2πiσ(RW) = 

0
, otherwise
W
The right side of (234), by a change of the order of summation
n(n + 1)
clearly reduces then to q
αq (S , T ) and (234) is established.
2
X
X
q
e2πiσ(S [A]W)
Since
e2πiσ(S [A]W) = ( )n
d
A mod d
A mod q
q
= ( )n g(S , W),
d
P
the right side of (234) can be rewritten as qmn d−mn g(S , W)e−2πiσ(T W) .
W
Hence (234) now reads as
q
n(n+1)
2
αq (S , T ) = q−mn
X
W[ε] mod 1
qE[ε]≡0(1)
d−mn g(S , W)e−2πiσ(T W)
(235)
175
This is true for every integral q = K! and as K → ∞ the right side of
P
(235) is just
d−mn g(S , W)e−πiσ(T W) provided this infinite series 186
W[∈] mod 1
converges. We shall show that this converges absolutely and then we
would have proved Lemma 20. We proceed thus the number of quadratic
forms W[∈] which are distinct mod 1 and have a given denominator
d can be roughly estimated from above by dn(n+1)/2 . Hence the series
under question can be majorised by
∞
X
d
n(n+1)
2
d−mn g(S , W)
d=1
which again can be further majorised by means of (229)
n(n + 1) m
−
P∞
2
2 with a suitable constant K. The last series is cleaK d=1 d
rly convergent under our assumption on m,n viz. m > n2 + n + 2 or
m n(n + 1)
−
> 1, and then the absolute convergence of the right side
2
2
of (233) which we wanted to establish is immediate.
We wish to have a partial sum representation for the infinite sum
n+1
ρ−
P
2 e2πiσ(T Z) (ρ > n + 1) in Lemma 21 and as a preliminary,
|T |
τ>0
integral
we quote Poisson’s Summation Formula, viz. that if f is any function
defined on the space L of all symmetric matrices Y, and T runs through
all symmetric integral matrices while F runs through all symmetric semi
integral matrices, then under suitable conditions,
X
XZ
f(T ) =
f(y)e−2πiσ(Fy) [dy]
(236)
T
F
L
A formal proof of (236) may be furnished as follows.
P
If f (y) = T f(T + y) then g(y) is a periodic function of y and can be
expanded in a Fourier series
187
P
2πiσ(Fy)
g(y) = F a(F)e
If H denotes the unit cube in L , then
R
P
α(F) = g(y)e−πiσ(Gy) [dy]. Replacing g(y) by its infinite sum T f
H
(T +Y) and changing the order of summation and integration (which can
176
be justified under suitable conditions) we have
XZ
a(F) =
f(T + y)e−2πiσ(Fy) [dy]
T
=
T
XZ
f(T + y)e−2πiσ(F(T +y)) [dy]
t
=
H
X Z
=
Z
T
T +H
L
Then
X
f(T ) = g(0) =
T
X
F
a(F) =
XZ
F
L
and this establishes (246).
We apply this result in
Lemma 21. Let Z ∈ fgn , ρ > n + 1 and let T (n) run through all positive
integral matrices while F (n) runs through all semi integral matrices.
Then
X
T
|T |ρ−
n+1
2
e2πiσ(T Z) = π
n(n − 1)
1
n−1 X
y(ρ)y(ρ − ).y(ρ −
)
|1πi(F − Z)|−ρ
4
2
2
F
(237)
where we put |2πL(F − Z)|−ρ = e−ρ log |2πi(F−Z)| and understand by log
|2πL(F − Z)| that branch of the logarithm which is real for
188
Proof. We first note that since F is real and Z ∈ Yn , |2πl(F − Z)| , 0 so
that the right side of (237) makes sense. Let us introduce the function
f(y)(y− symmetric) as

2πiσ(yz),


|y|p − n+1

2 e



f(y) = 
,ify > 0




−0, otherwise
and state that Poisson’s summation formula is valid for this function.
177
Then
X
f(T ) =
T
X
T >0
|T | p−
n+1
2
e2πiσ(T Z)
which is precisely the left side of (237) and by (236) this is equal to
XZ
XZ
n+1
−2πiσ(Fy)
|y| p− 2 e2πiσ(y(Z−F)) [dy]
f(y)e
[dy] =
F
F y>0
L
By Lemma 14 we have
Z
|y| p−
n+1 2πiσ(y(Z−F))
2 e
=π
n(n−1)
4
1
n−1
y(ρ)y(ρ − )y(ρ −
)|2πi(F − Z)|−ρ
2
2
y>0
and then (237) is immediate. We may note that the convergence of the
right side of (237) for ρ > n + 1 is a consequence of the convergence of
the Eisenstein series in this case. The proof of Lemma 21 is complete. In
the following Lemma we are concerned with a parametric representation
for integral matrices G with a given rank r.
Lemma 22. Let 1 ≤ r ≤ n ≤ m where r, n, m are all integral. Let
B = Bn,m run over all integral matrices of rank r and a = an,m run over
a complete set of right non associated primitive matrices.
Then every integral matrix G rank r is obtained as the matrix prod- 189
uct QB once and only once.
Proof. Let G be an integral matrix of rank n with suitable unimodular
matrices U1!, and U2 and a non-singular matrix D = D|r| we shall have
00
G = U1
U . Writing U1 = (Q∗) where Q = Q(n,r) and U2 = (R∗ )
00 2
where R = R(n,m) we have
!
D0 R
G = (a∗)
( ) = QDR = QB
DR = B
00 ∗
Since R is primitive and D nonsingular we conclude that B is integral, and by choice Q is primitive. We may observe that we can subsequently replace Q by any given element of the equivalence class of
178
Q by absorbing the right factor (unimodular) introduced thereby, into
B. To prove the uniqueness of the representation G = QB we proceed
thus. Let Q1 QB = Q2 B2 where Qν , Bν are matrices in the same sense
as Q, B are. We determine R1 such that U = (Q1 R1 ) is unimodular and
put U −1 Q2 = (AH ) with A = A(r) . Then since U −1 Q1 B1 = U −1 Q2 B2 we
have on substitution,
!
!
A(r)
E (r)
B2 or B1 = AB2 and 0 = HB2 Since rank B2 = r
B1 =
H
0
the last condition implies that H = 0 and consequently A is unimodular.
Since Q2 = U(0A ) = (Q1 R1 )(0A ) it now follows that Q2 = Q1 , A = E, and
it is immediate that B1 = B2 by one of the earlier conditions.
179
190
Missing page 190
180
191
We now introduce that ϑ series ϑ(S Z) for any S = S (m) > 0 integral,
and Z ∈ Yn with m ≥ n. By definition
X
ϑ(S , Z) =
eπiσ(S [G]Z)
(238)
G
where the summation for G is extended over all integral G = G(m,n) . It is
obvious that the series in (238) is convergent, and it is also obvious that
ϑ(S , Z) is a class invariant of S , in other words ϑ(S , Z) = ϑ(S , Z) if S ∗
is any element of the class of S . Besides the class invariant ϑ(S , Z), we
consider also the genus invariant
f(S , Z) =
X ϑ(S K , Z)
1
µ(S ) (S )r(S ) ε(S K )
(239)
K
This definition of f is analogous to the definition (226) of the mean
representation number of T by the genus of S .
Now ϑ(S , Z) can be rewritten as
ϑ(S , Z) = 1 +
n
X
X
eπiσ(S [G]Z)
r=1 G−integral
rankG=r
By Lemma 22, G′ has a representation of the form G′ = QB′ where
Q = Q(n,r) is a primitive matrix and B = B(m,n) is an integral matrix of
rank r. Then
σ(S [G]Z) = σ(S [BQ′ ]Z) = σ(S [B]Z|Q)
and consequently
ϑ(S , Z) = 1 +
n X
X
eπiσ(S [B]Z)[Q]
r=1 Q,B
192
Introducing
χ(S , Z) =
X
G
eπiσ(S [G]Z)
(240)
181
where the sum extends over all integral G with rank G = n (i.e. maximal
rank) the above series for ϑ(S , Z) can be written as
ϑ(S , Z) = 1 +
n X
X
χ(S , Z[Q])
r=1 Q
Since
χ(S , Z) =
X
X
eπiσ(S [G]Z) =
G
α(S , T )eπiσ(T Z)
T (n)>0
integral
it follows that
ϑ(S , Z) = 1 +
n X
X
α(S , T )eπiσ(T Z[Q])
(241)
r=1 Q,T
T = T (r) > 0, integral, Q = Q(n,n) primitive,
and consequently
f(S , Z) = 1 +
n X
X
α(S , T )eπiσ(T Z[Q])
r=1 Q,T
We now apply Siegel’s main result (227) and then (233) in an obvious way to obtain that, for m > n2 + n + 2,
f(S , Z) = 1 +
n X
X
r=1 Q,T
=1+
n X
X
α∞ (S , T ) lim
K→ν
π r(2m−r+l)
4
m−r+1
m−r+1
r=a T,Q y( 2 )y( 2)···y( m2 )
×
=1+
n X
X
αq (S , T )
X
qmn−r(r+µ/xxxx)
−r
|S | 2 |τ|
× eπiσ(T Z[Q])
m−r
2
dmn g(s.w)eπσ(T Z[Q] − 2w) w=wr rations
w
π(2m(−r+1/4))
π
m−r+1
−r
m−r+1
r=1 Q y( 2 )y( 2 ) . . . y( 2 )
W[ε]
|S |
−r
2
mod 1
182
×
X g(S , w) X m − r + 1
w
∂mr
τ
2
eπσ(τiσ(T z[Q]−2w))
the last result being obtained by a change of the order of summation the 193
justification for which will appear subsequently. By means of Lemma
21 we finally get
f (S , z) = 1+
n
X
r+1
−r
|S | 2
X X g(S , W) X
Q
W
dmr
F
m
|Π(2 f +2w−Z[(Q)])y 2 F = F (r)
semi integral
It is obvious that every quadratic form of r variables with rational
coefficients has a representation of the form (F + W)[∈]above and conversely, so that, in view of W[∈] and (F +W)[∈] having the same denominator d and g(S , W) being invariant for a replacement of W by (F + W)
we can finally write
f (s, z) = 1 +
n
X
r+1
194
|S |
−m
−r X X g(S , W)
|πi(dW − Z[Q]))| 2
mr
2 Q W
d
(242)
where W = W (r) now runs overall rational matrices. More explicitly we
obtain
Lemma 23. Let W (r) run over all rational symmetric matrices and Qn,r
run over a complete set of non right associated primitive matrices. Then
f (s, z) = 1 +
n
X
r=a
−r
cw
X
W,Q
−r
d 2 g(s, w)|z|Q| − 2w|
−m
2
(243)
provided m > n2 + n + 2 where d denotes the denominator of W[∈].
The reduction from (242) to (243) is straightforward and immediate.
P −mr
α g(S , W)|Z[Q] −
It remains only to show that the double series.
W,Q
m
2W|− 2 converges absolutely, to justify our earlier formal manipulations
with this series. In the case r = n, we have Q = E and this series reduces
P −mn
m
to
d g(S , W)|Z − 2W|− 2 . Since Z is fixed and ||Z − 2W|| ≥ |Y| the
W(n)
183
absolute convergence of this series can be clearly made to depend on
P −mn
d g(S , W) and the latter series does converge absolutely
that of
Wn
under our assumption m > n2 + n + 2, by means of Lemma 20. We have
then only to consider the case 1 ≤ r < n. Given the point, Z, we can
always assume Q so chosen that y[Q] is reduced. We can determine a
real non-singular matrix F (r) and a diagonal matrix, H = (δ( µν)) so that
Z[Q] − 2W = (H + iE)[F] and then
x[Q] − 2W = h[F] + y[Q] = F ′ F and |Y[Q]| = |F|2
(244)
If Q = (qµν) = (G1 G2 . . . G ), q = maxµν ±qµν and h = maxν ±hν we
have from (47-49),
195
r
Y
|Z|[Q] − 2W = |Y[Q]|
(i + hν ) and
ν=q
Ci |Y[Q]| ≤
Y
ν = 1]r Y[G ′ ν , Gν ]
(245)
[
where C1 , denotes a positive constant depending only upon n and λ
denotes the smallest characteristic root of Y . In stating the last of these
inequalities we have only to observe that G ′ ν Gν ≥ 1. If Q is a given
primitive matrix t, a given integer, we ask for the number of quadratic
forms W[∈] with rational coefficients and with a given denominator d,
consistent with the inequality t − 1 ≤ h ≤ t. From the relations (244) it
is easy to see that this number has the upper estimate (dtq2 )r(r+1)/2 .
By lemma (19)
g(s, W) < ℓdmn−m/2 and
||Z[Q] − 2w|| = |(y[Q])|Πrν=1 (i + hν)| ≤ b1 t|y[Q]| with
appropriate constants b, b1 , so that the absolute convergence of the series
under consideration can be reduced to the convergence of the series .
∞ X
∞ X
X
d=1 t=1
Q
m
m
(dtq2 )r(r+1)/2 d−mr dmr− 2 t− 2 |y[Q]|
184
=
∞
∞ X
X
X
m
m
qr(r+1) |y[Q]|− 2
(dt)r(r+1)/2− 2
Q
d=1 t=1
∞

X r(r+1) m 2 X
m
−
=  d 2 2 
qr(r+1) |y[Q]|− 2
Q
d=1
196
The series within the parenthesis is clearly convergent under our
assumption m > n(n + 1) + 2 > r(r + 1) + 2 so that we need only
−m
′
r(r+s)
confine ourselves to the series ΣQ q
|y[Q]| 2 . From (245) it is
clear that this last series, but for a constant multiplier, is majorised by
r(r + 1) m
−
2 , and this in turn is further majorised by a constant
ΣQ |y|[Q]| 2
multiple of
r
X
Y
r(r+1) −m r
r(r+1)
ΣGν ,0 ( )) 2 = ( (G ′ G )) 2 2 )
G ,1
ν=1
The number of integral columns G , 0 witht2 ≤ G ′ G < (t + 1)2
where t is a given integer, is of the order of tn−1 so that the last series
P
n−1+r(r+1)−m
can be compared with ∞
t=1 t
The series converges since m > n2 = (n − 1)n + n ≥ (r + 1) + n and
the proof of Lemma 23 is complete.
We now transform series (243) for the genus invariant in another
form by means of the calculus matrices. Towards this effect, we put
−2W = C1−1 D1 where C1(r) , D(r)
1 denote symmetric coprime matrices with
|C1 | , 0. The relation between W and the class {C1 , D1 } is bi-unique as
we have seen in pp. 44. According to Lemma 1, there is also is a 1 − 1
correspondence between the classes C1r Dr1 , Q(r,r) with |C1 | , 0, and
the classes C1r Dr1 with rank C = r. Since from (69),
|Z[Q] − 2W| = |Z[Q] + C1−1 = |C1−1CZ|Cz + D|
197
we conclude from (243) that
f (s, z) =
X
C (n) ,D(n)
n
h(c.d)|Czz + D| 2
(246)
185
with appropriate coefficients h(C, D)and it only remains to determine
useful expressions for h(C, D) .
We stop her to establish
Lemma 24. Let Cl(r) be an integral non-singular matrix and q a positive
multiple of |C1 |. If Gm,r runs through a complete set of integral matrices
distinct mod. q the GC1 runs through exactly qmn ||C1 ||−m of them , each
of these appearing the same number of times. i.e., ||C1 || times.
Proof. We can assume that C1 i is a diagonal matrix , as otherwise there
exists unimodular matrices u1 , u2 such that the product u1 , Cu2 is a
∗ −1
diagonal matrix C1∗ and then the products G1C1 = G1 u−1
1 C 1 u2 and
∗ −1
G2C1 = G2 u−1
mod .q for two matrices G∗1 , G∗2
1 C 1 u1 are distinct
which are themselves distinct mod .q. This in particular implies that
for the purpose of determining the number of matrices GC distinct
mod .q it is immaterial whether we argue with C1 or C1∗ . We therefore assume that C1 is a diagonal matrix, C1 = (δµνC{ ν}), C{ ν} > 0. Let
G = (gµν ) and G∗ = (g∗µν ) and assume GC1 ≡ G ∗ C1 (modq). Then
gµνCν ≡ gµνCν ( mod .q), µ = 1, 2, . . . , mν = 1, . . . r.
By assumption |C1 | is a divisor of q and a fortiori , each Cν is a 198
divisor of q. It therefore follows from the above congruence that
gµν ≡ g∗µν (q|cν )
(247)
The number of products G∗C1 which are congruent to GC1 (mod ·q)
is then just the number of solutions of the congruence (247) and this
number is clearly Πµ nu cν = ||C1 ||m . Consequently , the number of products GC1 distinct mod · q is exactly qmr ||Ci ||−m , amr being the total
number of G = G(m,r) distinct mod · q and the Lemma is proved.
Before resuming the main thread we note the following : If A, G are integral matrices , then
σ(S [A + GC1 ]C −1 D1 )pσ(S [A]C1−1 D1 )
= σ(Ci′G′ S GDi ) + σ(C1′ G′ S AC1−1 Di ) + σ(A′ S GDi )
= σ(Ci′G′ S GDi ) + Di ) + 2σ(A′ S GDi )
= σ(S [G]GiCi′ ) + 2σ(A′ S GDi )
(248)
186
In stating the above relations we have only made use of the fact that
C1−1 D1 is a symmetric matrix.
Consider now the sum
X
−1
e−πσ(S [A]C1 D1 )
(249)
A mod (2Ci )|
199
where the sum is extended over a complete set of integral matrices A
such that no two of them differ by a matrix of the form G2C1 having
2C1 as a right divisor. It is immediate from (248) that the sum (249) is
independent of the choice of representatives A mod.(2C1 |). Two cases
arise.
Case. i: Suppose that the congruence σ(s[G]DiCi′ ) ≡ 1( mod .2) is
solvable for an integral G. Then for that G, a replacement of A by A +
GC1 changes the sign of each term of the sum (249) as is seen from
(248) while the sum itself is left unaltered due to such a replacement. It
therefore follows that in this case
X
−1
A mod (2C1 )|eπiσ(S [A]Ci D1 ) = 0
(250)
Case. ii: Suppose on the other hand σ(S [G]D1C1′ ) ≡ 0( mod .2) for
every integral G. Then the general term of (249) depends only on A
mod (Cν and we can write the sum as
X
X
−1
−1
eπiσ(S [A]C1 D1 ) = 2mr
(251)
eπiσ(S [A]C1 D1 )
A mod (2Ci |)
A mod (2Ci |)
Let us now consider the Gaussian sums
1
g(S , w) = g(s, C −1 Di )
X2
−1
=
e−πiσS [A]C1 Di
mod .d
200
1
The denominator d of the quadratic form W[ε] = (Ci−1 D1 )[ε] is
2
obviously a divisor of 2|C1 | and let us assume that |2Ci | divides q. Then
A ≡ A∗ ( mod .q) implies that A − A∗ ≡ ( mod .12Ci |) in other words
187
(A − A∗ )|2Ci | ≡ 0( mod .1) and A ≡ A∗ mod .(2Ci )|.
Thus we obtain that
X
−1
−1
1
eσ(S [A]Ci D1 ) e−πiσ(S [A]Ci Di )
g(s, − C1−1 Di ) =
2
A mod .d
d mr X −πiσ(S [A]C−1 D1 )
i
=( )
e
q
A mod q
X
X
d
B mod q
πiσ(S [B]C1−1 Di )
= ( )mr
B≡A mod (2ci )|e
q
A mod (2Ci )
X
X
d mr πiσ(S [B]C−1 Di )
B mod q
1
=( ) e
B≡A mod (2ci )|
q
A mod (2C )
i
−1
s̈ince the term eπiσ(S [B]C1 Di ) is invariant for a replacement of B by
B∗ ≡ B mod (2, ci )| and in particular for a replacement of B by A. In
order to compute the inner sum we put B = A + 2GCi with an integral
G. If G runs over all cosets mod .q we obtain according to Lemma 24,
qmr ||2Ci−m distinct cosets. B mod .q, each one of them ||2C1 ||m times.
The inner sum is thus equal to qmr ||2Ci ||−m and we obtain that
1
dmr g(S − C1−1 D1 ) = ||2C||−m
2
A
X
−1 D
eπiσ(S [A]C1
i)
mod (2C1 )
In view of (250) - (251) it then turns out that



1 −1
0, or
mr
d g(S − C1 D1 ) = 

||Ci ||−m PA mod (2C ) eπiσ(S [A]C1−1 Di )
2
1
201
(252)
according as the congruence σ(S [G]DiC;i ) ≡ 1(12) is solvable for an
integral G or not.
From lemma (23) and (246) we now obtain
Theorem 18. Let C (m) , D(r) runs over a complete set of non associated
coprime symmetric pairs of matrices and let m > n2 + n + 2. Then
X
f (S , z) =
h(S , C, D)(CZ + D1)m/2
(253)
C,D
188
where h(s, oE) = 1 and for C , 0,

−m P
−r


i6mr/2 2 ||Ci || 2 A
h(S , C, D) = 

0
πiσ(S [A]C1−1 Di )
mod (ci ) , ore
(254)
according as the congruence σ(S [G]DiC1′ ) ≡ 1(2) is not or is solvable
for an integral matrix G where C1(r) D(r)1 denotes the unique class which
corresponds to C, D by lemma 1.
We have now expressed Siegel’s main theorem on the theory of modular forms in the shape of an analytic identity. It can be shown that
h(S , C, D) in (253) depends only on the cosets of C and D mod 4|5|
and consequently
X
X
|CZ + D|−m/2
f (s, z) =
h(S .C.D)
{C,D} mod 4|5|
202
C≡C|
{C,D}
mod 4|S |)
D≡Dc
In other words f (S , Z) is expressed as a finite linear combination of
the sums
X
{C.D}
C≡Co D≡Do ( mod 4|5|)
of the Einstein series which converges for m > 2n + 2. The fourth power
of these sums represent forms of degree n and weight 2m with respect
to the congruence group M(4|S |) consisting of all modular matrices,
M ≡ E0 E0 ( mod .4|S |)
The same is also true of the fourth power of the Theta- series ϑ(S , z)
A more detailed account of these results, in particular for n = 1, one
finds in a paper of Siegel. “Über die analytische theorie der quadratischen formen.” Ann. Math. 36(1935), 527 - 606.
Chapter 14
Indefinite Quadratic forms
and modular forms
We proceed to investigate how far the results of the last section can be 203
carried over to the case of indefinite quadratic forms. The very first
concept we introduced in the last section was that of the representation
number α(S , T ) which stood for the number of integral solutions of the
matrix equation S [x] = T where S = S (m) and T = T (m) are given
positive integral matrices. The problem of determining the number of
integral matrices x(m,n) satisfying the above equation in the case of an
indefinite S is much more difficult. In fact an indefinite rational symmetric matrix S is known to have in general an infinity of units, viz.
unimodular matrices u with S [u] = s. If now G is one integral solutions
of the equation S [X] = τ so is uG for every unit u of S so that the number of integral solutions of this equation will in general then be no more
meaningful, and we therefore replace it by the concept of representation
measures. If we divide the set of all integral solutions G of S [G] = T
into equivalence classes, stipulating that two different G′ s, say G1 , G2
belong to the same class if and only if G2 = uG1 for some unit u or S , the
resulting number of distinct classes can be shown to be finite. To each
to these classes we attach a positive weight in a particular way and then
the sum of these weights for all the classes will yield the representations
measures αA (S , T ). The following considerations are more general in
189
190
204
14. Indefinite Quadratic forms and modular forms
so far as they treat the notion of representation measure αA (S , T ) for the
set of solutions of the inhomogeneous equation S [G + A] = T .
Let S = S (m) be a real non-singular matrix and let (p, q) be the
signature of S . That is to say, for a suitable non-singular matrix C we
have
(p)
S = S o [C] with S o = −Σ0 Σ0(p)
While (p.q) is uniquely determined by S , C is not necessarily unique. Let us set G = C ∈, ∈= (xu ) and G = (ym )
Then we have S [∈] = −y21 − y22 − −y2p + y2p+1 + y2p+1 , p + q, p + q =
m Besides S [∈], we also consider the positive form P[∈] = y21 + y22 +
quady2p+q .
We call the symmetric matrix P defined by the above equation, a
majorant of S . This majorant is not unique as it evidently depends on
the choice of C. However, any majorant P of S is easily seen to be
characterized by the relation
PS −1 P = S , P > 0
(255)
The equation (255) is clearly invariant under the simultaneous transformation S → S [V], P → P[V] where V is an arbitrary non singular
real matrix. It is then immediate that the majorants Po of S o defined by
Po S o−1 Po = S o , Po > 0
205
(255′ )
yield all the majorants P of S in the form P = Po [C] where C, S o have
the same meaning as before. To determine a parametric representations
for the solutions of (255) it therefore suffices to consider the special case
(255′ ). In the border cases p = 0 or q = 0 we should have clearly S o = E
and S o = −E respectively so that the only possible solutions of (255)
are P = S and P = −S in the respective cases. For pq > 0, a parametric
representation for the solutions Po of (255′ ) is given by
Po = 2K − S o , K = Y −1 [x′ S 0 ], y = S o [x] > 0
(256)
where X is any real matrix of the type xµν with rank x = q and the
solutions of (255) are given by P = Po [C], Po , being represented by
(256). Writing x′ = (W1 W1 ) with and we observe that
′
′
1
y = S o [X] = (W ′ , W2′ )S (W
W2 ) = W1 W1 + W2 W2
191
so that the condition Y > 0 in (256) in particular implies that W2′ W2 > 0
and consequently |W2 | , 0. It is easy to see that two matrices X1 , X2
yield the same point in (256) if and only if X2 = X1 H with a nonsingular matrix H. We can therefore assume in (256) by replacing X by
1
XW21 = (W
E ) with W = W1 W2 that X is always of the normalized form
X = (W
E ) Given, X, Po is uniquely determined in (256) but not necessarily
is the converse true in general. However Po does determine W uniquely,
in other words X is uniquely determined by Po if we stipulate that X is
always of the above normalized form. What is more, the transformation
from W to P = Po [C] can be easily shown to be bi rational. Specially
with x = (W
E ) we have
′
−2w(E−W−W)1
( E+WW
E−WW ′
C, E − WW > 0
(257)
P=C
1
′
′
−2w(E−W−W) E+WW E−WW
Let u = CA DB with A = A(p) be a real matrix which transforms S o 206
into itself, i.e., S o [u] = S o . Then the transformation X → uX has on P0
and W the effect
Po → Po [u−1 ], W → u < w >= (AW + B)(CW + D)−1
(258)
The transformations W → W ∗ = u < W > constitute a group 1 − 1
transformations of the domains E−W ′ W > 0 onto itself and in this group
we have a sub-group formed by the units of S o viz., those among the u′ s
which are further unimodular. This group of units can be shown to be a
discontinuous group acting on the space E −W ′ W > 0 and an immediate
question is about a fundamental domain for this group or one of its sub
groups and the volume of the fundamental domain in an appropriate
sense. Towards this effect we note that the space E − W ′ W > 0 can be
considered as a Riemannian space with a metric, invariant relative to the
transformation (258), given by
8ds2 = σ(p−1 dp)2 = σ(P−1
o dPo )
= σ((E − WW ′)−1 dW(E − W ′ W)−1 dW ′ )
(259)
The invariant volume element in this metric can also be computed
to be
m
dϑ = |E − WW ′ 1 2 [dW]
(260)
192
207
All these considerations are valid for every real matrix S with signature p, q. We now restrict the domain of S to suit our needs.
Consider the inhomogeneous equation
S [G + A] − T = 0, G = G(m,n) integral (m ≥ n)
(261)
and require that the left side of this equation should represent an integral
valued matrix with the elements of G as variables, via., a matrix whose
elements are polynomials in the elements of G with integral coefficients.
This means the following requirements on S , A and T
(i) S is semi integral
(ii) S A is integral for n > 1 and 2S A is integral for n = 1
(iii) T ≡ S [A]( mod 1)i.e., S [A] − T is integral.
Confining ourselves to the above case, let us introduce the theta series
fA (Z, P) =
X
e2πiσ(S [B]x+iP[B]y ), z = x + iy
(262)
B≡( mod 1)
208
where P denotes an arbitrary majorant of S . In the case p = 0 we
should have P = S and then fA (Z, P) represents a theta series in the
usual sense. The case q = 0 presents no new difficulty either so that we
confine our attention to the case pq > 0. Since fA (Z, P) depends only
upon the coset A( mod 3) and A[ε] has a bounded denominator, it is
clear that we only we have a finite number of different series fA (Z, P)
corresponding to a given majorant P of S . Let f(Z, P) denote the column
with the f(Z, P)′ s as elements in a !certain order. Due to an arbitrary
A B
unimodular substitution M =
∈ M on f (Z, P), one obtains
C d
P
q
|CZ + D|− 2 |C Z̄ + D|− 2 f (M < Z >, ρ) = L M f (Z, P)
(263)
with a non-singular matrix LM . The matrices LM belong to the modular group in case p ≡ q ≡ o( mod 2), and (263) is quite meaningful for
193
positive definite S too.
! Finally, it can be shown that LM is the unit maE o
trix for M ≡
(mod λ) for a suitable integer λ, so that the matrio E
ces LM define a representative system for the quotient group M/M(λ)
where M(λ) denotes the main congruence group to the level λ i.e. M(λ)
!
E o
consists of the substitutions M satisfying the congruence M ≡
o E
(mod λ)
In the group of units U of S , viz. unimodular matrices U with
S [U] = S , let yA (S ) be the sub group defined by UA ≡ A( mod 1).
We denote by FA (S ) a fundamental domain in E − W ′ W > o relative to
this sub-group and introduce the volume
Z
VA (S ) =
dϑ
(264)
FA (S )
on FA (S ) computed with the volume element (260). This volume can be
shown to be finite in all cases with
√ one exception-the exceptional case
being one for which m is 2 and −|S | is rational. From (262) it is easily
seen that
FA (Z, P[U]) = F (Z, P)
(265)
for U ∈ yA (S ) Therefore it makes sense to form the integral mean value 209
Z
1
fK (Z, P)dV
(266)
gA (Z, S ) =
VA (S )
FA (S )
This integral certainly exists provided 2n < m − 2
It is remarkable that the column G (Z, s) with elements gA (Z, s) (in
some order) also satisfies a transformation formula of the same kind as
ρ(Z, P) does in (263).
Specifically we have
p
q
|Cz + D|− 2 |C Z̄ + D|− 2 G (M < Z >, S ) = LM G (Z, S )
(267)
Developing gA (Z, S ) into a Fourier series there results an expansion
which differs from the usual ϑ- series expansion (241) for the definite
194
case only in so far as the representation numbers α(S , T )
in (241)
are to be replaced by the representation measures, αA (S , T ) and the exponential function e−2πσ(tY[Q] ) by certain generalized confluent hyper
geometric functions.
If we replace the representation measures in the Fourier series of
gA (z, s) by the products of the p-adic solution densities, consequent on a
main theorem of Siegel concerning indefinite quadratic forms, we obtain
a new representation of gA (z, S ) which yields a partial fraction decomposition of the kind
X
−q
−p
(268)
gA (Z, S ) =
hA (C, D, S )|CZ + D| 2 |C Z̄ + D| 2
C,D
210
where (C, D) runs over a complete set of non associated symmetric coprime pairs and hA (C, D, S ) denotes a certain sum of the kind (254). As
against the definite case we have in (268) not only a means of expressing
Siegel’s main theorem as an analytic identity but actually it permits us to
prove Siegel’s result by analytical means. Towards this effect, one has to
show that the Eisenstein series on the right of (268) have the same transformation properties relative to the modular substitutions as gA (z, S ) and
that the Eisenstein series can be developed into a Fourier series of the
same kind as that of gA (z, S ). These facts coupled with some special
properties of the generalized confluent hyper geometric function yield,
as M.Koecher could prove (unpublished) that the two sides of (268) are
identical upto a constant factor, which factor becomes 1 after a suitable
normalization.
Now the question presents itself whether functions of the type (268)
which no longer depend analytically on the elements of Z can be characterized in any way. As we shall see in the next section, such a characterization is possible by a system of partial differential equations with
the invariance properties we have got to require of them. Another fact
which appears at the outset in the case n = 1 can also be reasonably
fused into this differential equation-theory. It concerns the following
problem:In the case n = 1 if we apply the Mellin’s transform to Siegel’s
zeta functions of indefinite quadratic forms with the signature (p, q) we
195
obtain a function of the type
X
hA (C, D, S )|cz + D|−α |cz̄ + D|−β
(269)
(C,D)
instead of (268), with certain exponents α, β which satisfy the relations
α ≡ P/2, β ≡ q/2(
mod 1), d + β = 1/2(p + q)
We now ask for a process which yields a direct correspondence between the type (268) and (269) without the use of Dirichlit series. We 211
shall in the following section that such a correspondence can be defined
through certain differential operators.
For further details on some of the points raised in this section we
refer to
1. H.Maass, Die Differentialgleichungen in der Theorie der elliptiochen Modulfunktionen, Math. Ann. 125(1953), 235-263.
2. C.L.Siegel, On the theory of indefinite quadratic forms, Ann.
Math. 45(1944), 577-622.
3.
′′
′′
4.
′′
′′
, Indefinite quadratische Formen und Modulfunksionen,
Studies Essays, Pres. to Courant, New York 1948, 395-406.
, Indefinite quaderatische Formen und Funtionen theoie
I, Math. Ann. 124(1951), 17-54.
Chapter 15
Modular Forms of degree n
and differential equations
Let Z = (zµν ) and Z̄ = (z̄µν ) and consider the elements zµν , z̄µν as in- 212
dependent complex variables. We shall require however that zµν + z̄µν
and i(zµν , Z̄µν ) are real. Let α, β be arbitrary complex numbers. By
|CZ + D|−α and |C Z̄ + D|−β where Z ∈ Y , and (C, D) represents the second matrix row of a symplectic substitution we always understand the
functions e−αlog|CZ+D| and e−α log |CZ̄+D| with the principal value for the
logarithm defined by
log z = log |z| + i arg z, log |z| real , −π < arg z ≤ π
for complex numbers z , o
We now ask for differential operators Ωαβ which annihilate the
Eisenstein series
X
h(C, D)|CD + D|−α |C Z̄ + D|−β
(270)
g(Z, Z̄, α, β) =
C,D
for an arbitrary choice of the constants h(C, D) where (C, D) denotes
the second matrix row of a modular substitution or more generally of a
symplectic substitution of degree, n. Thus we have to require that
Ωαβ |CZ + D|−α |C Z̄ + D|−β = o
197
(271)
198
15. Modular Forms of degree n and differential equations
for all pairs (C, D) such that rank (C, D) = n, CD′ = DC ′
We shall also demand that the manifold of the functions satisfying
Ωαβ f (Z, Z̄) = o
213
(272)
is invariant relative to the transformations
f (Z, Z̄) → f (Z, Z̄)|M = |CZ + D|−α |C Z̄ + D|−β f (M < Z >, M < Z̄ >)
(273)
!
A B
for any M =
∈ S In other words if f (z, z̄) is a solution of
C D
(271) so is f (z, z̄)|M. In view of this requirement, (272) now takes a
particularly simple form. Prima facie, there is no loss of generality if
we assume (271) to hold only for pairs (C, D) with |C| , O as the submanifold defined by |C| = 0 in the manifold of all (C, D)′ s occurring in
(271) is one of lower dimension. Then with P = C1−1 D1 we have P = P′
and (271) is equivalent with
Ωαβ |Z + P|−α |Z̄ + P|−β = o
(271)
If we further assume that (272) is left invariant by the transformations (273) as we did, and in particular by the! transformations
E −P
f (z, z̄) → f (z, z̄)|M where M =
then (271) can be further
o E
simplified into
Ωαβ |z|−α |z̄|−β = o
(274)
and this is the equivalent form of (271) we sought for.
In order to construct the operators Ωαβ with the desired properties
we introduce the matrix operators
∂
∂
∂
∂
= (eµν
),
= (eµν
)
∂z
∂zµν ∂z̄
∂z̄µν
(275)
1
with eµν = (1 + δµν ) and
2
Kα = αE + (z − z̄)
∂
Λβ = −βE + (z − z̄)∂/∂z̄
∂z
(276)
199
214
Then
Ωαβ = Λβ− 1 (n+1) kα+α(β− 1 (n+1))E
2
2
(277)
has the desired properties as we shall see presently.
We also show that Ωαβ and
1
Ω̃αβ = Kα− 1 (n+m) Λβ + β(d − (n + m))E
2
2
annihilate the same functions.
We first prove the following formulae.

∂
∂
∂
∂


Ωαβ = (Z − Z̄)((Z, Z̄) ∂Z̄ )′ ∂Z + α(Z − Z̄) ∂Z̄ − β(Z, Z̄) ∂Z


Ω̃αβ = (Z − Z̄)((Z, Z̄) ∂ )′ ∂ + α(Z − Z̄) ∂ − β(Z, Z̄) ∂
∂Z ∂Z̄
∂Z
∂Z̄
(278)
(279)
∂
∂
f)
f (Z, Z̄) = (
∂Z
∂z µν
∂
∂f
∂
∂
f = f
+
while in other places
f =
that as an operator
∂z µν
∂z µν ∂z µν
∂z µν
∂f
. The meaning of the symbol will be clear from the context.
∂z µν
We take up the proof of (279). We first need to establish the identities
We may remark here concerning the use of
n+1
∂
∂
(z − z̄) =
E + ((z − z̄) )′
∂z
2
∂z
∂
n+1
∂
(z − z̄) = −
E + ((z − z̄) )′
∂z̄
2
∂z̄
Indeed by a simple transformation we have
X
∂
∂
(z̺ν − z̺̄ν ))
(z − z̄) = ( (eµ̺
∂z
∂z µ̺
̺=1
n
n
X
X
∂
= ( (z̺ν − z̺̄ν )eµ̺
+ ( eµ̺ δµ̺ )
∂z µ̺
̺=1
̺=1
n
X
n+1
∂
= ( (z̺µ − zµ̺ )e̺ν
)+
δµν )
∂z ̺ν
2
̺=1
(280)
200
n
X
n+1
∂
)+
E
= ( (zµ̺ − z̄µ̺ )e̺ν
∂z
2
̺ν
̺=1
= ((z − z̄)
∂ ′ n+1
) +
E
∂z
2
215
The other half of (280) follows on similar lines.
Now consider Ωαβ . By means of (280) we have
∂
n+1
∂ n+1
αE + (z − z̄)
+ α(β −
)E + (z − z̄)
)E
Ωαβ = − (β −
2
∂z̄
∂z̄
2
∂
∂
∂
n+1
∂
= (z − z̄) (z − z̄) + α(z − z̄) − (β −
)(z − z̄)
∂z̄
∂z
∂z̄
2
∂z
∂
∂
∂ ∂
= (z − z̄)((z − z̄) )′ + α(z − z̄) ) − β(z − z̄)
∂z̄ ∂z
∂z̄
∂z̄
which is the first part of (279). The proof of the other part is exactly
similar.
We now show that Ωαβ annihilates the Eisenstein series (270). A
formula proof only requires that it annihilates each term of (270) separately, in other words that (271) is true. Let A = (aµν ) be a square matrix
the elements of which belong to a commutative ring, and Aµν the algebraic minor corresponding to aµν . With Ã = (Aνµ ) we have in general
AÃ = |A|E.
Then
∂
∂
∂ −α
|z|−α ) = −α|z|−α−1 (eµν
|z|)
|z| = (eµν
∂z
∂z µν
∂z µν
= −α|z|−α−1 (zµν ) = −α|z|−α−1 z̃
216
∂ −β
|z̄| can be similarly computed and we have
∂z̄


∂/∂z |z|−α = −α|z|−α−1


z̃
−β
−β−1 

∂/∂z̄ |z| = −β|z̄|
z̃
Since from (279)
(281)
201
(z − z̄)−1 Ωαβ = ((z − z̄)∂/∂z̄ )′
that
∂
∂
∂
+ α − β it follows by means of (281)
∂z
∂z̄
∂z
(z − z̄)−1 Ωαβ |z|−α |z̄|−β = − αβ|z|−α |z̄|−β−1 z̃ + αβ|z|−α−1|z̄| −βz̃
∂
− α((z − z̄) )′ |z|−α−1 z̃|z̄|−β .
∂z̄
The last term on the right can be rewritten as
∂
−α((z − z̄) |z|−β )′ |z|−α−1 z̃ with the terms outside the parenthesis
∂z̄
not depending on z̄. Using (281) this is easily seen to be equal to
αβ|z|−α−1 |z̄|−β−1 z̄(z − z̄)z̃
Thus
(z − z̄)−1 Ωαβ |z|−α |z̄|−β = −αβ|z|−α |z̄|−β−1 z̄+
+ αβ|z|−α−1 |z̄|−β z̃+
+ αβ|z|−α−1 |z̄|−β−1 Z̃(z − z̄)z̃
and the right side is now easily seen to vanish, there by establishing 217
∂ ∂
∂ ∂
(271). Since ((z − z̄) )′ is the transpose of ((z − z̄) )′ we verify
∂z̄ ∂z
∂z ∂z̄
without difficulty that
Ω̃αβ = (z − z̄)((z − z̄)−1 Ωαβ )′
and this says that Ωαβ and Ω̃αβ annihilate the same functions as we
wanted them to do.
We have still to prove the invariance of the manifold formed by the
solutions of
Ωαβ F (z, z̄) = o
(272)′
under the transformations
f (z, z̄) → f ∗ (z, z̄) = |cz + D|−α |cz̄ + D|−β f (M < z > M < z̄ >) (273)′
!
A B
where M =
∈ S. The proof is rather long and mainly consists
C D
in securing the following operator identity:
|cz + D|−α |cz̄ + D|−β Ω∗αβ |cz + D|α |cz̄ + D|β = (zc′ + D′ )−1 ((cz + D)Ω′αβ )′
(282)
202
218
where Ω∗αβ is the operator which results from Ωαβ on replacing z, z̄ by
Z ∗ = M < Z >, Z̄ = M < z̄ > respectively.
Assuming (282) for a moment, to obtain our main result, we argue
as follows:We have to show that, under (272), Ωαβ f∗ (z, z̄ = o
This is equivalent to showing that the right side of (282), or equivalently, the left side annihilates f∗ (z, z̄.
The left side of (282) applied to f ∗ (z, z̄ gives
|cz + D|−α |cz̄ + D|−β Ω∗αβ f (z∗ , z̄∗ )
and this clearly vanishes under (272). This proves the desired result. We
have then only to establish (282).
By means of the general rule relating to any three square matrices
M1 , M2 , M3 which are such that the elements of M1 . commute with
those of M2 , viz.


(M1 M2 )′ = M2′ M2′ , (M1 (M2 M3 )′ )′ = M2 (M1 M3′ )′ 

(283)


σ(M1 M2 ) = σ(M2 M1 ) 
It is possible to reduce the proof of (282) to showing that

∂ ′
∂ ′
α
α
α ′



((cz + D) ∂z ) |cz + D| = |cz + D| ((cz + D) ∂z ) + α|cz + D| c



((cz + D) ∂ )′ |cz̄ + D|β = |cz̄ + D|β ((cz̄ + D) ∂ )′ + β|cz̄ + D|β c′
∂z
∂z̄
(284)
It suffices of course to establish the first part of (284) and that too under the assumption |C| , 0 as in the alternative case, the corresponding
(C, D)′ s form a sub-manifold of lower dimension as stated earlier.
In this case, (284) can be reduced to its special form corresponding
to C = E, D = o
viz.
(z
∂ ′ α ∂
) |z| (z ) + α|z|α E
∂z
∂z
(z
∂ ′ α ∂
) |z| (z ) + α|z|α E
∂z
∂z
or equivalently
203
219
by a substitution of the form Z → Z + S with an appropriate symmetric
matrix S , and the last relation is immediate by a proof analogous to that
of (281).
To deduce (282) as a consequence of (283) and (284) one need first
∂ ∂
relato determine the transformation properties of the operators ,
∂z
∂z̄
!
A B
tive to symplectic substitutions. Let
denote a symplectic matrix
C D
and let
z∗ = (AZ + B)(cz + D)−1 , z̄∗ = (AZ̄ + B)(cz̄ + D)−1
From (122) we deduce that
αz∗ = (zc′ + D′ )−1 dz(cz + D)−1 , αz̄∗ = (z̄c′ + D′ )−1 dz̄(cz̄ + D)−1
If f = f (z, z̄) is an arbitrary function of z, z̄ its total differential α f
can be represented in the form
d f = σ(dz
∂
∂
f ) + σ(dz̄ f )
∂z
∂z̄
on the one hand, and on the other,
∂
∂
d f = σ(dz∗ ∗ f ) + σ(dz̄ ∗ , f )
∂z
∂z̄
∂
= σ dz(cz + D)−1 ( f )(zc′ + D′ )−1
∂z
∂
+ σ dz̄(cz̄ + D)−1 ( ∗ f )(z̄c′ + D′ )−1
∂z̄
Comparing the last with (285) one deduce that
∂
∂
f = (cz + D)−1 ( ∗ f )(zc′ + D′ )−1
∂z
∂z
∂
∂
f = (cz̄ + D)−1 ( ∗ f )(z̄c′ + D′ )−1
∂z̄
∂z̄
(285)
204
Consequently we have the operator identities
220
∂ ′
∂


=
(cz
+
D)((cz
+
D)(
)


∂z∗
∂z 


∂
∂ ′



=
(cz̄
+
D)((cz̄
+
D)(
)
∗
∂z̄
∂z̄
(286)
With these preliminaries we take up the proof of (282). For convenience we shall denote the product |cz + D|α |cz̄ + D|β by ϕ. Then by
means of (283), (284), (286) and (122) we have
∂
∂
∂ ∂
Ω∗αβ α = (z∗ − z̄∗ )(z∗ − z̄∗ )( ∗ )′ ∗ + α(z∗ − z̄∗ ) ∗ − βz∗ − z̄∗ ) ∗
∂z̄ ∂Z
∂z̄
∂z̄
∂
∂ =(zc′ + D′ )−1 (z − z̄)(cz̄ + D)−1 (z − z̄)((cz̄ + D) )′ ((cz + D) )′ α
∂z̄
∂z̄
∂
+ α(zc′ + D′ )−1 (z, z̄)((cz̄ + D) )′ − β(z̄c′
∂z̄
∂ + D′−1 (z − z̄)((cz + D) ) α
∂z̄
=|cz + D|α (zc′ + D′ )−1 (z, z̄)(cz̄ + D)−1
∂
∂ (z, z̄)((cz̄ + D) )′ |cz̄ + D|β ((cz + D) )
∂z̄
∂z
∂ + α|cz + D|α (zc′ + D′ )−1 (z, z̄)(cz̄ + D)−1 (z, z̄)((cz̄ + D) )′ |cz̄ + D|β c′
∂z̄
∂ ′
′
′ −1
′
′ −1
+ αϕ(zc + D ) (z, z̄)((cz̄ + D) ) + αβϕ(zc + D ) (z, z̄)c′
∂z̄
∂
− βϕ(zc′ + D′ )−1 (z, z̄)((cz̄ + D) )′
∂z̄
− αβϕ(z̄c′ + D′ )−1 (z, z̄)c′
∂ ∂
= ϕ(zc′ + D′ )−1 (z, z̄)(cz̄ + D)−1 (z − z̄)((cz̄ + D) )′ ((cz + D) )′
∂z̄
∂z̄
∂
+ βϕ(zc′ + D′ )−1 (z, z̄)(cz̄ + D)(c(z − z̄)((cz̄ + D) )′
∂z̄
∂ + αϕ(zc′ + D′ )−1 (z, z̄)(cz̄ + D) (c(z − z̄)((cz̄ + D) )′
∂z̄
+ αβϕ(zc′ + D′ )−1 (z, z̄)((cz̄ + D)−1 c(z, z̄)c′
∂
+ αψ(zc′ + D′ )−1 (z, z̄)((cz̄ + D) )′ + αβϕ(zc′ + D′ )−1 (z − z̄)c′ + ..
∂z̄
∂
− βϕ(z̄C ′ + D′ )−1 (z − z̄)((cz + D) )′ − αβϕ(z̄C ′ + D′ )−1 (z − z̄)C ′
∂z
205
221
Collecting the like terms together it is seen that the terms involving
αβϕ together and to zero while those involving ϕ, αϕ and βϕ by themselves, all survive. These terms ultimately turn out to be respectively
∂
∂ ′
) ((CZ + D) )′ ,
∂z
∂z
∂
αϕ(ZC ′ + D′ )−1 (z − Z̄)((CZ + D) )′ and
∂z
∂
′
′ −1
−βϕ(ZC + D ) (Z − Z̄)((CZ + D) )′
∂z
(ZC ′ + D′ )−1 (Z − Z̄)((z − Z̄)
Thus
∂
∂ ∂
Ω∗αβ ϕ = ϕ(zc′ +D′−1 , (z− z̄) ((z− z̄) )′ +α((cz+D) )′ −β((cz+D) )′
∂z̄
∂z̄
∂z̄
It then follows that
′
∂ ′
−1
′
′
+
cz − z̄) (zc + D , ζαβ ϕ = ϕ(cz + D) ((z − z̄)
∂z
∂
∂
+αϕ(cz + D) − βϕ(cz + D)
∂z̄
∂z
∂ ∂
∂
∂ ′
=ϕ(cz + D) ((zz̄)′ )′ + α − β
∂z ∂z
∂z
∂z
′
=ϕ(cz + D) (z − z̄)−1 Ωαβ
Consequently
′ ′
(cz + D)−1 (z − z̄)−1 (zc′ + D′ )Ω∗αβ ϕ
= ϕ(z − z̄)−1 Ωαβ
or
′ ′
(cz + D)−1 (zc′ + D′ )Ω∗αβ ϕ
= ϕΩαβ
Hence finally
′
ϕ−1 Ω∗αβ ϕ = (zc′ + D′ )−1 (cz + D)Ω′αβ
222
206
and this precisely is the assertion of (282). The proof is now complete.
1
1
Let us introduce the variables X, Y as x = (z + z̄) and Y = (Z, Z̄
2
2ℓ
∂ ∂
,
as in (275) with Z = (zµ ) replaced by X = (xµν ) and
and define
∂x ∂y
Y = (yµν ) respectively. One then obtains the transformation formulae
∂
1 ∂
∂
∂
1 ∂
= ( −
),
= (
∂z 2 ∂x ∂Y ∂z 2 ∂x
1 ∂
∂
∂
∂
= (
),
=
−ℓ
∂z µν 2 ∂x µν
∂y µν ∂z µν

∂



),



∂Y


1 ∂
∂ 


(
)
+ℓ

2 ∂x µν
∂y µν 
+
Consider the operator ∆ defined by
∂ ′ ∂
∆ = −σ(z, z̄) (z, z̄)
∂z̄ ∂z
In terms of X and Y, ∆ takes the form
∂ ′ ∂ ∂ ∂
)
)
∆ = −σ (Y(y )′ + Y(Y
∂ x̄ ∂x
∂Y ∂Y
(287)
(288)
(289)
This is immediate from the relations
∂
∂ ′ ∂
∂ ∂ ′ ∂
= σY Y( + Y )
−
)
−σ(z − z̄) (z − z̄)
∂z̄ ∂z
∂x
∂y
∂x ∂Y
∂ ′ ∂
∂ ′∂
∂ ∂
∂ ′ ∂
+ σ Y(Y )
+ Y(Y )
− Y(Y )
= σ Y(Y )
∂x ∂x
∂y ∂y
∂y ∂x
∂x ∂y
223
and the fact that
∂ ′ ∂
∂ ∂
S = (Y
)
(Y )′
is a skew symmetric matrix so that
∂Y ∂x ∂x ∂y
σ(YS ) = o
One interesting fact about ∆ is its invariance relative to the symplectic substitutions. Let the substitution Z, Z̄ → Z ∗, Z̄ ∗ carry ∆ into ∆∗
∗
∗
−1
where
! Z = (AZ + B)(CL + D) and Z̄ = (AZ̄ + B)(C Z̄ + D) with
AB
∈ S . We observe from (122) that
CD
Z ∗ − Z̄ ∗ = (ZC 1 + D′ )−1 (Z, Z̄)(C, Z̄ + D)−1
207
= (Z̄C ′ + D)−1 (Z, Z̄)(CZ + D)−1
(122)′
Consequently, by means of (286) and (283) we have
∂ ∂
f
∂Z̄ ∗ ∂Z ∗
n
= −σ(ZC + D)−1 (Z − Z̄)(C Z̄ + D)−1 (C Z̄ + D′ )−1 (Z − Z̄)
∆∗ f = −σ(Z ∗ − Z̄ ∗) (Z ∗ Z̄ ∗)
)
∂ ′
∂
′
′
(C Z̄ + D)
} × {(CZ + D)( f)(ZC + D )
∂z
∂Z̄
)
(
∂ ′
−1
′
′ −1
= −σ(Z − Z̄)(C Z̄ + D)
(ZC + D ) (Z − Z̄)((C Z̄ + D) )
∂z̄
(
)
∂
× (CZ + D)( f)
∂z
∂
∂
′
= −σ(Z − Z̄) (ZC ′ + D′ )−1 (Z − Z̄)
(CZ + D)( f)
∂Z
∂z
∂
= −σ(Z − Z̄) (ZC ′ + D′ )−1 (Z − Z̄) f = ∆f
∂z
and it is immediate that ∆∗ = ∆
We with to identify ∆ with the Laplace Beltrami operator associated with the symplectic metric. We may recall that the Laplace Beltrami operator in a Riemannian space with co-ordinate systems χ1 , χ2 ·
P
χN and the fundamental metric form ds2 =
gµ,ν dχµ dχν is given by
µ,ν
P 1 ∂ √ µν ∂ gg
where g = g(gµν ) and (gµν )(gµν ) = E. With 224
√
g ∂χµ
∂χν
µ,ν
respect to a geodesic co-ordinate system at a given point, this operator
takes in this point the simple form
X
µ,ν
gµ,ν
∂2
∂λN ∂χν
(290)
and can be easily computed. The Laplace Beltrami operator in any Riemannian space is invariant relative to the movements of the space. In
particular the Laplace Beltrami operator associated with the symplectic
metric is invariant under symplectic movements. As this has been shown
208
to be true of ∆ also, to prove the equivalence of the two, it suffices to
verify that they define the same operator at a special point. We choose
this special point to be Z = (E = Z̄). At this point let us introduce the coordinate system W, W̄ by W = (Z−1E)(Z+1E)−1 , W̄ = (Z̄+iE)(Z̄−(E)−1
which is a geodesic one relative to the symplectic metric. Analogous
(286) we have
∂
−1
∂ ′ ∂
∂ ′
1
(291)
=
(W − E) (W − E)
= (W̄ − E) (W̄ − E)
∂Z
Z
∂W ∂Z̄ 2
∂W̄
It terms of W, W̄ we have
∂ ′ ∂
∆ = σ(E − W W̄) (E − W W̄)
∂W̄ ∂W
The special point Z = iE = −Z̄ transforms into W = 0 = W̄ and to
compute the Laplace Beltrami operator at this point, one needs only to
determine the coefficients of the fundamental metric form
ds2 = 4σ dW(E − W W̄)−1 dW̄(E − W W̄)−1
225
at this point. Then in (290) one obtains the same operator as ∆. We
proceed to consider modular forms associated with arbitrary sub-groups
of the symplectic group. Let G be a group of symplectic substitutions
and ν(M) a multiplicator system of G(M ∈ G)α, β being arbitrary complex numbers, a function f(Z, Z̄) is said to be a modular form of the type
(G, α, βϑ)
if
1
(i) f(Z, Z̄) is regular in the domain where Z + Z̄ is real and (Z − Z̄)
2i
is real and positive
(ii) Ωαβ f(Z, Z̄) = 0
(iii)
f(Z, Z̄|M = ν(M)f(Z, Z̄), M ∈ G,
(292)
209
Some restriction on the behaviour of f(Z, Z̄) on the boundary of Y is
perhaps necessary but we assume none. We shall denote by {G, α, β, ν}
the linear space of all modular forms of the type (G, α, β, ν). It is an easy
consequence of our definition that for every symplectic substitution M,
{G, α, β, ν}|M = {M −1GMα, β, ν∗}
(293)
with an appropriate multiplicator system ν∗ depending on M, where the
left side just means the set of all f(Z, Z̄)|M for f(Z, Z̄) ∈ {G, α, β, ν}
We would like to treat the question, whether it is possible to set
up a correspondence between {G, α, β, ν} and {G, α ± 1, β ∓ 1, ν}. Such
a correspondence, we shall see, will be defined by certain differential 226
operators. Consider the Eisenstein series (270), viz.
X
g(Z, Z̄, α, β) =
h(C, D)|CZ + D1−α |C Z̄ + D|−β
(270)′
C,D
and introduce differential operators Mα , Nβ with the properties
where


Mα g(Z, Z̄, α, β) = εn (α)g(Z, Z̄, α + 1, β − 1)



Nβ g(Z, Z̄, α, β) = εn (β)g(Z, Z̄, α + 1, β − 1)
n−1
1
).
εn (λ) = λ(λ − ) · · · (λ −
2
2
(294)
(295)
The truth of (294) for all g(Z, Z̄) or equivalently for all coefficients
h(C, D) in (270)′ clearly requires that
Mα |CZ + D|−α |C Z̄ + D|−β = εn (α)|CZ + D|−α−1 |C Z̄ + D|−β+1 ,
and as in earlier contexts, it suffices to suffices to require this for (C, D)
with |C| , 0. Then by a replacement of the form Z → Z + S with a
suitable symmetric matrix S , the above can be reduced, assuming that
Mα is invariant for such a replacement, to
Mα |Z|−α |Z̄|−β = εn (α)|Z|−α−1 |Z̄|−β+1
210
If further we are sure that in Mα only
is independent of
∂
appears explicitly and it
∂Zµν
∂
as will be the case, then the last relation simpli∂Zµν
fies further into
Mα |Z|−α = εn (α)|Z|−α−1 |Z̄|
227
(296)
It is now our task to construct Mα with these required properties,
viz:(i) Mα depends only on
∂
∂
and does not involve
∂Zµν
∂Z̄µν
(ii) Mα is invariant relative to a replacement of Z by Z + S with an
arbitrary symmetric matrix S .
(297)
(iii) Mα satisfies (296)
For arbitrary integers I ≤ L1 < L2 < · · · < lh ≤ n, 1 ≤ k1 < k2 <
kh ≤ n we get
!


l1 , l2 · · · lh



= |Ziµ kν |


k1 , k2 · · · kh z



(298)

#
"


∂ 
l1 l2 · · · lh


= |eiµ kkν
|

k1 k2 · · · kh z
∂zνµ lν 
Also let
∂
sh (Z − z̄, ) =
∂z
X
1≤l1 <···<lh ≤n
1≤k1 <···<kh ≤n
for h = 1, 2, . . . n and so (Z − z̄,
Finally we introduce
Mα
l1 l2 · · · lh
k1 k2 · · · kh
!
z−z̄
"
li l2 · · · ll
k1 k2 · · · kh
#
(299)
z
∂
)=1
∂z
n
X
∂
εn (α)
sh (Z − Z̄, )(εo (α) = 1)
ε (α)
∂z
h=0 h
(300)
That Mα satisfies the first two conditions in (297) is a matter of
easy verification. Besides, in the case n = 1, Mα is identical with Kα
211
introduced earlier. The proof that Mα satisfies (296) is a consequence of
the following result
"
l1 l2 · · · lh
k1 k2 · · · kh
#
−α
Z
|Z|
h
−α−1
= (−1) εh |α||Z|
l1 l2 · · · lk
k1 k2 . . . kh
!
(301)
Z
!
!
l1 l2 · · · lh
l1 l2 · · · lh
where
.
228
denotes the algebraic minor of
k1 k2 · kh Z
k1 k2 · · · kl Z
This minor differs from the determinant of the submatrix which
arises from Z on cancelling the rows and column corresponding to the
indices l1 , l2 , . . . lh and k1 , k2 , . . . kh respectively by the sign
(−1)l1 +l2 +k1 +k2 ···kh . In the case h = n, we define the algebraic minor
to be 1. The relation (301) can be proved by resorting to induction of f
but we leave it here for fear of the length of such a proof. Certainly a
simpler proof of (301) will be desirable.
Assuming (301) we proceed to establish (296). It is immediate from
(301) that
sh (Z, Z̄,
X
∂
)|Z|−α = (−1)h εh (α)|Z|d−1
Dl1 l2 ...lh
∂z
1≤i <l ≤n
1
h
where
Dli l2 ···lh =
X
1≤k1 <k2 <···kh
l1 , l2 . . . lh
k1 k2 · · · kh
!
Z−Z̄
l1 , l2 . . . lh
k1 k2 · · · kh
!
By a standard development of a determinant (Laplace’s decomposition theorem) it is clear that Dl1 ,l2 ···l2 is the determinant of the n- rowed
matrix which arises from Z on replacing its rows with the indices
l1 , l2 , . . . l2 by the corresponding rows of Z − Z̄. We split up this resulting
matrix into a sum of matrices each row of which is upto a constant sign 229
a row of either Z or Z̄, and the summands which result are 2h is number.
Let ∆g1 g2 ...gr denote the determinant of the matrix which arises from
Z on replacing its rows with the indices g′′ g2 , . . . gn by the corresponding
rows of Z̄. For a given ∆g1 g2 ...gn to appear in the above decomposition of
Di1 ,i2 ,...ih it is necessary and sufficient that the set ( f, r2 . . . gr ) is a subset
212
of the set of indices (l1 l2 . . . l0 ) so that a fixed determinant ∆g1 g2 ...gr oc-!
n−r
curs in the above decomposition of Dl1 l2 ...lh for a fixed h exactly
h−r
times, and the sign which it takes is (−i)r . Consequently we have
X
Di1 ,i2 ,...uh =
1≤i1 <···<lh ≤n
h X
X
!
n−n
∆
(−1)
h − r g1 g2 ,...g
r
r=0 1≤r1 <
!
h
X
r n−n
=
(−1)
A
h−r n
r=0
with
An =
X
1≤g1 <···<gn ≤n
∆y,g2 ϕr > 0 and Ao = |Z|
Clearly An = |Z̄|. Then
Mα |Z|−α =
n
X
εn (α)
∂
sh (Z − Z̄, )|Z|−α
ε (α)
∂z
h=0 h
!
n
h
X
X
h
h n−n
= εn (α)|Z|
(−1)
(−1)
A
h−r n
h=0
h=0
!!
n
n
X
X
−α−1
h−r n − n
= εr (a)|Z|
(−1)
An
h−r
−α−1
r=0
230
h=n
The sum within the parenthesis can be rewritten as
r−n
P
(−1)h
h=0
n−1
h ) and
this latter sum is equal to 0 or 1 according as n−r > 0 or n−r = 0
The above now reduces to
0.
Mα |Z|−α = εn (α)|Z|−α−1 An = εn (α)|Z|−α−l |Z̄|
as was desired. The deduction of the first half of (294) at this stage is
similar to the deduction from (274) of the corresponding result for (270).
We have still to introduce the operator Nβ . We introduce the operator
λ by the requirement
λf(z, z̄) = f(−z̄ − z)
(302)
213
and set Nβ = λMβ λ. It is immediate that for n = 1 we have Nβ = −∧β
while we had Mα = Kα in this case. It is easily seen that Nβ satisfies the
second half of (294) in view of Mα satisfying the first half. In general it
can be conjectured that
(Nβ−1 Mα − εn (α)εn (β − 1)){G, α, β, ν} = 0
(Mα−1 Nβ − εn (β)εn (α − 1)){G, α, β, ν} = 0
(303)
Mα {G, α, β, ν} ⊂ {G, α + 1, β − 1, ν}
Nβ {G, α, β, ν} ⊂ {G, α + 1, β − 1, ν}
This has been proved however only in the case n = 1 and 2 and the 231
proof makes use of certain operator identities.
We turn new to a different problem, viz. that of finding a Fourier development for modular function defined in (292). Net much progress has
so has been recorded in this direction. We assume that! the given group
E O
G contains symplectic substitutions of the form
(S Symmetric),
0 E
P
say, as is the case with M. If f(z, z̄) ∈ {G, α, β, ν} and f(z, z̄) = α(y, <
T
πT ) ∈2πiσ(T χ) , the summation for T being over all semi-integral matrices, the hypothesis, viz. Ωα,β f = 0 implies that Ωαβ annihilates each
term of the above sum.
Thus Ωαβ 0αz2πT eπiσ(T χ) which on replacing 2πT by T gives
Ωαβ eiσ(T χ) a(y, T )T 0. This represents a system of differential equationshere we can consider T to be an arbitrary real symmetric matrix-for the
functions a(y, T ), and we have to determine all the solutions of this system which are regular in y > 0. Only in the case n = 2 some real
progress has been achieved and the main result in this case is that linear space {αβT } of the solutions is finite dimensional for T such that
|T | , 0. Before we give a detailed account of the results in this case
it will be useful to make the following remarks on a special parametric
representation of the matrices y(z) > 0
Every 2 X 2 positive matrix y has a parametric representation of the
form
!
p (x2 + y2 )y−1 xy−1
(304)
Y = |y|
xy−1 g1
214
232
233
and the requirement y > 0 is equivalent with y > 0. Let Z = X + iy.Or
the surface |y| = L, the correspondence y → y is y − 1. This surface
can be considered as a Riemannian surface, as in y > 0 we have the
fundamental metric form ds2 = σ(y−1 τy)2 . Relative to this metric, the
mappings y → UyU, U unimodular, are movements (and the quadratic
form ds2 is invariant relative to these movements). The space |y| p
= 1 is
nothing else than the hyperbolic plane in this metric. For if ω = |y| it
can be shown that
!
dω2 dx2 + dy2
2
ds = Z 2 +
(305)
ω
y2
On |y| = 1 we have dω = 0 and then in (305) one has clearly the
fundamental metric form for the hyperbolic plane. If U is a proper
unimodular matrix (ϕ|U| = 1) then y → UyU ′ and Z → U < Z > are
representations of the same hyperbolic movement. We shall have more
to say one the determinantal surface |y| = 1 later.
For the special case n = 2 we are considering, we state the following
specific results. Introduce x, y by (304) and set U = (τ(y0))2 − 4|yτ |
Case i T = 0.
In this case the solutions a(y, 0) can be written as
3
1
a(y, 0) = ϕ(x, y)|y| 2 (1α+β) + C1 |y| 2 −d−β + 02
(306)
where ϕ(x, y) represents an arbitrary regular solution in y > 0 of the
wave equation
y2 (ϕχχ + ϕyy ) − (α + β − 1)(α + β − 2)ϕ = 0
and C1 , C2 are arbitrary constants. This is the general solution for α+β ,
3
3
, 2, 1. The singular cases α + β = , 2, 1 are those at which at least two
2
2
1
3
of the three exponents (1 − α − β), − α − β, 0 occurring in (306) are
2
2
equal and require some modifications. We exclude these cases here.
Case ii : T ≥ 0, rank τ = T .
Here we shall have
3
a(y, T ) = ϕ(u)i|y| 2 −α−β + ψ(u)
(307)
215
where ϕ(u) and ψ(u), denote confluent hypergeometric functions satisfying the differential equations
′′
Uϕ + (3 − α − β)ϕ′ + (α − β − U)ϕ = 0
′′
Uψ + (α + β)ψ′ + (α − β − u)ψ = 0
We have two independent solutions each for Uϑ and the dimension
of {α, βT } is consequently equal to 4 in this case. Here again (307)
3
is the general solution for α + β , and we exclude the exceptional 234
2
case from our considerations. We remark that the condition T ≥ 0 is
irrelevant here. We can also take T ≥ 0 and in general,
{α, β, T } = {β, α, −T }.
Case iii T > 0
This is the first, general case we are dealing with in so far as the
solutions a(y, T ) here depend on functions of more than one variable
(unlike the ϕ and ψ of the earlier case). We have in this case
a(y, T ) =
∞
X
gx (u)ϑν , (|ν| < u2 )
(308)
ν−0
the functions gν (U) are defined recursively by
′′
′
4(ν + 1)2 µgν+1 + Ugν + 2(2ν + α + β)gν + (2α − 2β − u)gν = 1U ≥ 0
and
1
go (v) = u1−α−β ψ(U), ψ(U) = ϕ(u)
u
′′
2(β − α) (α + β + 1)(α + β − 2) ϕ.
+
ϕ = 1+
U
U2
Every possible function gν (U) leads to a series (308) converging in
the whole domain y > 0 so that the dimension of {α, β, T } is easily seen
to be 3. As in the earlier case we can also consider T with −T > 0
Case iv : rank T = 2, T indefinite
216
This is the last case and we have here
a(y, T ) =
∞
X
hν (α)uν , (u2 < ν)
(309)
ν=0
The functions hν (ϑ) are recursively defined by
235
′′
(ν + 2)(u + 1)hν+2 + 4νhν + 4(α + β + ν)h1ν − ℓν = 0, ν ≥ 0
and
′′
′′
′
8ν2 ho + 4(α + 3β + 3o )ϑpo (4(α + β)3 + 2(α + β − 1) − 2ν)ho +
′
−(α + β)ho = (α, β)hi
8νhi + (α − β)r j = (β − α)ho
Every permissible ho , h1 leads to a series (309) which converges in
y > 0 and the dimension of {α, β, T } is four.
It remains to study these functions a(y, T ). These are generalisations
of the confluent hypergeometric functions of 2 variables. We can also
obtain them in the following way. We develop the Eisenstein series.
P
|CZ + D|−α |C Z̄ + D|−β as a power series,
X
X
|CZ + D|−α |C Z̄ + D|−β =
a(y, 2πT ) ∈kπiσ(T χ) m
T
236
If we compute the coefficients by means of the Poisson summation
formula, we obtain representations of a(y, 2πT ) by integrals. It is desirable to give a characterisation of these special functions a(y2πT ) ∈
{α, β2πT } and this has to be done by studying the behaviour of these
functions in neighbourhood of the boundary of the domain y > 0.
For a detailed account of some of the topics treated in this section
we refer to :H. Maass, Die Differentialgleichungen in der Theorie der Siegelachen
Modul functionen, Math. Ann. 126 (1553), 44-68.
Chapter 16
Closed differential forms
Our aim in this section is to generalise to the case of non-analytic forms 237
some of the concepts associated with the analytic forms of Grenzkreis
group of the first kind considered by Petersson, and study their applications to the space of symplectic geometry. Let G denote a Grenzkreis
group of the first kind. With respect to a multiplicator system ϑ, let
< G, K, ϑ > denote the space of all analytic forms of G of weight Kanalytic in the usual sense, viz. that the only singularities if any, are
poles in the local uniformising variables of the Riemann surface f defined by G (We can assume that f is a closed Riemann surface). In
conjunction with < G, K, ϑ > we have got to consider also the space
< G, 2 − K, 1/ϑ >, a sort of adjoint to the first. For r ≥ 2 the weight
of g(z) ∈< G, 2 − K, 1/ϑ > is zero or negative so that its dimension (
= - weight) is positive or zero. Since it is known that an automorphic
form of zero or positive dimension which is everywhere regular vanishes identically, it follows that in this case g(z) is uniquely determined
by its principal parts if all the principal parts are zero g(z) itself is identically zero. The question now arises whether there always exists an
automorphic form of a given weight 2 − K with arbitrarily given principal parts. While this is true for K = 2 in the case of the Gaussian spherea Riemann surface of genus zero-this is not unreservedly true either if
K > 2 or if the genus is strictly positive. Nevertheless it is possible to
give a necessary and sufficient condition for the given principal parts to
217
16. Closed differential forms
218
238
satisfy, in order that they should be all the principal parts of an automorphic form of weight K of G. We state this condition presently. For
f (z) ∈ hG, K, µ and g(z) ∈ hG, 2 − K, 1/µi introduce
ω(f, g) = f(z)y(z)dz
(310)
which is a closed meromorphic differential form on the Riemann surface
defined by G. That ω(f, g) is invariant relative to G is immediate if we
observe that the product f(z)g(z) is an automorphic form of weight 2 (the
sum of the weights of f and g) while dz behaves as an automorphic form
of weight −2, and the multiplicator systems for f, g are ν and 1/ϑ respectively. If f denotes a fundamental domain of G in Y1 , by integrating
ω(f, g) over the boundary of f it is easily seen that
X
Residue ω(f, g) = 0
(311)
Z∈f
239
On the left side of (311) we face only a finite sum as we assume
that the Riemann surface is closed and there is only a finite number
of singularities in the fundamental domain f. For K > 2, as f(z) runs
over a basis of everywhere regular form in hG, K, µ the left side of
(311) depends only on the principal parts of g(z) ∈ hG, 2 − K. 1/µh and
the conditions (311) themselves are called the principal part conditions.
These are necessary and sufficient for the existence of a form g(z) with
assigned principal parts.
We now wish to generalise the notion of ω(f, g) to the case of the
linear spaces {G, α, β, µ} of non analytic automorphic forms of degree
n, and in particular we have to settle when two such spaces can be considered adjoint like the spaces hG, K, µi and hG, 2 − K, 1/µi. The two
important properties of ω(f, g)h are
(i) it is completely invariant relative to G
(ii) it is closed.
The last condition is trivial in the earlier case of analytic forms and
is not so in the present case.
219
First we have to introduce the concept of the dual differential forms
in the ring of exterior differential forms. Consider a Riemannian manifold R in which local complex coordinates Y ′ = (Z1 , Z2 , . . . , Zn ), z̄ =
(z̄, z̄z , . . . , z̄n ) are introduced such that a neighbourhood of each point is
mapped pseudo conformally onto a neighbourhood of the origin of the
corresponding coordinate space. We consider z and z̄ as formally independent complex variables, and the above means that if
′
′
z∗ = (z∗1 z∗2 , . . . , z∗n ), z̄∗ = (z¯∗1 , z¯∗2 , . . . , z¯∗n )
is any other system of complex coordinates at the given point then we
have
z∗ = f(z), z¯∗ = f̄(z̄)
(312)
with the elements of the columns f(z), f̄z̄ representing regular functions
of (z1 , z2 , . . . , zn ), (z¯1 , z¯2 , . . . , z¯n ) respectively, which vanish at zz̄ = 0.
In the local co-ordinate system the metric fundamental form ds2 is a
Hermitian form and let us assume
ds2 = dz′ Gdz̄
(313)
with a Hermitian metric G.
In the ring of exterior differential forms let us introduce
240
[dz] = dz1 dz2 . . . dzn ,
[dz̄] = dz̄1 dz̄2 . . . dz̄n ,
ων = (−1)ν−1 dz1 . . . dzν−1 dzν+1 . . . dzn
ω̄ν = (−1)ν−1 dz̄1 . . . dz̄ν−1 αz̄ν+1 . . . dz̄n
(314)
[dz] = dz2 ω2 ; [dz̄] = dz̄, ω̄ν
(315)
Then
Denote with W , W¯ the columns with the elements ων , ω̄ν respectively, i,e.
W ′ = (ω1 , ω2 , . . . ωn );
W¯ ′ = (ω̄1 , ω̄2 , . . . ω̄n )
(316)
220
Consider a differential form θ fo degree 1. It can be represented as
θ = Y ′ dz + U ′ dz̄
(317)
where Y , U are columns with functions of zz̄ as elements. We also
introduce the dual form
θ̃ = |G|(Y ′Ḡ−1 W [dz] + U ′G−1 W [dz̄])
(318)
We shall show that θ̃ is uniquely defined by θ or what is the same, it
does not depend on the local coordinate system. We have only to prove
that θ̃ is invariant relative to the pseudo conformal mappings of the kind
(313), in the z∗ , z̄∗ system θ have the representation
′
′
θ = Y ∗ dz∗ + U ∗ dz̄∗
241
From (313) we have, with a certain non singular matrix T ,
dz∗ = T dz, T dz̄∗ = T̄ dz̄
and then
Y = T ′ Y ∗ and U = T̄ ′ U ∗
A simple computation shows that
T ′ W ∗ = |T |W , T̄ ′ W¯ ∗ = |T̄ |W¯ ;
[T ′ z∗ ] = |T |[dz], [dz̄∗ ] = |T̄ |[dz̄]
where the elements with a star denote the corresponding elements in the
new coordinate system z∗ , z̄∗ . Besides, we also have
′
ds2 = dz′ Gdz̄ = dz∗ G∗ dz̄∗
and it is easy to deduce that
T ′G∗ T̄ = G
The invariance of (318) relative to the transformations (313) is now
immediate. If
∂
∂ ∂
∂
∂
∂ ∂
∂
,(
,
,··· ,
), , (
,
,··· ,
)
∂z ∂z1 ∂z2
∂zn ∂z̄ ∂z̄1 ∂z̄2
∂z̄n
(319)
221
we have
∂
∂
|G|Y ′Ḡ−1 W¯ [dz] + dz′ |G|U G−1 W [dz̄]
∂z̄
∂z
∂′
∂
=
|G|G−1 Y [dz̄][dz] + |G|G−1 U [dz][dz̄]
∂z̄
∂z
′
∂′
∂
= {(−U
|G|G−1 )Y +
|G|G−1 U }[dz][dz̄]
∂z̄
∂z
dθ̃ = dz̄′
(320)
We therefore infer that the condition for the form θ̃ to be closed is 242
that
(−1)n
∂′
∂′
|G|G−1 Y +
|G|G−1 U = 0
∂z̄
∂z
(321)
We now compute the invariant volume element in the metric (312).
Let zν = r2 + iyν , z̄ν = rν + iyν (ν = 1.2 · · · n) and z = ε + iY , z̄ = E + iY .
Then dzν dz̄ν = −zidxν dy so that [dz][dz̄] and [dE][dY ] differ only by
a constant factor. Using the transformation formulae for these differential forms for a change of coordinate systems and the transformation
properties of |G| it is easily seen that |G|[dz][dz̄] is invariant for the coordinate transformations. This is therefore true of |G|[dE][dY ] and this
is precisely then the invariant volume element, viz.
dU = |G|[dE][dY ]
(322)
We wish to interpret these results in the space Y of symplectic geometry. The fundamental metric form here is
ds2 = σ(dzy−1 dz̄y−1 ), z = z + iy, z̄ = x − iy
If we set |y|y−1 = (yµν ), then yµν is just the algebraic minor of Yµν in
y = (†µν ) and we have
X
dS 2 = |y|−2
dzµν yν̺ dz̺̄σ yσµ
µ,ν
̺,σ
= |y|−2
X 1
dzµν yν̺ dz̺̄σ yσµ
e
µ≤ν µν
̺,σ
222
= |y|−2
=
X
µ≤ν
̺≤σ
X
µ≤ν,̺≤σ
1
(yµσ yν̺ + yνσ y̺σ )dzµν dz̺̄σ
2eµν e̺σ
Gµν,̺σ dzµν dz̺̄σ
(323)
1
|y|−2 (yµσ yν̺ + yνσ y̺σ )
2eµν e̺σ
(324)
with
Gµν,̺σ =
If G = (Gµν,̺σ ) and G−1 = (Gµν,̺σ ), we find from the relation
= E that
1
Gµν,̺σ = (Yµ̺ Yνσ + Yµσ Yν̺ )
(325)
2
GG−1
243
The transformations we have to consider are the symplectic substitutions
Z ∗ = (AZ + B)(CZ + D)−1 , Z̄ ∗ = (AZ̄ + B)(C Z̄ + D)−1
and these are pseudo conformal mappings of Y in the sense of (313).
Let us introduce as in the earlier case,
Y
Y
[dz] =
dzµν , [dz̄] =
dz̄µν
µ≤ν
µ≤ν
with the lexicographical order of the factors and
Y
dz̺σ = ωνµ
ωµν = ±
̺≤σ
(̺,σ),(µ,ν)
ω̄µν = ±
244
Y
for µ ≤ ν,
dz̺̄σ = ω̄νµ
̺≤σ
(̺,σ),(µ,ν)
the ambiguous sign being fixed in each by stipulating that
[dz] = dzµν ωµν ; [dz̄] = dz̄µν ω̄µν
(326)
In the place of W , W¯ we introduce
Ω = (eµν ωµν , Ω̄ = (eµν ω̄µν .
(327)
223
As before we start with a differential form of degree 1 given by
θ = σ(pdz) + σ(θdz̄)
(328)
where p, θ are n−rowed symmetric with functions as elements. We have
to compute the dual form θ̃.
In the earlier notation,
θ̃ = (Y′ Ḡ−1 W¯ [dz] + U ′ G−1 W [dz̄]).
((318)′ )
|G| can be fixed as follows. We know that the invariant volume element
in symplectic geometry is
dϑ = |y|−n−1 [dx][dy]
From (321) it is also given by
dϑ = |G|[dE][dY ].
A comparison of the two shows that |G| is equal to a constant multi- 245
ple fo |Y|−n−1 and this constant factor can be assumed to be 1 for the purposes of θ̃. Then |G| = |Y|−n−1 . We now compute the product Y ′ θ̃−1 Y¯ .
X 1
1
Pµν (Yµ̺ Yνσ + Yµ̺ Yν̺ )ω̺̄σ
Y′ Ḡ−1 W¯ =
e
2
µ≤ν µν
̺≤σ
=
1
Pµν (Yµ̺ Yνσ + Yµ̺ Yν̺ )ω̺̄σ
2
µ≤ν
X
̺≤σ
=
X
µ,ν;̺,σ
eµν Pµν Yµ̺ Yνσ ω̺̄σ = σ(ypyΩ̄)
The other product U ′G−1 W similarly reduces to σ(yθyΩ̄) and from
(318) we have
θ̃ = |Y|−n−1 (σ(ypyΩ̄)[dz] + σ(yθyΩ)[dz̄]).
(329)
Then form (320) we will have for the differential of θ̃,
dθ̃ = {(−1)
n(n+1)
2
σ(
∂
∂
ypy|Y|−n−1 ) + σ( yθy|Y|−n−1 )}[dz][dz̄].
∂z
∂z
(330)
224
We now specialise θ by appropriate choices of P and Q and investigate in detail the cases in which dθ̃ vanishes.
Let {α, β} denote the linear space of all regular functions f(z, z̄) in Y
which satisfy the differential equation (272). Let α, β; α′ , β′ be complex
numbers, arbitrary for the present, and choose
f = f(z, z̄) ∈ {α, β}, G = G(z, z̄) ∈ {α′ , β′ }
(331)
p = E|Y|y y−1 GKα f, Q = |Y|y y−1 F Λβ G
(332)
246
We then set
where we assume
E2 = 1, y = α + α′ = β + β′
(333)
and the differential operators Kα and Λβ are defined by (276). With this
P and Q, (328) defines θ, and we set
ω(f, g) = θ
(334)
We shall study the behaviour of this differential form relative to the
symplectic substitutions
z∗ = (az + B)(cz + B)−1 , z̄∗ = (az̄ + B)(cz̄ + B)−1 .
The replacement of z, z̄ by z∗ , z̄∗ in any operator or function shall in
general be denoted by putting a star (∗) over it and in particular
f∗ = f(z∗ , z̄∗ ), G∗ = G(z∗ , z̄∗ )
!
A B
If M =
is any symplectic metric, we also introduce
C D
f(z, z̄)|M = |cz + D|−α |Cz̄ + D|−β f(z∗ , z̄∗ )
′
′
G(z, z̄)|M = |cz + D|−α |Cz̄ + D|−β G(z∗ , z̄∗ )
225
and for any differential form µ(z, z̄),
µ(z, z̄)|M = µ(z,∗ z̄∗ )
(335)
247
By means of (286), (122)′ and (284) we have
(z∗ − z̄∗ )
∂
∂
f = (z̄C ′ + D′ )−1 (z, z̄)((cz + D) )′
∗
∂z
∂z
α
|ez + D ||cz̄ + D|β (f|M)
= |ez + Dα ||cz̄ + D|β (z̄C ′ + D′ )−1 (z, z̄)((cz + D)
∂
{αC ′ + ((cz + D) )′ }(f|M)
∂z
= |ez + Dα ||cz̄ + D|β (z̄C ′ + D′ )−1 [{α(zc′ + D′ )
∂
− α(z̄c′ + D′ )}(f|M) + (z − z̄ f|M)(zc′ + D′ )]
∂z
and
Kα∗ f∗ = |Cz + Dα ||cz̄ + D|β (z̄C ′ + D′ )−1 (Kα f|M)(zc′ + D′ )
Further,
dz∗ = (zc′ + D′ )−1 dz(zc + D)−1
Y∗−1 = (cz + D)Y −1 (z̄c′ + D′ )
|Y ∗ν | = |Y|y |cz + D|−y |Cz̄ + D|−y
and this yields
|Y ∗ |y y ∗−1 g∗ (Kα∗ f∗ )dz∗ = (cz + D)|Y|y Y −1 (g|M)(kα f|M)dz(cz + D).
In our notation (355) the left side is just
1
(|Y|y Y −1 G(Kα f)dz)|M = ( pdz)|M
E
Hence we deduce that
248
226
1
σ( pdz)|M = σ{(cz + D)|y|y y−1 (g|M)(K − αf|M)dz(cz + D)−1
E
= σ{|y|y y−1 (giM)(Kα f|M)dz}
One proves analogously that
σ(Qdz̄) = σf|Y|y y−1 {|M)(Λβ G|M|dz̄)}
and then it is immediate that
ω(f, G)|M = ω(f|M.G|M).
(336)
This is naturally true of the dual form ω̃(f, G) also so that
ω̃(f, G)|M = ω̃(f|M.G|M).
(337)
Let us now compute dω̃(f, G) form (330). In doing this we shall
need the following identity
((z − z̄)
∂
1
1
∂ ′
) = {((z − z̄) )′ W}y − σ(yw)E − w′ y
∂z̄
∂z̄
2
2
(338)
where W = (ωµν ) denote an arbitrary matrix its elements all functions.
In fact, since
∂Yν,µ
1
= − (∂̺τ ∂µν + ∂̺ν∂µτ )
e̺µ
∂z̺̄,µ
4i
we have
((z − z̄)
X
∂ X
∂ ′
) (wy) = ( (zν̺ − z̄ν̺ )e̺µ
( ωµτ Yτν )
∂z̄
∂z̺̄,µ τ
ϕ
X
∂ X
= ( (zν̺ − z̄σ̺ )e̺µ
( ωµτ Yτν )
∂z̺̄,µ τ
̺στ
1 X
∂
= [((z − z̄) )′ W]y − ( (z̺σ − z̺̄σ )ωστ (δ̺τ δµν + δµτ δ̺ν )
∂z̄
4i ̺στ
= [((z − z̄)
∂ ′
1
1
) W]y − σ((z − z̄)W)E W ′ (z, z̄)
∂z̄
4i
4i
227
249
which is just another form of (338).
Similarly one obtains the analogous identity, viz.
((z − z̄)
∂
1
1
∂ ′
) (Wy) = [(z − z̄) )′ Wy]y + σ(yW)E + W ′ Y
∂z
∂z
2
2
(339)
Finally one also has

∂


f)y 



∂z



∂


= β|y|µ−n−1 yfG + |y|µ−n−1 F (z − z̄)( G)y
∂z
EyP|y|−n−1 = α|y|µ−n−1 yfG + |y|µ−n−1 G(z − z̄)(
yθ|y|−n−1
(340)
Using (280) and (338-340) we can now state that
∂
ε yP|y|−n−1 =
∂z̄
y−n−1
∂
y f g+
|y|y−n−1 y−1 + |y|y−n−1
=α −
2ℓ
∂z̄
y−n−1
∂
∂
+ −
g.(z − z̄)( f )y
|y|y−n−1 y−1 + |y|y−n−1
2ℓ
∂z̄
∂z
∂
dℓ
= (y − n − 1)|y|( y − n − 1) f gE − (y − n − 1)|y|( y − n − 1)g( f )+
2
∂z
dℓ (y−n−1) n + 1
∂ − |y|
f g+
−
E + (z − z̄)
2
2
∂z̄
∂
∂
+ |y|(y−n−1) ( g)(z − z̄)( f )y
∂z̄
∂z̄
n+1
∂ ′ ∂
( f )y
E + (z − z̄)
+ g|y|(y−n−1) −
2
∂z̄
∂z
di
n + 1 y−n−1
n yn−1 ∂
= (y −
)|Y|
fGE − (y − )|y|
G( f)+
2
2
2
∂z
di
n + 1 y−n−1
n yn−1 ∂
= (y −
)|Y|
fGE − (y − )|y|
G( f)+
2
2
2
∂z
∂
∂
α|y|yn−1 G( f)y + α|Y|y−n−1 f( G)Y+
̺z
∂z
∂
∂
2i|y|y−n−1 ( G)y( G )Y
∂z
∂z
∂ ∂
1
∂
+ G|y|y−n−1 [{((Z − Z̄, )′ f}y − σ(y f)E]
∂z̄ ∂z
2
∂z
228
=
n + 1 y−n−1
n
∂
αi
(y −
)|y|
fGE − (y − − β)|y|y−n−1 G( f)y+
2
2
2
∂z
∂
y−n−1 ∂
yn−1 ∂
+ α|y|
f( G)y + 2i|y|
( )y( f)y+
∂z
∂z̄ ∂z
∂
1 y−n−1
Gσ(y f)E
− |y|
2
∂z
(341)
250
In stating the last relation we have used the fact that
((z − z̄)
∂ ∂
∂
∂
f) f = −α f + β f
∂z̄ ∂z
∂z̄
∂z
which is easily proved.
We have similar lines the dual formula also, viz.
∂
yQ|y|y−n−1
∂z
n
∂
n + 1 y−n−1
)|y|
fGE + (y − − y)|y|y−n−1 f( G)y+)
2
2
∂z̄
∂
∂
∂
−β′ G|y|y−n−1 (( f)y + 2i|y|y−n−1 (( f)y(( )y
∂z
∂z
∂z̄
1 y−n−1
∂
+ |y|
fσ(y( G)E
(342)
2
∂z̄
= (β′ (y −
251
In view of our assumption (333), setting E = −(−1)
obtain that
dω̃(f, G) =
n(n + 1)
we now
2
in ′
n + 1 y−n−1
(β − α)(y −
)|y|
fG[dz][dZ̄]
2
2
(343)
In particular we shall have
dω̃(fG ) = 0
(344)
in the two cases
y=
n+1 ′ n+1
n+1
α =
− α, β′ =
−β
2
2
2
(345)
and
y = α + β, α′ = β, β′ = α
(346)
229
We shall look into these cases a little more closely.
Let f ∈ {G, α, β, µ}, G ∈ {G, α′ , β′ , µ′ } where we can assume u′ =
1/ν. From (336) and (337) it is seen that the differential forms ω(f, g)
and ω̃(f, g) are invariant relative to the substitutions of G. With regard
to the case (346) we can now make the following remark, viz. if α, β
are real and |µ| = 1, the linear spaces {G, α, β, µ} and {G; β, α, 1/µ} are
mapped onto each other by the involution
f(z, Z̄) = G(z, z̄) = f̄(z̄, z).
Indeed, if f(z, z̄) ∈ {G, α, β, µ} then Ωαβ f(z, z̄) = 0 so that, taking!comA C
plex conjugates, ΩβαG(z, z̄) = 0 On the other hand, if M =
∈ G,
B D
then
f(Mhz̄i)|cz + D|−α |cz̄ + D|−β = ϑ(M)f(zz̄),
f̄(Mhz̄i, Mhz̄i)|Cz̄ + D|−α |Cz̄ + D|−β = µ̄(M)f̄(z̄, z).
252
1
µ (M)G(z, z̄) and our assertion
This means that G(z̄, z)|M =
We can now state if α, β are real and |µ| = 1 then


ω̃( f (z, z̄), Ḡ(z̄, z))|M = ω̃( f (z, z̄), Ḡ(z̄), z)



dω̃( f (z, z̄), Ḡ(z̄, z))
is proved.
(347)
for all f (z, z̄), G(z, z̄) ∈ {G, dβ, µ}.
Applications of (347) have so far been made only in the case when
n = 1. The next case of interest will be when n = 2. In this case, an
explicit expression for ω̃(f, G) can be given as follows. Let Lq denote
the intersection of the fundamental domain f of the modular group of
degree n with the domain defined by |y| ≤ q. In either of the cases (345),
(346) whence dω̃ = 0, we will have
Z
ω(f, G) = 0
Kq
230
where Rq denotes the boundary of Qq . In view of the invariance properties of ω(f, G) set out earlier, this reduces to
Z
ω(f, G) = 0
(348)
fq
253
where fq denotes that part of Rq which lies on |y| = q. While this is all
true for any n, in the case of n = 2, we have the parametric representation
(304) for matrices y > 0, viz.
#
" 2
p
(x + y2 )y−1 xy−1
, u > 0, y > 0; u = |y|
y=u
−1
−1
xy y
One 7q of course we shall have du = 0 and then it can be shown that
∂( f g)
∂( f g)
∂( f g)
ω̃( f, g) = 4u2ν−2 i(x2 + y2 )y−1
+ iy−1
+ iy−1
∂x1
∂x12
∂x22
∂g
∂f
−f
+ z(α − β)u−1 f g dxu dx12 y−2 dxdy (349)
+g
∂u
∂u
The explicit computation of the integral on the left side of (348)
appears to be possible only by a detailed knowledge if the Fourier expansions of the forms f and g.
The following reference pertain to the subject matter of this section:1. H. Mass Über eine neue Art von nichtanalytischen automorphen
Funktionen and die Bestimming Dirichlet scher Reihen durch
Funktionalgleichungen, Math. Ann. 121 (1949), 141 - 193.
254
2. H. Petersson , Zur analytischen Theorie der Grenzkeris gruppen,
Teil I bis TeilV, Math. Ann 115 (1938), 23 - 67, 175-204, 518572, 670-709, Math. Zeit 44(1939), 127-155.
3. H. Petersson Konstruktion der Modulformen and der zu gewissen
Grenzkreisgruppen gehorigen automorphen Formen von positiver
reeller Dimension and die vollstandinge Bestimmning ihrer
Fourie - koeffizienten, Sitz. Ber Akad. Wiss. Heidelberg 1950,
Nr. 8
Chapter 17
Differential equations
concerning angular
characters of positive
quadratic forms
The notion of angular characters was first introduced by Hecke in the 255
study of algebraic number fields. These are functions u defined on the
non zero elements of a give algebraic number field K such that
1. u(α) = u(rαε) for rational numbers r , 0 and units ∈ of K
2. For a given α ∈ K, α , 0, the set of values u(α) (for the different
u′ s) and the norm Nα of α determine the principal ideal (α) of α
It is possible to realise the angular characters u (α) with the variable
α as solution of a certain eigen value problem and this enables us to
carry out certain explicit analytic computations with u(α). Analogous
considerations can be developed for positive quadratic forms. Here we
ask for functions u(y) defined on the space of positive matrices y = y(n)
which are invariant relative to the transformations y → ry[u] where r is
a positive real number and u, an unimodular matrix, such that the determinant |y| and the set of all values u(y) for a given y determine uniquely
231
17. Differential equations concerning ....
232
256
the class of matrices y[u] which are equivalent with y. It can be expected
that suitable functions u(y) appear again as solutions of an eigen value
problem. The use of such angular characters will be particularly felt
in determining a set of Dirichlet series equivalent with a give modular
form of degree n in analogy with a theory of Hecke. A satisfactory treatment of the theory of angular characters for positive quadratic forms has
so far been possible only in the case n = 2. We now make our above
statements precise.
Consider a space of n real coordinates yν > 0, ν = 1.2 . . . n, and the
linear differential operators Ω in this space,
Ω=
X
C ν 1 ν 1 . . . νn
∂νn
∂ν1
ν2 . . .
∂y2
∂yνnn
(350)
We shall denote by ψ the set of all such operators Ω which are invariant relative to the group of mappings
Yν → Yν∗ = aν yν , ν = 1, 2 . . . n
where there a′ν S are arbitrary positive real numbers. Given any algebraic number field K− for simplicity we assume K to be totally real we can associate with K a space of n real coordinates as follows. Let
the dimension of K over the field of rationals be n, and for α ∈ K, let
α(1) , α(2) , . . . α(n) denote the conjugates of α. We then set Yν = |α(ν) |, ν =
1, 2 . . . n, |α(ν) | denoting the absolute value of α(ν) .
The angular characters of K are functions u(α) = u(α(1) , . . . α(ν) ) of
n real variables which depend only on the absolute values
|α(1) |, |α(2) |, . . . |α(n) | of the variables, and which satisfy the following
conditions:1. u is an eigen function of every differential operator Ω ∈ P
2. u is invariant relative to the mappings Yν → rYν where r is an
arbitrary positive real number.
3. u is invariant relative to the unit group of K
257
We shall look into these defining properties of u a little more closely.
233
∂
, ν = 1, 2 . . . n lie in ψ, and it is easily
∂yν
seen that these operators actually generate ψ, meaning that any operator
in ψ is a polynomial of the type
Clearly all the operators Yν
σν1 ν2 ..νn aν1 ν2 ...νn (Y1
∂ ν1
∂ νn
) . . . (Yn
)
∂y1
∂yn
with constant coefficient aν1 ν2 ...νn . Then the first condition on u re∂
duces to the requirement that u is an eigen function of each Yν
,ν =
∂yν
1, 2 . . . n, in other words that
(Yν
∂
+ λν )u = 0, ν = 1, 2 . . . n
∂yν
with appropriate λ′ν s. Hence we conclude that
u=C
n
Y
Yν−λν
(351)
ν=1
The second condition on u requires that u is homogeneous of degree
0 in Y1 , Y2 , . . . Yn and then
n
X
yν
ν=1
∂
u=0
∂Yν
(352)
P
form which it follows that nν=1 λν = 0
We now come to the last condition on u. It is well known that any
unit of K is a power product of n − 1 fundamental units E1 , E2 . . . En−1
multiplied by a certain root of unity. Hence the last condition only
amounts to requiring that u is invariant under the mappings
Yν → Yν |Eµ(ν) |, µ = 1, 2 . . . n − 1. Then
n
Y
ν=s
(yν |Eµ(ν) |)λν =
n
Y
ν=1
n
Y
ν
so that
y−λ
ν
ν=s
|EK(ν) |−λν = 1, µ = 1, 2, . . . n − 1
234
Together with (352) this can be written as
n
X
258
λν = 0
ν=1
n
X
ν=1
λν log |ε(n)
µ | = 2Kµ πi iµ = 1, 2, . . . n − 1
For a given set of integers Kµ these n equations define uniquely the
n eigen values λ1 , λ2 . . . λn and hence also u by (351). Varying systems
of K′ s determine the different u′ s.
We now generalise these considerations for positive quadratic forms
or what is the same for positive matrices. Here ψ will stand for the set of
all linear differential operators Ω on the space of positive matrices Y (n)
which are invariant relative to the group of mappings T → Y[R], R being
an arbitrary non singular matrix. As in the earlier case we can show that
∂
the operators σ(Y )K ∈ ψ, K = 1, 2 . . . n and they generate ψ where as
∂Y
∂
∂
usual,
= (eµ,ν
). Let Y ∗ = Y[R], R non singular. For an arbitrary
∂y
∂yµ,ν
function f (y) we have for the total differential,
d f = σ(dy
∂
∂
f ) = σ(dy∗ ∗ f )
∂y
∂y
Also dy∗ = (dy)[R] so that
σ(dy
∂
∂
∂
f ) = d f = σ(dy∗ ∗ f ) = σ(dyR ∗ f R′ )
∂y
∂y
∂y
and consequently,
∂
∂
∂
∂
= R ∗ R′ ∗ = [R−1 ]
∂y
∂y
∂y
∂y
Then
(Y ∗
259
and
∂ ∗ ′ ∂ R −1
) R (Y ) R , K = 1, 2, . . . n,
∂y∗
∂y
(353)
235
∂
∂ K
) = σ(Y )K
∗
∂y
∂y
σ(Y ∗
(354)
∂ K
) ∈ ψ, K = 1, 2 . . . n. Consider now any
∂y
operator Ω ∈ ψ. As a linear operator it is of the form
This proves that σ(Y
Ω = Ω(y,
n
Y
X
∂
∂µν
CR1 (y)
)=
sµν ̺µν≥0 integral,
∂y
∂yµν
R
µ,ν=1
1
and R1 = (̺µν ) = R′1 .
Since Ω is invariant relative to the mapping Y → y∗ = y[R], R nonsingular, we have
Ω(Y[R],
∂
∂ −1
[R ]) = Ω(y, ), |R| , 0
∂y
∂y
(355)
Let Y0 by an arbitrary point an let V be such that Y0 [V] = E. Let u
be an arbitrary orthogonal matrix (u′ u = E) and set R = Vu. Then
Y0 [R] = (Yo [V])[u] = E.
(356)
Let f (y) be an arbitrary function and define
g(y) = Ω(y
With Y ∗ = y[V] we have
∂
) f (y)
∂y
(357)
∂
∂
= [V −1 ] so that
∂y∗ ∂y
∂
∂
∂ −1
[R ] = ( [V −1 ])[u] = ∗ [u]
∂y
∂y
∂y
Then from (355), (356) and (357) we have
∂
) f (y)}y=yo
∂y
∂
= {Ω(E, ∗ [u]) f (y∗ [V −1 ])}y∗ =E
∂y
g(y0 ) = {Ω(y0 ,
(358)
260
236
Since this is true for all orthogonal matrices u we conclude that the
right side of (358) is independent of u, in other words,
Ω(E,
∂
∂
[u]) = Ω(E, ∗ ),
∂y∗
∂y
or equivalently,
Ω(E,
∂
∂
[u]) = Ω(E, )
∂y
∂y
(359)
∂
is isomor∂yµν
under the isomorphism
Since the ring generated by the differential operators
phic with the polynomial ring generated by Yµν
∂
Yµν ↔ eµν
, the truth of (359) for every orthogonal matrix u is equiv∂yµν
alent with the relation
ω(E, Y[u]) = ω[E, y]
for these u′ s. In particular, choosing u such that Y[u] = Λ = (δµν , ) we
shall have
Ω(E, Λ) = Ω(E, Y)
(360)
The left side is a symmetric function of the λ′ν s, ν = 1, 2 . . . n in fact
symmetric polynomial in the λ′ν s so that it is a polynomial of the ‘power
P
sums’ nν=1 λKν = σ(Λ)K ), K = 1, 2 . . .. Thus Ω(E, y) = Ω(E, Λ) =
P(G(Λ) . . . σ(Λ)n ), a polynomial with constant coefficients. Since u is
orthogonal and y[u] = Λ we have σ(Λ) = σ(y[u]) = σ(y), and then
Ω(E, Y) = P(σ(y), σ(y)2 , . . . σ(y)n ),
261
where P is a polynomial with constant coefficients.
Then form (358),
g(y0 ) = {Ω(E,
= {P(σ(
∂
}y∗ =E
∂y∗ ) f (y∗ [V −1 ]
∂
∂
∂
, σ( ∗ )2 , . . . σ( ∗ )n s(y∗ [V −1 ]}y∗ =E
∗
∂y )
∂y
∂y
(361)
237
∂
∂ K
) and σ( ∗ )K , and in
∗
∂y
∂y
particular their highest degree terms at the point y∗ = E, we can write
Comparing the two operators σ(y∗
∂
∂
), . . . σ(y∗ ∗ )n f (y∗ [V −1 ]}y∗ =E
∂y∗
∂y
∂
+{Ω∗ (y∗ ∗ ) f (y∗ [V −1 ])}y∗ −E
∂y
g(y0 ) = {P(σ(y∗
∂
∂
is smaller than that of Ω in ∗ . Due to
∗
∂y
∂y
∂ K
the invariance of the operators σ(Y ) for the mappings y → Y ∗ , the
∂y
above gives
where the degree of Ω∗ in
∂
∂
), . . . σ(y )n f (y[V]}y=y0
∂y
∂y
∂
+{Ω1 (y ) f (y)}y=y0
∂y
g(y0 ) = {P(σ(y
with Ω1 having the same degree in
arbitrary by choice,
g(Y) = P(σ(y
∂
∂
as Ω∗ in ∗ and since y0 is
∂y
∂y
∂
∂
∂
), . . . σ(y )n f (y) + Ω1 (y ) f (y).
∂y
∂y
∂y
∂
) f (y) and the above is true for every function 262
∂y
f (y) it is immediate that
Since g(y) = Ω(y,
Ω(y,
∂
∂
∂
∂
∂
) = P(σ(y ), σ(y )2 . . . σ(y )n )+ Ω1 (y )
∂y
∂y
∂y
∂y
∂y
∂
∂
is less than that Ω in
) and P is a
∂y
∂y
polynomial with constant coefficients. By resorting to induction on the
∂
∂
∂
degree of Ω in
we conclude that Ω( ) is a polynomial in σ(y )K ,
∂y
∂y
∂y
R = , . . . n, and this was what we set out to prove.
We now define the angular characters as functions u(y) which satisfy
the following requirements.
where the degree of Ω1 in
238
(1) u is an eigen function of each operator Ω ∈ ψ
(2) u is homogeneous of degree o in y, in other words, u(ry) = u(y) for
every real r > 0,
(3) u is invariant relative to the group of mappings y → y[u] where u is
an arbitrary unimodular matrix.
(4) u is square integrable over a fundamental domain of the group of
mappings y → y[u], u unimodular, in the determinant surface |y| = 1
with a certain invariant volume element dϑ1 .
Let us examine the consequences of these defining properties of u.
∂
Since we have shown that ψ is generated by the elements σ(y )K , K =
∂y
1, 2 . . . n, the first condition on u can be replaced by a system of differential equations
(σ(y
263
∂ K
) + ηK )u(y) = 0k = 1, 2 . . . n
∂y
(362)
The second condition is then nothing else but the requirement that
η1 = 0. The third condition means that u is a kind of automorphic
function, and it is a pertinent question as to the existence of non-trivial
solutions of (362) which satisfy the second and third condition above.
We shall subsequently show that such non trivial solutions u always exist. But these special functions u fail to be angular characters in that they
do not need the fourth requirement.
The equation in (362) corresponding to K = 2 is of particular inter∂
∂
est, as in this case σ(y )K = σ(y )2 can be proved to be the Laplace
∂y
∂y
Beltrami operator of the space y > 0, considered as a Riemannian space
relative to the metric
dr s = σ(y−1 dy)2
(363)
We prove this as follows. The Laplace - Beltrami operator is invariant relative to the transformations y → y[R], R non-singular which are
∂
movements of the space y > 0, and the operator σ(y )2 is also invari∂y
ant relative to these transformations as is seen from (354). In view of
239
the transitivity of the group of movements then, it suffices to show that
the two operators are equal at the special point γ = E. At this point we
choose the coordinate system
x = (−y + E)(y + E)−1
(364)
The substitution y → x is a 1 − 1 mapping of the space y > 0 onto
the space E − x′ x > 0, x′ = x, and a simple computation shows that
ds2 = 4σ((E − xx′ )−1 dx)2
(365)
264
It is then immediate that our coordinate system at E is a geodesic
one. Clearly the point y = E corresponds to x = 0. By means of the
relations
1
∂
∂
= − (E + x)((E + x) )′
∂y
2
∂x
n+1
∂
∂
x=
E + (x )′
∂x
2
∂x
∂
∂ ∂ ′ 1 ∂
1 ∂
∂
(x ) = (x
) +
+ σ( )E
∂x ∂x
∂x ∂x
2 ∂x 2 ∂x
which are easily proved, we obtain that at x = 0
∂
1 ∂ ∂
1 ∂
1 ∂
n+1 ∂ (y )2 =
,
+
+ σ E−
∂y
4 ∂x ∂x 2 ∂x 2 ∂x
2 ∂x
and then
σ(y
∂ 2 1 ∂ 2 X 2 ∂2
) = σ ) =
eµν 2
∂y
4 ∂x
∂xµν
µ,ν
=
1 X ∂2
1 X ∂2
+
.
4 ν ∂x2µν 8 µ<ν ∂x2µν
On the other hand, at x = 0, we have
X
X
ds2 = 4σ(dx)2 = 4
dx2νν + 8
dx2µν
ν
µ<ν
(366)
240
∂ 2
) is identical with the
∂y
Laplace - Beltrami operator at x = 0 and hence also at any other point.
We are also interested in the determinant surface |y| = 1 and we can
introduce the Laplace - Beltrami operator ∆1 in this surface. We wish
to determine the relationship between ∆ and ∆1 . We first observe that in
(363).
dr2 = dz log |y|
(367)
and we can then conclude from (290) that σ(y
265
For, −d2 log|y| = −d(d log |y|) = −d(
1
d|y|)
|y|
∂
1
, σ( |y|) = −d(σ(dyy−1 ))
|y|
∂y
= −d(σ(ydy−1 )) = σ((dy−1 )dy)
= d(
= −σ(−y−1 dyy−1dy ) = σ(y−1 dy)z
= ds2
The relationship between ∆ and ∆1 is now provided by the following
general lemma, viz.
Lemma 25. Let R be a domain in the space if n real coordinates
y1 , y2 , . . . yn , and x = x(y1 , y2 , . . . yn ), a positive homogeneous function on K of degree K > 0. Denote by M the surface defined by
x(y1 , y2 , . . . yn ) = 1. Let R, M be considered as Riemann space relative to the metric ds2 = d2 ϕ where ϕ = − log x, and let ∆, ∆1 denote the Laplace Beltromi operator in R and M respectively. If h1 =
h1 (y1 , y2 , . . . yn ) is an arbitrary function in M and
y2
yn
y1
h = h1 ( √ , √ , . . . √ )
n x K x
K x
266
the homogeneous function of degree 0 in R which extends h1 then we
have
∆1 h1 = ∆h
(368)
Proof. We chose in R a special coding system (x1 , x2 , . . . xn ) as follows.
Let yµ = ψµ (x1 , x2 , . . . xn−1 )(1 ≤ µ ≤ n) be a parametric representation
241
of the surface M so that in particular, rank (
equations
∂ϕµ
) = n − 1. Then the
∂xν
yµ = yµ (x1 , x2 , . . . xn−1 )xn , i ≤ µ ≤ n
(369)
define the desired coordinate system (x1 , x2 , . . . xn ). To prove our as∂ϕµ
sertion we need only show that the square matrix (
) has the rank n
∂xν
∂ϕµ
∂ϕµ
∂ϕµ
=
×n for ν < n and
= ϕµ, ϕ = 1, 2 . . . n, it suffices
Since
∂xν
∂xν
∂xν
to show that the homogeneous linear equations
n−1
X
ν=1
ξν
∂ϕµ
× n + ξn ϕµ = 01 ≤ µ ≤ n
∂xν
(370)
admit of only the trivial solution. Since x(ϕ1 , ϕ2 . . . ϕn ) = 1 identically
in x1 , x2 , . . . xn−1 , we have
n
X
∂ϕµ
∂X
(ϕ)
= 0, 1 ≤ ν < n
∂ϕν
∂xν
µ=1
(371)
Besides,
n
X
∂λ
(ϕ)ϕµ = Kλ(ϕ) > 0
∂ϕ
ν
µ=1
(372)
by a standard result on homogeneous functions. Multiplying both sides
of (371) by ξxn and of (372) by ξ and adding, we obtain in view of (370) 267
that ξKλ(ϕ) > o whence it follows that ξn = o Since we know that rank
∂ϕµ
µ=1,2,...n
(
) = n − 1,ν=1,2,...n−1, we conclude from (370) that ξo = o for all
∂xν
ν = 1, 2, . . . n. We have now shown that (369) gives a parametric representation of the whole space R in terms of that in |v|. Let us compute
fundamental metric form ds2 in this special coordinate system. We have
n
n
X
X
∂ϕ
∂2 ϕ dxµ
ds2 = d2 ϕ = d(
dxν ) =
∂ dxν
∂xν
∂xµ xν
ν=1
µ,ν=1
242
=
X
dxµ dxν
(373)
µ,ν
with gµν =
Also
∂z ϕ
(1 ≤ µ, ν ≤ n)
∂xµ ∂xν
n
∂ϕ
∂(− log χ
1 ∂(χ(y)
1 X ∂χ(y) ∂yµ
=
=−
=−
∂xν
∂xν
χ ∂xν
2 µ=1 ∂yµ ∂xν
n
1 X ∂χ(y) ∂ϕµ
=−
xn
χ µ=1 ∂yµ ∂xν
(374)
∂χ
is homoge∂yµ
neous of degree K− 1 in y1 , y2 , . . . yn Also, all the y′ s are linear functions
∂ϕ
is a homogeneous function
in χn and it then follows from (374) that
∂xν
∂ϕ
is independent
of degree o in xn for ν < n. In other words, for ν < n,
∂χν
of xn and consequently
gnν = gνn = o
(375)
Since χ is homogeneous of degree K in y1 , y2 , . . . yn ,
268
for these values of ν.
Also,
n
1 X ∂χ(y) ∂yµ
∂ϕ
=−
∂xn
χ µ=1 ∂yµ ∂xn
=−
−K
1 X ∂χ(y) yµ
=−
2 µ ∂yµ xn
xn
so that
gnn =
∂2 ϕ
K
= 2.
2
∂x
xn
(376)
Then from (373), ds2 is given by
ds2 =
n−1
X
µ,ν=1
gµν dxµ dxν +
K 2
dxn .
x2n
(377)
243
On N of course we have xn = 1 so that dxn = o, and then
ds2 =
n−1
X
gµν dxµ dxν
(378)
µ,ν=1
gives the metric form.
Let (gµν ) (µ, ν < n) denote the inverse of the matrix (gµν ) and let
∂ϕ
is independent of xn for
g denote the determinant of (gµν ). Since
∂xν
ν < n it follows that gµν (µ, ν < n) and consequently g are independent
of xn . Then by the definition of △, using (375) and (376), we have
n
X
x2 ∂
1
∂
∂ √
1
∂ √
△= √
ggnn n
( ggnn gµν
)√
ggn n µ,ν=1 ∂xµ
∂xν ggnn ∂xn
K ∂xn
n−1
1
∂
∂
1 X ∂ √ µν ∂
( gg
) + xn
χn
= √
g µ,ν=1 ∂xµ
∂xν
K ∂xn ∂xn
= △1 +
1
∂ 2
(xn
)
K ∂xn
(379)
Since h(y1 , y2 , . . . yn ) = h1
√
K
χ=
!
y1
yn
,... √
by assumption and
√
K
K
χ
χ
p
p
K
K
χ(ϕ.xn ) = xn χ(ϕ) = xn
269
(380)
in an obvious notation, we obtain
h(y1 , y2 , . . . yn ) = h1 (ϕ1 , ϕ2 · · · ϕn )
∂h1
∂h
=
= o. It now follows from (379) that △h = △1 h1 and
∂xn
∂xn
Lemma 25 is proved. Finally we remark that if dϑ, dϑ1 denote the invariant volume elements of R and H relative to the metrics (377) and
(378) respectively, then
√
n
n
Y
K√ Y
√
dϑ = ggnn
g
dxν =
dxν ,
xn
ν=1
ν=1
and
244
dϑ1 =
n−1
√ Y
g
dxν
ν=1
so that dϑ and dϑ1 are connected by the relation,
√
dϑ = Kx−1
n dxn dϑ1
(381)
We now interpret (368) and (381) in the particular space y > o. In
view of (367) we can state
270
Lemma 26. Let △ denote the Laplace-Beltrami operator of the space
y > o relative to the metric ds2 = σ(y−1 dy)2 and △1 the LaplaceBeltrami operator of the determinant surface |y| = 1 relative to the induced metric. If h1 (y) is an arbitrary function on this surface and h(y)
the homogeneous function of degree 0 on the whole space y > 0 defined
by h(Y) = h1 (|Y|1/n Y) then we have △h = △1 h1 .
This is how (368) reads in the space y > 0. We now take up (381).
The fundamental metric in the space of real matrices y = y(n) was given
by ds2 = σ(y−1 dy)2 , and the L − 1 transformations of this space onto
itself which leave the metric invariant are given by y → y∗ = y[R] with
an arbitrary real non singular matrix R. From (138) we know that
n+1
n+1
∂(y∗
= |R|n+1 = |y∗ | 2 |y|− 2
∂(y)
n+1
and then then |y| 2 [dy] is left invariant by the transformations
y → y∗ . It follows that the invariant volume element dϑ in this case is
given by
√
n+1
dϑ = C|y|− 2 [dy]
−
with an arbitrary constant C. We fix C by requiring dϑ in the standard
√
form, viz. dϑ = gπ p≤ν dyµν in the usual notation. Then g = c|y|(n+1)
and in particular at y = E we have g = C. But at this point
ds2 = σ(dy)2 =
n
X
µ=1
dy2µµ + 2
X
µ<ν
dy2µν
245
so that the matrix (grs ) whose determinant is g is diagonal with η of
η(n − 1
as 2. Thus, at Y = E,
the diagonal elements 1 and the rest
2
g = zn(n−1)/2 and this is then the value for c Thus
dϑ = 2n(n−1)/4 |y|−
n+1
2
[dy]
(382)
1
From (381) we have χ−1
n dxn dϑ1 = √ dϑ where χn is given by
K
p
K
χ(y).
In
the
present
case,
χ(y) = |y| and K = n so that
(380), viz.
χ
=
n
p
xn = n |y| = y (say), and then from (381) and (382),
n+1
1
1
y−1 dydϑ1 = √ 2n(n−1)/4 |y|− 2 [dy] = √ dϑ.
n
n
(383)
271
We can now compute the volume of the fundamental domain of the
unimodular group acting on the determinant surface |y| = 1. If K denotes
R
the space of all reduced matrices, this volume is given by Vn =
dϑ1 .
y1 ∈K
Now
n(n + 1
=
Vn Λ(
2
Z∞
e y
Z
e
−y
n(n+1
2 −1
dy
o
=
√
n
Z
dϑ1
y1 ∈K
|y1 |=1
|y|
|y|
n+1
2
y−1 dydϑ1
y∈K
1
= √ 2n(n−1)/4
n
Z
e−
√
n
|y|
[dy].
y∈K
The last integral has already been explicitly computed in (149) and
putting it in, we have
Vn =
n + 1 √ n(n−1)/4
n2
ϑn
2
(384)
In particular we have shown that the volume of the fundamental
domain is finite. We can therefore state that if u(y) is homogeneous
246
272
of degree 0, is invariant relative to the modular group and satisfies the
equations (362) and if further U is bounded in the fundamental domain,
then U is square integrable and is hence an angular character.
We return now to the question of existence of solutions of (362) and
examine how far they meet the other three requirements for an angular
character. We first observe that if U(y) is a solution of (362) with the
set of eigen values λ1 , λ2 , . . . λn , then U is a homogeneous function of
degree −λ1 so that |Y|λ1 |n U(y) is homogeneous of degree 0. We show
presently that for every solution U(y) of (362), |Y| s U(y) (for an arbitrary complex constant s) is also a solution with certain eigen values
λ∗1 , λ∗2 , . . . λ∗n depending on s and then in particular |y|λ1 |n U(y) will be a
solution of (362) with λ∗1 = o. In other words, given any U(y) which
satisfies the first condition for an angular character, we can determine
one which, besides the first, fulfills the second condition too.
That |Y| s U(y) is a solution of (362) is an immediate consequence of
the following operator identity,
K
(Y
∂ K s X ν K s ∂ K−ν
) |y| =
s ν |y| (y )
∂y
∂y
ν=0
(385)
where we remark that |Y| is to be treated as an operator. For K = 1 it is
directly verified that
(y
∂
∂ s
|y| = |y| s y + s|y| s E,
∂y
∂y
and then (385) is proved by induction on K. One also obtains the analogous identity
K
(y
273
∂ ∂ ′ K s X ν K s
) |y| =
s ν |y| (y )′ K−ν
∂y
∂y
ν=0
(386)
and this we need later.
We now show that (362) has non trivial solutions which are invariant
∂
relative to the unimodular substitutions y → y[U]. We decompose y,
∂y
as

 ∂
!
1 ∂


∂
y y
3
y= 1 3 ,
=  1∂y∂1 2 ∂y

∂ 
y3 y2
∂y
2 ∂y3
∂y2
247
with y1 = yr1 , o < r < n, and prove by induction on K that
!
n − r K−1 −s E
∂ K −s
o
K
|y1 | = (−1) s s −
|y1 |
.
y
y′3 y−1
o
∂y
2
1
(387)
Indeed,
y
∂
∂ −s
|y1 | = −s|y1 |−s−1 y |y1 |
∂y
∂y
!
−1 o
−s−1 |y1 |y1
= −s|y1 |
y
o
o
!
E
o
= −s|y1 |−s ′ −1
y3 Y1
o
and then we have only to assume (387) for a particular value of K ≥ 1
and prove it for the next higher value.
Now
!
n − r K−1 ∂
∂ K+1 −s
∈
0
K
−s
) y |y1 |
(388)
(y ) |y1 | = (−1) s(s −
y′3 y−1
0
∂y
2
∂y
1
by the induction assumption. Also,
!  ∂
 ∂y1
∂
E
o
−s
|y1 |
1 ∂ ,
′ y−1 o = 
y
∂y
3 1
2 ∂y3
1 ∂ 

2 ∂y3 

∂ 
∂y2
|y1 |−s E
|y1 |−s y′3 y
1
−s ∂ ′ −1 o
−s|y1 |−s y−1
1 + 2 |y1 | ( ∂y3 y3 )y1
=
o
o
!
1
−1
−1
−sy1 + 2 (n − r)Ey1 o
= |y1 |−s
o
o
!
−1
n−r
y
o
= −(s −
.
)|y1 |−s 1
o
o
2
o
−1
1
o
!
=
!
274
Putting this in (388) we have
n − r K −s −s y−1
∂
o
) |y1 | |y2 | y 1
(y )K+1 |y1 |−s = (−1)K+1 s(s −
o o
∂y
2
!
248
K+1
= (−1)
!
n − r K −s E,
o
s(s −
) |y1 |
y′3 y−1
o
2
1
and the proof of (387) is now complete.
It follows from (387) that
(
)
∂ K
n−r
− s)K−1 ; K ≥ 1.
σ(y ) + λK |y1 |−s = o, λK = rs(
∂y
2
(389)
In other words we have shown that |y1 |−s is a solution of (362) belonging to the set of eigen values.
ηK = rs(
n−r
− s)K−1 , K = 1, 2, . . . n.
2
(390)
But itself |y1 |−s is not invariant relative to the unimodular substitu(n,r)
tions; in fact if U = (QR) is any unimodular matrix
Q = Q ,
with
(
275
then due to a transformation y → y[U] of y, y1 = y oE r) goes over into
( (y[U] oE r) = y[Q], and |y2 |−s goes into |y[Q]|−s . However it is easily
seen from (389) that |y[Q]|−s also satisfies the same equation with the
same set of eigen values and then
X
E (y, s) =
|y[Q]|−s
(391)
Q
where Q runs through a complete representative system of the classes
{Qn,r } of primitive matrices is again a solution of (362) belonging to the
above system of eigen values (390). Of course, we have E (y[U], s) =
E (y, s) for any unimodular matrix U; in other words E (y, s) is a solution
of (362), which is invariant relative to the unimodular transformations.
The function E (y, s) is a generalisation of the usual Epstein’s zeta
function. It follows from (387) that
)
(
n−r
∂
∂ 2
)(y ) |y1 |−s = o
(y ) + (s −
∂y
2
∂y
and hence we can conclude that
)
(
n−r
∂
∂ 2
)(y ) E (y, s) = o
(y ) + (s −
∂y
2
∂y
(392)
249
It is possible to characterise E (y, s) as a solution of (392), whose
Fourier coefficients (in a certain sense) have specified properties.
Finally we prove that for every solution U(y) of (362), U ∗ (Y) =
U(Y −1 ) also satisfies a differential equation of the same kind,
)
(
∂ K
∗
σ(y ) + λK U ∗ (y) = o, K = 1, 2, . . . n,
∂y
where λ∗K , K = 1, 2, . . . n are uniquely determined by the eigen value s 276
λ1 , λ2 , . . . λn of U. We need a few operator identities as preliminaries.
∂
Let y = △ = (ωµν ). We prove first that
∂y
1
1
ωµν ω xλ = ωµλ ωµν + δνµ − δλµ ωµν
2
2
Since eµν
(393)
∂yxλ 1
= (δµν,xλ + δµ,ν,λν ), δµν,xλ = δµx δνλ
∂yµν 2
We have
ωµν ω xλ =
=
=
=
n
X
ρ=1
n
X
ρ,σ=1
n
X
ρ,σ=1
n
X
yµρ eρν
n
∂ ∂ X
yxσ eσλ yσλ
∂yρν σ=1
∂
yµρ eρν
∂
∂
yxσ eσλ
∂yρν
∂yσλ
yµρ yxσ eρν
n
∂ 1 X
∂
yµρ yµρ (δρν,xσ + δρν,σ )eσλ
∂yσλ 2 ρ,σ=1
∂yσλ
n
∂
∂ 1 X
yxσ eσλ yσλ yµρ eρν yρν
yµρ (δρ,νxσ + δρν,σx )
∂
∂ 2 ρ,σ=1
ρ,σ=1
n
1 X
∂
∂
−
yxσ (δσλ,µρ + δσλ,ρµ eρν yρν
eσλ
∂yσλ 2 ρ,σ=1
∂
1
∂
1
= ω xλ ωµν + yµx eνλ eνλ
+ δνλ ωµν
2
∂yνλ 2
∂
1
1
− δλµ ω xν
+ yxµ eλν
2
∂yλν 2
250
1
1
= ω xλ ωµν t δνx ωµλ − δλµ ωµν ,
2
2
277
and (393) is established.
We use (393) to show that for any operator matrix A = (αµν ),
σ(△K A) =
X
νν2 ···νK+1
ων2 ν3 · · · ωνg−1 νg ων1 ν2 ωνg νg+1 · · · ωνK νK+1 ανK+1 ν1
n
1
− σ(Λg−2 )(ΛK+g+1 A) + σ(ΛK−1 A)
2
2
(394)
The proof is by induction on j. For j = 2 (394) is obviously true.
Assume then j ≥ 2. By means of (393), the induction assumption gives
that
X
σ(ΛK A) =
ων1 ν2 ··· ωνK νK+1 ανK+1 ν1
ν1 ,nu2 ···νK+1
=
X
ν1 ,ν2 ,...νK+1
+
1 X
2 ν ,ν ,...ν
1
−
=
ων2 ν3 · · · ωνg νg+1 ων1 ,ν2 ωνg+1 νg+2 · · · ωνK νK+1 ανK+1 ν1
1
2ν
2
K+1
ων2 ν3 · · · ωνg+1 νg δnu2 νg ων1 νg+1 ωνg+1
ωνg+1 νg+2 · · · ωνK νK+1 ανK+1 ν1
X
1 ,ν2 ,...νK+1
ων2 ν3 · · · ωνg−1 νg+1 ν1 ωνg ν2
ωνg+1 νg+2 · · · ωνK νK+1 αK+1ν1
1
n
− σ(Λg−2 )σ(ΛK−g+1 A) + σ(ΛK−1 Λ)
2
X2
ων2 ν3 · · · ωνg νg+1 ων1 ν2 ωνg+1 νg+2 · · · ωνK νK+1 ανK+1 ν1
ν1 ,ν2 ,...νK+1
n
1
− σ(Λg−1 σ(ΛK−g Λ) + σ(ΛK−1 A)
2
2
278
and this proves (394). Putting j = k + 1 in (394) we obtain that
X
σ(ΛK A) =
ων2 ν3 · · · ωνK νK+1 ων1 ν2 ανK+1 ν1
ν1 ,ν2 ,...νK+1
251
1
n
σ(ΛK−1 )σ(A) + σ(ΛK−1 A)
2
2
n
1
= σ(ΛK−1 (Λ′ A′ )) − σ(ΛK−1 )σ(A) + σ(ΛK−1 A).
2
2
Choosing for A the special operator matrix (Λ′ )ℓ ′ this yields,
σ(ΛK (Λ′ )ℓ ′ = σ(ΛK−1 (Λ′ )ℓ+1 )′
1
n
− σ(ΛK−1 )σ(Λ′ )ℓ σ(ΛK−1 (Λ′ )ℓ ′
(395)
2
2
As a consequence of (395) we can conclude by induction on K that
X
σ(ΛK = σ(Λ′ )K +
eKν1 ν2 ..νr σ(Λ′ )νs σ(Λ′ )ν2 · · · σ(Λ′ )νr (396)
ν1 ν2 ···νr
ν1 +···+νr <K
Inverting this system of equations one also has
X
dνK1 ν2 ..νn σ(Λ)ν1 σ(Λ)ν2 · · · σ(Λ)νr
σ(Λ′ )K = σ(ΛK ) +
(397)
ν1 ν2 ···νn
ν1 +···+νr <K
279
with certain constant coefficients dνk1 ν2 ...νr .
Let now y∗ = y−1 and U ∗ (y) = U(y∗ ). Then dy∗ = −y−1 dyy−1 , and
for any function ϕ we have
dϕ = σ(dy
∂
∂
∂
ϕ) = σ(dy∗ ∗ ϕ) = −σ(dyy−1 ( ∗ ψ)y−1 )
∂y
∂y
∂y
and consequently
∂
∂
∂
∂
ϕ = −y−1 ( ∗ ϕ)y−1 , on y = −(y∗ ∗ )′ .
∂y
∂y
∂y
∂y
Replacing y by y∗ in (397) we can state now that
∂ K ∗
∂ ) U (y) = (−1)Kσ (y∗ ∗ ) K U(y∗ )
∂y
∂y










X


K
∗ ∂ ν1
∗ ∂ νr 
K
∗ ∂ K
d
σ(y
)
+
)
·
·
·
σ(y
)
= (−1) 
U(y∗ )
σ(χ

ν1 ν2 ···νr


∗
∗
∗


∂y
∂y
∂y


ν1 ν2 ···νr




ν +ν <K
σ(y
1
r
252






K
−λK +
= (−1) 





X
ν1 ν2 1···νr
ν1 +···+νr <K
= −λ∗K U ∗ (y)
with







r K
U(y∗ )
(−1) dν1 ν2 ···νr λν1 λν2 · · · λνr 















X




∗
K
r K
λK (−1) 
(−1)
d
λ
λ
·
·
·
λ
ν
ν
ν
ν
ν
···ν
1
2
r


1
2
n






1 ν2 ···νr
ν ν+···+ν

<K
1
(398)
r
A detailed calculation shows that
280
σ(Λ′ ) = σ(Λ), σ(Λ′ )2 = σ(Λ)2
n
1
σ(Λ′ )3 = σ(Λ)3 − σ(Λ)2 + (σ(Λ))2 ,
2
2
1
1
σ(Λ′ )4 = σ(Λ)4 − nσ(Λ)3 + σ(Λ)σ(Λ)2 +
2
2
2
1
n
n
+ σ(Λ)2 σ(Λ) σ(Λ)2 − (σ(Λ))2
2
4
4
and it follows then that
1
n
λ∗1 = λ1 , λ∗2 , λ∗3 = −λ3 + λ2 + λ21 ,
2
2
2
n
n
λ∗4 = λ4 − nλ3 − λ1 λ2 + λ2 + λ21 .
4
4
It would be of interest to determine the cases when we shall have
λ∗ν = λν for each ν.
When all is said and done, the question remains still open whether
angular characters according to our definition actually exist. We only
know that if we are given one angular character we can obtain others
from it; for instance, if U(y) is angular character then so is U ∗(y) =
U(y−1 ). For n > 2 the existence question is still a problem. For n = 2
however, the angular characters are completely determined and they are
just solutions of the wave equations relative to the modular group.
Let us introduce the operators HK by
HK =
∂
∂ 1
(σ(y )K + σ (y i)K
2
∂y
∂y
(399)
253
and observe in view of (397) that
HK = σ(y
∂ K 1
) +
∂y
2
X
281
ν1 ν2 ···νr
ν1 +···+νr <K
dν1 ν2 ···νr σ(y
∂
∂ ν1
) · · · σ(y )νr
∂y
∂y
for K = 1, 2 · · · n.
It is easily seen that by means of these n equations, σ(y
∂ K
) ,K =
∂y
1.2. · · · n can be expressed in terms of the HK′ s.
It follows therefore that HK , K = 1.2 · · · n also generate the space of
∂
invariant (linear) differential operators as σ(y )K − K = 1.2 · · · n do.
∂y
The angular characters U(y) are then eigen functions of these operators
HK , K = 1.2 · · · n, and in place of (362) we have an equivalent system of
differential equations for U(y), viz.
(HK + µK U(y) = o, K = 1.2, . . . n
(400)
with certain eigen values µK (K = 1.2 · · · n) which are uniquely determined by the eigen values λK of U(y) and conversely. We shall give an
interesting integral formula (407) involving these operators.
First we have to generalise the method of partial integration. Let A =
(aµν ), B = (ℓµν ) be arbitrary matrices whose elements are all functions
n+1
−
n(n−1)/4
of the elements of y We know that dϑ = 2
|y| 2 [dy] is the
volume element in the space of matrices y > o, invariant relative to the
transformations y → y[R], |R| , 0. Let
Y
ωµν = ±
dyρσ ,
ρ,σ,µ,ν
the ambiguous sign being determined by the requirement that
dyµν ωµν = [dY]and let
Ω = 2n(n−1)/4 |Y|−
n+1
2
a matrix of differential forms of degree
Y(eµν ωµν ),
n(n+1)
2
− 1. We wish to prove 282
254
that
Z
∂
B{Y A}dϑ = −
∂Y
Z
∂
{Y( )′ B′ }′ Adϑ +
∂Y
G
G
Z
BΩA
(401)
ℓG
where G denotes a compact domain with a given orientation and bg is
its boundary, assumed to be piece wise smooth. Now
∂
∂
A) + ((Y )′ B′ )A =
∂Y
∂Y
X
X
∂
∂
aτν )) + ( eρτ (
ℓµσ )aτν )
= ( )ℓµρ yρσ eστ (
∂y στ
∂Y ρτ
ρστ
ρστ
X
∂
∂
= ( yρσ eρστ (ℓµσ
aτν + aτν
ℓµσ ))
∂Y
∂Y
ρτ
ρτ
ρστ
B(y
(402)
Also
n+1
d(BΩA) = d(B2n(n+1) /4)|Y|− 2 Y(eµνωµν )A)
X
n+1
= 2n(n+1)/4 d(
ℓµν |Y|− 2 yρσ rστωστ aµν )
ρστ
X
n+1
∂
= 2n(n+1)/4 ( eστ
|Y|− 2 yρσ )ωστ )
∂y στ
ρστ
X
n+1
∂
(ℓµρ aτν |Y|− 2 yρσ )ωστ )[dY]
= 2n(n+1)/4 ( eστ
∂y
στ
ρστ
X
∂
(ℓµρ aτν )dϑ+
= ( yρσ eστ
∂yστ
ρστ
X
n+1
∂
+ 2n(n−1)/4 ( eστ ℓµρ aτν
(|Y|− 2 yρσ )[dy]
∂y στ
ρστ
283
(403)
n+1
yρσ
−
∂
We shall show that σ eστ
) = 0, and then the last
(|Y| 2
∂y στ
term on the right side of (403) vanishes. Let Yµν denoted the algebraic
minors of the elements of Y so that |Y|Y −1 = (Yµν )
P
255
Then
n+1
∂
(|Y|− 2 yρσ ) =
∂y
στ
σ
X
n+1
n + 1 − n+1 X
|Y| 2
=
Yστ yρσ +
|Y|− 2 δρτ eστ
2
σ
σ
X
=
eστ
n + 1 − n+1
n + 1 n+1
|Y| 2 δρτ +
|Y| 2 δρτ = 0
2
2
as desired. (403) now reduces, in view of (402), to
X
∂
(ℓµρ aτν ))dϑ
d(BΩA) = ( yρσ eρσ
∂ στ
ρστ
= (B(Y
∂
∂
A) + ((Y )′ B′ , A)dϑ
∂Y
∂Y
and then, applying Stoke’s formula by which
Z
Z
ω
dω =
G
ℓG
for an arbitrary exterior differential from, ω we have
Z
Z
Z
∂ ′ ′
∂
BΩA.
B(Y A)dϑ = − ((Y ) B) Adϑ +
∂Y
∂Y
G
G
ℓG
More generally we can show that
Z Z K−1
X
∂
∂
(y )K A′ dϑ = (−1)K
(y )′ )K B′ Adϑ +
(−1)ν
∂y
∂y
ϑ=0
G
Z G
∂ K−1−ν ∂ ν ′
(404)
((y ) B Ω Y( )
∂y
∂y
ℓG
For K = 1, (404) reduces to (401) proved above and then (404) is
established by restoring to introduction on K
284
256
Setting A = ϕE and B = ψE in (404) where ϕ and ψ are arbitrary
functions, and taking the trace of both sides, we obtain that
Z
G
Z
K=1
X
∂ K
∂
K
ψσ(y ) ϕdϑ = (−1) =
ψσ(y )′ )K ψdϑ +
(−1)ν
∂y
∂y
ν=0
Z G
∂
∂
(405)
σ ((y )ν ) Ω (y )K−1−ν ϕ
∂
∂y
ℓG
Interchanging ϕ and ψ and replacing ν by K−1−ν in (405) we obtain
by transposition that
Z
G
Z
K
X
∂ ′K
∂ K
K
ψσ((y ) ) ψdϑ = (−1) =
ψσ(y ) ψdϑ +
(−1)ν
∂y
∂y
ν=0
Z G
∂ ′ K−1−ν ∂ ν
ϕ
(406)
σ (y ) Ω ((y ) )
∂
∂y
ℓG
It follows by addition, from (405) and (406) that
Z
G
K−1
Z
1X
(−1)ν
ϕHK ψdϑ +
ψHK ψdϑ = (−1)
2 ν=0
Z G
∂ ′ ν ∂ K−1−ν 1 K−1
ϕ + Σν=0 (−1)ν
σ ((y ) ) ψ Ω (y )
∂y
∂y
2
ℓG
Z ′ ∂
∂
σ ( )ν ψ Ω ((y )′ )K−1−ν ϕ
(407)
∂y
∂y
K
ℓG
285
where HK is defined in (399).
By means of the formula (407) one can prove the orthogonal relations for angular characters. For we require a special representation for
the scalar product of the angular characters given below. Let u, ũ, be
two angular characters and introduce the scalar product (u, ũ) by
Z
u(r1 )ũ(y1 )dϑ1
(408)
(u, ũ) =
yi ∈K|y1 |=1
257
Then,
(u, ũ) =
Z
u(y1 )ũ(y1 )dϑ1
y1 ∈K
|y1 |=1
=
Ze
y−1 dy, y = |y|1/n
1
Z
u(y)ũy−1 dydϑ1
Z
u(y)ũ(y)dϑ
y∈K
i≤|y|≤en
=
y∈K
i≤|y|≤en
We now take for G in (407) the domain {y ∈ K, 1, |y| ≤ en } .
This domain is not compact and this presents some difficulty. We
have to approximate this by compact domains G i = 1, 2, . . ., apply (407)
to each Gi and resort to a limiting process. Of course all this needs 286
justification.
A detailed account of some of the results quoted in this section one
finds in the following references.
1. E. Hecke , Eine neue Art von Zetafunktionen und ihre Beziehungen
Zur Verteilung der Primzahlen, math.Zeit. 1 (1918), 357-376.
2. W. Roelcke , Über die Wellengleichung bei Grenzkreisgruppen erster Art, Act. Math (in print).
Chapter 18
The Dirichlet series
corresponding to modular
forms of degree n
We wish to determine a set of Dirichlet series which is equivalent with a 287
given modular form of degree n. We need as pre liminaries the followings the following lemma and a host of other operator identities.
P
Lemma 27. Let f(Z) = T geO a(T )e2πiσ(T Z) (T semi integral) (98) be a
modular from of degree n and weight K ≡ (2). Then
|a(T )| ≤ ϑo |T |K
(409)
for T > o with a certain constant ϑo
Proof. In view of (95) both sides of (409) are invariant relative to a modular substitution T → T [u].Hence we need prove (409) only for T > o
which are further reduced. Clearly a(T ) has the integral representation,
Z
−1
a(T ) =
f (× + iT −1 )e2πiσ(T ×i ) [d×]
H
as in (pp.64) and then
|a(T )| ≤ max | f (× + iT −1 )|e2πn
×εH
259
18. The Dirichlet series corresponding ....
260
Hence to infer (401) we need only prove that
| f (× + iT −1 )| ≤ C1 |T |K
288
and this again is ensured from (49) as by assumption T is reduced provided we show that
|f(× + iT −1 )| ≤ C2 (t11 t22 . . . tnn )K
where we assume T = (tµν )
(410)
!
A B
We now prove (410). Let Z ∈ Yn and determine M
∈ Mn
C D
!
S −E
such that Z1 = M < Z >∈ fn Let N
∈ Mn and Za N −1 < Z >=
E D
(−Z + S )−1 .
!
∗
∗
Then Z1 = MN < Zo > and MN =
by the property of
cs + d −ε
modular forms we have f(Zo ) = | − z + s|K f(z), and f(z1 ) = |(cs + D)zo −
Co | f (zo ) and consequently
f(z) = | − z + s|−K |(cs + D)zo − C|f(z1 )
(411)
Choose S = (Sµν ) as an integral symmetric matrix with ±Sµν ≤ n
and |cs+D| , 0. Such a choice is clearly possible as |CS +D| is a polynomial of degree almost n. Not vanishing identically in the elements Sµν
of S . Since C, S , D are all integral matrices, we have then in particular
kCS + Dk ≥ 1.
289
(412)
Let Y, Yo denote the imaginary parts of Z, Zo . Since ||Z + S || ≥ |Y| for
any real symmetric matrix S and |Yo | = |Y||−Z +S |−2 for zo = (−Z +S )−1
as is seen from (72) we conclude from (411) - (412) that
| f (Z)| ≤ C3 || − z + s||K ||zo − (cs + D)−1 c||−K
≤ C3 || − z + s||−K |yo |−K
= C3 || − Z + S ||K |y|−K |
(413)
261
If now elements of X are reduced modulo 1 as in (410) and Y =
T −1 , T > 0, and semi integral, then the elements of −X and of B =
−X + S are bounded , and
(−Z + S )y−1 = (BT − iE), Z = × + iY
(414)
P
The elements of (BT − iE) are of the form Cµν tρν − iρµν and the
P
absolute value of { Cµν tρν − iρµν } is at most equal to C4 tνν as tνν ≥ 1.
Hence ||BT − iE|| ≤ C5 t11 t12 . . . tnn and (410) now follows as a consequence of (413) and (414).
Let us split up the series on the right of 98′ according to the rank of
T and write
n
X
X
f(iy) =
fr (y), fn (y) =
a(T )e2πσ(T Y) .
r=0
t≥0
ranKT =r
Set Y = yY1 , y > 0 and |Y1 | = 1, and introduce the integral
R(S , Y1 ) =
Z∞
fn (yY1 )ynS −1 dy
(415)
o
R(S , Y1 ) is a function on the determinant surface |Y1 | = 1 and is
invariant relative to the unimodular substitutions, viz.
R(S , Y1 [u]) = R(S , Y1 )
for any unimodular matrix u. Let u(y) be any angular character and let 290
Z
R(S , Y1 )u(Y1 )dϑ1
(416)
ξ(S , u) =
Y1 εK
|Y1 |=1
where as usual, K is the space of reduced matrices.
Since y−1 dydϑ1 = √1n dϑ from (383), we have
ξ(S , u) =
Z
Z∞
Y1 εK y=0
|Y1 |=1
fn (Y)u(Y)|Y|n−1 ydydϑ1
262
1
= √
n
Z
fn (Y)u(Y)|Y|S dϑ
YεK
In this we substitute for fn (Y) its representation by a series and integrate termwise. All these and the subsequent transformations can be
justified if u(Y) is bounded and the real part of s is sufficiently large, and
we assume this to be the case. Then
Z
1 X
a(T )
e−2πσ(T Y) u(Y)|Y|S dϑ
(417)
ξ(S , u) = √
n T >o
YεK
In the right side (417) we have a sum of the type
P
F(T ) where T
T >o
runs over all semi integral matrices.
If { T } denotes the class of all matrices T [u] for a given T (u)
(-unimodular) then in general we have
X
X 1 X
> 0F(T ) =
FF(T [u])
ε(T
)
u
T
{T }
291
where ∈ (T ) denotes the number of units of T , i.e., of unimodular matrices u with T [u] = T this number is finite as T > 0 and u runs over all
unimodular matrices while {T } runs over all distinct classes of positive
semi integral matrices T . Since by assumption K ≡ 0(2) we have from
(95) that a(T [u]) = a(T ) and then from (417),
Z
1 X a(T ) X
e−2πσ(T [u]Y) u(Y)|Y|S dϑ
ξ(S , u) = √
n {T } ε(T ) u
YεK
Z
′
1 X a(T ) X
e−2πσ(T [u ]Y) u(Y[u′ ])|Y[u′ ]|S dϑ
= √
n {T } ε(T ) u
Z YεK
1 X a(T )
e−2πσ(T Y) u(Y)|Y|S dϑ
= √
n {T } ε(T )
Y>0
The substitution Y[R] → Y where T = RR′ then yields
Z
2 X a(T ) S
ξ(S , u) = √
|T |
e−2πσ(Y) u1 (Y)|Y|S dϑ
ε(T
)
n {T }
Y>o
(418)
263
where
u1 (Y) = u(Y[R−1 ])
Where
W(S , u1 ) =
Z
(419)
Y>o
that gives
2 X a(T ) S
|T | W(S , u1 )
ξ(S , u) = √
n {T } ε(T )
(420)
We may remark that u1 is a solution of (362) belonging to the same
set of eigen values as u and if u is bounded so is u1 . We have to compute 292
W(S , u1 ) and before we do this we need some preparations. For an
arbitrary function matrix A = (aµν ) we prove that
(Y
∂
1
1
∂ ′
) AY = (Y(Y )′ A)′ )′ → A′ Y + σ(AY)E
∂Y
∂Y
2
2
Indeed we know that
eρµ
∂y
1
= (δρτ δµν + δρν δµτ )
∂yρµ 2
and then
(Y
X
∂ X
∂ ′
) AY = ( yνρ eρµ
)( aµτ yτν )
∂Y
∂yρµ τ
ρ
X
∂
= ( )yσρ eσµ
aστ yτν )
∂yρµ
ρστ
X
∂
1 X
= ( yµτ yσρ eρν
aστ )′ + ( yσρ aστ δρτ δµν )
∂yρν
2 ρστ
ρστ
X
1
+ ( yσρ aστ δρτ δµν )
2 ρστ
= (Y((Y
∂ ′ ′′ 1
1
) A) ) + σ(AY)E + A′ Y.
∂Y
2
2
(421)
264
By induction on K and by means of (421) one easily proves now that
K
(Y
∂ ′ K K K 1X
)Y = Y +
σ(Y ν )Y K−ν (K ≥ 1)
∂Y
2
2 ν=1
(422)
A direct computation yields that
293
(Y
and
∂ ′
) σ(Y K ) = KY K (K > 0)
∂Y
(423)
∂ ′ −2πσ(Y)
−2πσ(Y)
)e
=−2πe
Y
∂Y
By repeated applications of (422) – (424) we obtain that
(424)
(Y
∂
((Y )′ )K e−2πσ(Y) (−2π)K Y K + e−2πσ(Y)
∂Y
X
aKν1 ν2 ...νr ν νσ(Y ν 1) . . . σ(Y νr )Y ν
(425)
ν1 +···+νr +ν<K
with certain constant coefficients aK ν1 ν2 ...νr ν . Taking traces, we have
P
∂ ′ K −2πσ(Y)
)) e
= e−2πσ(Y) {(−2π)K + ν1 +···+νS <K
σ((Y ∂Y
mathscrCνK1 ν2 ...νS σ(Y1ν ) . . . σ(Y νS )} and this is easily generalised (by inductions on ρ) to the following result, viz.
∂
∂
∂ ′ νρ
) ) σ((Y )′ )νρ−1 . . . σ((Y )′ )ν1 e−2πσ(Y)
∂Y
∂Y
∂Y
o
n
=e2πσ(Y) (−2π)ν1 +···+νρ σ(Y ν1 ) . . . σ(Y νρ ) + Cν1 ν2 ...νρ (Y)
σ((Y
(426)
with
Cν(Y)
=
1 ν2 ...νρ
X
µ1 ...µr
µ1 +···+µr <ν1 +···+νρ
294
ν ν ...ν
µ
µ
Cµµ1 22...µrρ σ(Y µ1 )σ(Y2 ) . . . σ(Yr )
The interesting fact is that the terms on the right side of (426) are
all homogeneous (of different degrees) and the first term is of the highest degree, viz ν1 + ν2 + · · · + νρ while the degrees of the other terms
265
ν ...ν
are strictly less. The CνK1 ν2 ...νr ′ S and eµ11 ...µρr ′ S occurring above are all
constants.
We now return to the integral formula (405) and specialise that functions ϕ and ψ occurring there as ϕ = u1 (Y), ψ = f(Y)|Y|S where u1 (Y) is
a bounded solution of the differential equation system (362) and f is an
arbitrary function at our disposal. For G we now take the whole domain
Y > 0. By a suitable choice of f (Y) we can obtain the integral over
the whole domain as a limit of integrals over appropriate subdomains G
such that in the limit, the integrals over the boundary of G vanish. If
λ1 . . . λn are the eigenvalues of u1 (Y), using (383) we obtain- from (405)
for K ≥ n that
Z
f(Y|Y|S u1 (Y)dϑ) =
− λK
Y>o
K
= (−1)
Z
Y>o
K
X
K
= (−1)
u1 (Y)σ((Y
S ν (Kν )
ν=o
= (−1)K
K
X
Z
∂ ′K S
) ) |Y| f (Y)dϑ
∂Y
{σ((Y
∂ ′ K−ν
) ) f(Y)}|Y|S u1 (Y)dϑ
∂Y
Y>o
S K−ν (Kν )
ν=o
Z
{σ((Y
∂ ′ K−ν
) ) f(Y)}|Y|S u1 (Y)dϑ
∂Y
(427)
Y>o
Finally by induction on ρ we obtain by means of (427) that for in- 295
dices K1 , K2 , . . . , Kρ ≥ n
Z
λK 1 λK 2 . . . λK ρ
f(Y)|Y|S u1 (Y)dϑ =
y>o
K1 ,K2 +···+Kρ +ρ
(−1)
K1 X
K2
X
ν1 =0 ν2 =0
where
K
[∗] = S K1 +K2 +···+Kρ −ν1 −ν2 ...νρ (Kν11 )(Kν22 ) . . . (νρρ )
···
Kρ
X
[∗]
νρ =0
266
×
Z
{σ((Y
∂
∂
∂ ′ νρ
) ) σ((Y )′ )νρ−1 . . . σ((Y )′ )ν1 f(Y)}|Y|S
∂Y
∂Y
∂Y
Y>o
×u1 (Y)dϑ}
(428)
choosing in particular f(Y) = e−2πσ(Y) and assuming that the real part
of S is sufficiently large, all the above steps can be justified, and using
(426) we finally obtain that
λK 1 λK 2 · · · λK ρ e2πσ(Y) |Y|K u1 (Y)dϑ
= (−1)K1 +K2 +···+Kρ −nu1 −ν2 −···−νρ
K2
XX
ν1 =0 ν2 =0
···
Kρ
X
ν =0
ρ
K
(Kν11 )(Kν22 ) · · · (νρρ )S K1 +···+Kρ −ν1 ...νρ [∗]
where
ν1 +ν2 +···+νρ
[∗] = (−2π)
Z
e−2πσ(Y) σ(Y ν1 ) . . . σ(Y νρ )|Y|S u1 (Y)0
Y>o
+
X
ν ν ...ν
Cµ11 µ22 ...µρ2
µ1 ...µr
µ1 +···+µr <ν1 +···νρ
Z
e−2πσ(Y) σ(Y µ1 ) . . .
Y>0
σ(Y νr )|Y|K u1 (y)dϑ
(429)
296
Introduce again y and Y1 in place of Y by
Y = yY1 , |Y| = yn , y > o
and denotes as usual by dϑ1 the invariant volume element of the determinant surface Y1 > 0|Y1 | = 1. Then from (383)
dϑ =
√
ny−1 dydϑ1 .
In the sequel we consider only homogeneous functions u1 (Y) of degree 0 i.e., we assume λ1 = 0. In (429), after the above substitutions,
267
the integral in y is a y integral. Carrying out this integration over y we
obtain that
Z
e−2πσ(Y) σ(Y ν1 ) . . . (σ(Y νρ )|Y|S u1 (y)dϑ
Y>o
√
= ny(nS + ν1 + · · · + νρ )(2π)nS −ν1 ···νρ
Z
ν
×
(σ(Y1 ))−nS −ν1 ···vνρ σ(Y1ν1 ) . . . σ(Y1 ρ )u1 (y1 )dϑ1
Y1 >0
Let
Λ(nS + ν1 + · · · + νρ )
Jν1 ν2 ...νρ (S , u1 ) =
Λ(nS )
Z
ν
×
u1 (Y1 )(σ(Y1 ))−nS −ν1 −···−νρ σ(Y1ν1 ) . . . σ(Y1 ρ )dν1
(430)
Y1 >O
and
J(S , u1 ) =
Z
297
−nS
u1 (Y1 )(σ(Y1 ))
dν1
(431)
Y1 >O
Then (429) implies that
λK1 λK2 . . . K(S , u1 ) = (−1)K1 +···+Kρ +ρ
K2
K1 X
X
...
ν1 =O ν2 =O
!
!
!
Kρ
X
Kρ
K1 K2
S K1 + · · · + Kρ − ν1 − · · · − νρ ×
...
...
νρ
ν1 ν2
νρ =O
X
× (−1)ν1 +···+νρ Jν1 ···νρ (S , u1 ) +
µ1 ...µr
µ1 +···+µr <ν1 +···+νρ
ν ν ...ν
Cµ11 µ22 ...µρr (2π)−µ1 −···−µr Jµ1 ...µr (S , u1 )
(432)
We therefore obtain for the integral corresponding to the indices
(ν1 , ν2 . . . νρ ) = (K1 , K2 , . . . Kρ ) a representation of the kind
JK1 K2 ···Kρ (S , u1 ) = (−1)ρ λK1 λK2 . . . λKρ J(S , u1 )
268
X
µ1 ...µr
µ1 +···+µr <K1 +...+Kρ
298
r
PKµ11µK22...µ
(S )Jµ1 µ2 ...µr (S , u1 )
...Kρ
(433)
µ ...µ
where PK11 ...Krρ is either 0 or is a polynomial in S of degree atmost K1 +
ν ν ...ν
K2 + · · · + Kρ − µ1 − µ2 − · · · µr . Since the coefficients Cµ11 µ22 ...µρr by their
very definition, do not depend on the eigen values λ1 , λ2 . . . λn , this is
µ ...µ
true also of the polynomials PK11 ...Kρρ (S ). The method of induction on
K1 + K2 + · · · + Kρ yields by virtue of (433) that
JK1 K2 ...Kρ (S , u1 ) = qK1 K2 ...Kρ (S )J(S , u1 )
(434)
where qK1 Kρ (S ) is again either 0 or is a polynomial in S of degree at
most K1 + K2 + · · · + Kρ . We now show that
qK1 K2 ...Kρ (S ) = nρ S K1 +···+Kρ + lower powers of S ,
(435)
in other words that qK1 ···Kρ (S is of degree exactly K1 + · · · + Kρ and
that the coefficient of the highest degree term is precisely nρ . The proof
is again by induction on K1 + · · · + Kρ . If K1 + · · · + Kρ = O whence
K1 = K2 = · · · = Kρ = O we have obviously qK1 ...Kρ = nρ , ensuring that
the start is good. Assume now that K1 + · · · + Kρ > O and
qµ1 µ2 ...µr (S ) = nr S µ1 +···+µr +
lower powers of S .
(436)
for
µ1 + µ2 + · · · µr < K1 + K2 + · · · + Kρ .
If
qK1 K2 ...Kρ (S ) = CS K1 +···Kρ +
299
lower powers of S
(437)
we have to show that C = nρ .
The application of (434) – (437) in (432) yields a polynomial identity. The power S K1 +K2 +···+Kρ appears only on one side of this identity so
that the coefficient of this power must necessarily vanish. This requires
that
!
!
Kρ
K1
X
X
Kρ ρ
ν1 +···+νρ K1
(−1)
···
n
···
νρ
ν1
ν1 =O
ν1 +···+νρ <K1 +···+Kρ
νρ =O
269
+ (−1)K1 +···+Kρ C = O
Since this equation determines C uniquely and if we substitute C by
nρ this equation is satisfied - at least one Kν being greater than 0 - we
conclude that C = nρ .
We observe more over that the coefficient of S K1 +···+Kρ −1 in
qK1 K2 ...Kρ (S does not depend on the eigen values λ1 , λ2 . . . λn . While
this is evident in case K1 +· · ·+Kρ = 1, as in this case qK1 ...Kρ (S ) = nρ S ,
the general result follows by induction on applying (434) in (433).
Now for any positive matrix Y, |Y| is a rational function of the sums
of powers of the characteristic roots λ1 , λ2 · · · λn of Y, and by a standard
result on elementary symmetric functions, it is then a rational function of
P
σ(Y ν )(= nµ=1 λνµ ), ν = 1, 2, . . . n. In particular σ(Y n ) can be expressed
as a function of |Y| and σ(Y ν ), ν < n. If we take Y = Y1 with |Y1 | =
1, then σ(Y1n ) is a function of σ(Y1ν ), ν < n, and this function can be
determined to be of the form
X
(438)
σ(Y1n ) = (−1)n+1 n +
dν1 ν2 ···νr σ(Y1ν1 ) · · · σ(Y1ν1 )
ν1 ···νn <n
ν1 +···+νr =n
with constant coefficients dν1 ν2 ...νr .
Now by definition
Z
Λ(nS + n)
u1 (Y1 )(σ(Y1 ))−nS −r σ(Y1n )dν1
Jn (S , u1 ) =
Λ(nS )
300
(439)
Y1 >O
and using (438) we have
Λ(nS + n)
Jn (S , u1 ) = (−1)n+1 n
J(S + 1, u1 )
Λ(nS )
X
dν1 ν2 ...νr Jν1 ν2 ...νr (S , u1 )
+
ν1 ν2 ...νr <n
ν1 +···+νr =n
Hence
Λ(nS + n)
J(S + 1, u1 ) = q(S )J(S , u1 )
Λ(nS )
(440)
270
with
1
q(S ) = (−1)n+1 (qn (S ) −
n
X
ν1 ...νr <n
νρ +···+νr =n
dν1 ···νr qν1 ···νr (S
(441)
q(S ) is polynomial in S of degree n. The power S n appears in q(S )
with the coefficient
X
1
(−1)n+1 n −
dν1 ν2 ...νr nr
n
ν1 ···νr <n
νρ +···+νr =n
301
and it is immediate from (438) (setting Y1 = E) that this coefficient is
exactly equal to 1. Hence q(S ) can be factored as
q(S ) = (S − d1 )(S − d2 ) · · · (S − dn )
(442)
with certain complex constants d1 , d2 . . . dn , and from our earlier statements concerning the coefficient of S ν1 +ν2 +···+νρ −S in qν1 ν2 ...νρ (S ) we
infer that the coefficient of S n−1 in q(S ) viz. ±L1 + L2 + · · · + dn does
not depend on the eigen values λ1 . . . λn . By means of the functional
equation for Λ(S ), viz. Λ(S + 1) = S Λ(S we finally obtain from
(440) that
J(S + 1, u1 ) =
(S − α1 ) · · · (S − αn )
J(S , u1 )
+ 1n ) · · · (S + n−1
n )
nn S (S
(443)
We shall use this transformation formula for computing Λ(S , u1 )
explicitly, by a method which Huber developed recently for the case
n = 2. Clearly the function
G(S ) =
Λ(S − d1 )Λ(S − d2 ) . . . Λ(S − dn )
nnS Λ(S )(S + 1n ) . . . Λ(S + n−1
n )
(444)
satisfies the same transformation formula as Λ(S , u1 ) does in (443) so
that, if we set
J(S , u1 )
(445)
H(S , u1 ) =
G(S )
271
then H(S , u1 ) is a periodic function of S and
H(S + 1, u1 ) = H(S , u1 )
302
(446)
For values of S for which the real part of S is sufficiently large,
J(S , u1 ) and G(S ) are regular functions of S and then this is true also
of H(S ). Since H(S ) is further periodic, it is an entire function. We
wish to show that this entire function is actually a constant depending
only upon u1 (E). For this we need some asymptotic formulae.
It is well known that
√
1
(S → ∞)
Λ(S − α) ∼ 2πS S −α− 2 e−S
and it follows that
G(S ) ∼ n−nS S −α1 −α2 −···−αn −
n−1
2
(S → ∞)I.
(447)
We shall now determine the value of the sum α1 + α2 + · · · + αn .
We know that this sum is independent of the eigen values λ1 , λ2 . . . λn ,
in other words it is independent of the function u1 . We can then choose
u1 ≡ 1 and obtain by means of Lemma 14 in view of (383) that
Z
1
n−1
−nS + n(n−1)
4 Λ(S )Λ(S −
) · · · Λ(S −
)=
e−2πσ(Y) |Y|S dν
(2π)
2
2
Y>O
Z Z
√
e−2πyσ(Y1 ) ynS −1 dydϑ1
= n
=
√
Y1 >O Y>O
−nS
n(2π)
Λ(S , 1).
Hence
303
J(S , 1) = (2π)
n(n−1)
4
Λ(S )Λ(S − 12 ) · · · Λ(S −
√
nΛ(nS )
n−1
2 )
Since by the Gaussian multiplication formula,
1
Λ(nS ) =
nnS − 2
n−1
)
Λ(S
)Λ(S
1 ) · · · Λ(S −
n
n
(2π)(n−1)/2
(448)
272
the above gives that
J(S , 1) = (2π)
(n+2)(n−1)
4
Λ(S )Λ(S − 21 · · · Λ(S −
nnS (S )Λ(S + 21 ) · · · Λ(S
n−1
2 )
+ n−1
n )
(449)
and consequently
S (S − 21 ) · · · (S − n−1
J(S + 1.1)
n )
=
J(S , 1)
nn S (S + 1n ) · · · (S + n−1
n )
A comparison with (443) now shows that
α1 + α2 + · · · + αn =
n(n − 1)
.
4
(450)
Substituting this in (447) we have
G(S ) ∼ n−nS S −n+z)(n−1)/4
(S → ∞)
(451)
and in particular, the asymptotic value of G(S ) is independent of the
eigen values λ1 , λ2 . . . λn .
Also it follows from (449) that
J(S , 1) ∼ (2π)n+2)(n−1)/4 n−nS S −(n+2)(n−1)/4
304
(452)
and against (451) this gives
J(S , 1) ∼ (2π)(n+2)(n−1)/4 G(S )(S → ∞)
(453)
It is also clear from (451) that
G(S + K) ∼ n−nS G(K)
(K → ∞)
(454)
where we let K to tend to ∞ through all rational values. With
−S we can now state from (446) that H(S +K, u ) =
ω(Y) = u1 (Y)( σ(Y)
1
n )
H(S , u1 ) for every integer K. If now S → ∞ through all rational
values, say, it is clear in view of (454) that
H(S , u1 ) = lim H(S + K, u1 ) = lim
K→∞
K→∞
J(S + K, u1 )
G(S + K)
273
J(S + K, u1 )
K→∞ n−nS G(S )
= lim
(455)
Now
nnS J(S + K, u1 ) = nnS
Z
σ(Y1 )−nS −nK u1 (Y)dϑ1
Y1 >O
=
Z
u1 (Y)(
σ(Y) S
) (σ(Y1 ))−nK dϑ1 = J(K, ω).
n
Y1 >O
Hence we conclude from (455) by means of (454) that
J(K, ω)
J(K, ω)
= (2π)(n+2)(n−1)/4 lim
K→∞ G(K)
K→∞ J (K, 1)
H(S , u1 ) = lim
(456)
We now prove for continuous and bounded functions ω(Y) that
J(K, ω)
= ω(E)
K→∞ J(K, 1)
lim
(457)
In view of the linearity of J(K, ω) in ω we can further assume that
ω(E) = 0 as in the alternative case we need only argue with the function
ω(Y) − ω(E) in the place of ω(Y). Representing σ(Y) and |Y| in
√ terms
of the characteristic roots of Y one easily sees that σ(Y) ≥ n |Y| for
Y > 0. In particular, σ(Y1 ) ≥ n for Y1 > 0, |Y1 | = 1, and the equality
is true only for Y1 = E. This implies that if σ(Y1 ) → n and Y1 → A
then σ(A) = n and consequently A = E. Hence given ε > 0 we can find
δ = δ(ε) > 0 such that |ω(Y1 )| <∈ for σ(Y1 ) < n(1 + δ). Let |ω(Y1 )| ≤ ϑ
for all Y1 . (By assumption ω is bounded).
Then,
R
ω(Y1 )σ(Y1 )−nK dϑ
J(K, ω) y1 >O
R
,
=
J(K, 1)
σ(Y1 )−nK dϑ1
Y1 >O
and hence
R
R
(σ(Y1 ))−nK dϑ1
(σ(Y1 )−nK dϑ1
σ(Y
)>n(1+δ)
J(K,
ω)
σ(Y
)≤n(1+δ)
1
1
R
R
+ϑ
···
J(K, 1) ≤ ε
(σ(Y1 ))−nK dϑ1
(σ(Y1 ))−nK dϑ1
Y1 >O
Y1 >O
305
274
nK
nK
≤ ε + ϑ(1 + δ)− 2 n− 2
J( K2 , 1)
J(K, 1)
306
From (452) we have
− nK
2
n
and then
In other words,
J( K2 , 1)
∼ 2(N+2)(N−1)/4 (K → ∞)
J(K, 1)
J(K, ω) ∼ 2ε for K ≥ K (ε)
O
J(K, 1) lim
K→∞
J(K, ω)
= O = ω(E).
J(K, 1)
Applying (457) in (456) we obtain that
H(S , u1 ) = (2π)(n+2)(n−1)/4 ω(E).
But ω(E) − µ1 (E) and thus
H(S , u1 ) = (2π)(n+2)(n−1)/4 u1 (E).
(458)
After all these, we are in a position to face the integral W(S , u1 ) in
(419). We have
Z
W(S , u1 ) =
y>O
=
√
√
n(2π)−nS Λ(nS )J(S , u1 )
n(2π)−nS ΛG(nS )H(S , u1 ) · · ·
Λ(S − d1 ) · · · Λ(S − dn )
= (2π)−nS Λ(nS )n−nS
Λ(S )Λ(S + n1 ) · · · Λ(S + n−1
n )
(n + 2)(n)
(2n)
4
√
nu1
=
275
307
We know from (418) that u1 (E) = u(T −1 ) = u∗ (T ) in the notation
in (p. 251). Using again the Gaussian multiplication formula (448) the
above gives that
W(S , u1 ) = (2π)−nS +
n(n−1)
4
Λ(S − d1 ) · · · Λ(S − dn )u∗ (T )
(459)
Then from (420)
2 X a(T ) −S
|T | W(S , u1 )
ξ(S , u) = √
n {T } ε(T )
n(n−1)
2
= √ (2π)−nS + 4 Λ(S − d1 ) · · · Λ(S − dn)D(S , u) (460)
n
where
D(S , u) =
X a(T )u∗ (T )
ε(T )
{T }
|T |−S .
(461)
The next question is whether the function D(S , u) defined by the
above Dirichlet series for values of S whose real parts are sufficiently
large, can be continued analytically in the whole plane. We have
Z
1
fn (Y)u(Y)|Y|S dϑ.
ξ(S , u) = √
n
y∈K
We can take in the place of K a domain which is invariant relative
to the transformation Y → Y −1 . Since the volume element dϑ is also 308
invariant relative to this transformation we can write
Z
1
fn (Y −1 )|Y|−S u∗ (Y)dϑ
ξ(S , u) = √
n
Y∈K
|Y|≥1
1
+ √
n
Z
Y∈K
|y|≥1
fn (Y)|Y|S u(Y)dϑ
(462)
276
By the transformation formula for modular forms we have f(−Z −1 ) =
and in particular, f(iY −1 ) = inK |Y|K f(iY). Then,
|Z|K f(Z)
n
X
r=o
fr (Y −1 ) = f(iY −1 ) = inK |Y|K
n
X
fr (Y)
r=o
and consequently
−1
nK
K
fn (Y ) = i |Y| fn (Y) +
n−1
X
r=o
inK |Y|K fr (Y) − fr (Y −1 )
Substituting this in (462) we have
1
ξ(S , u) = √
n
Z
fn (Y){inK |Y|K−S u∗ (Y) + |Y|S u(Y)}dϑ +
Y∈K
|Y|≥1
Z
Y∈A
|Y|≥1
inK |Y|K fr (Y) − fr (Y −1 ) |Y|−S u∗ (Y)dϑ
n−1
X
1
√
n
r=o
(463)
Here we have assumed that the order of summation and integration
can be inverted. We now compute explicitly the integral corresponding
to r = o in the sum on the right side of (463). With fo (Y) = ao , this
reduces to
Z
√
a(o)
inK yn(K−S ) − y−nS u∗ (Y1 ) ny−1 dydϑ1
√
n
Y∈K
|Y|≥1
= a(o)
Z
Y1 ∈K
|Y1 |=1
=−
309
inK
1
−
u (Y1 )dϑ1
n(S − K) nS
∗
a(o)(1, u∗ ) 1
inK
+
n
S K−S
!
!
(464)
where (1, u∗ ) is the scalar product of the two angular characters introduced in (408).
277
The first term on the right side of (463) is an entire function of S .
In the special cases when n = 1, 2, all the other integrals occurring there
can be shown to be meromorphic in S and it will follow that in these
cases ξ(S , u) and consequently D(S , u)′ can be continued analytically
in the whole domain. It is quite likely that this is true for arbitrary n too.
Further, due to a transformation
S → K − S , u → u∗ ,
the first integral on the right side of (463) gets multiplied by a factor inK . 310
For n = 1, 2, it is known that all the integrals occurring on the right side
of (463) have the above property so that in these cases we have
ξ(K − S , u∗ ) = inK ξ(S , u)
(465)
It is to expected that this transformation formula is also valid in the
general case and our conjectures are further supported (1) by the fact
that one of the integrals computed in (464) (corresponding to r = 0)
verifies all these properties.
Bibliography
[1] E. Hecke Über die Bestimmung Dirichletscher Reihen durch ihre
Funktionalgleichung, Math. Ann. 112 (1936), 664-700.
[2] H.Maass Modulformen Zweiten Grades und Dirichletreihen,
Math. Ann. 122 (1950), 90-108.
279
Bibliography
[1] H. Braun: Zur Theorie der Modulformen n-ten Grades, Math. Ann. 311
115 (1938), 507-517.
[2] H.Braun: Konvergenz veralgemeinerter Eisensteinscher Reihen,
Math. Z. 44 (1939), 387-397.
[3] M.Koecher: Uber Thetareihen indefinitter quadratischer Formen
Math. Nachr. 9.(1953) 51-85.
[4] M. Koecher: Uber Dirichlet-Reihen mit Funktionalgleichung,
Crelle Journal 192(1953), 1-23.
[5] M. Koecher: Zur Theorie der Modulformen n-ten Grades.I. Math.
Z. 59 (1954), 399-416.
[6] H. Maass: Uber eine Metrik im Siegelschen Halbraum, Math. Ann.
118 (1942), 312-318.
[7] H. Maass: Modulformen zwoiten Grades and Dirichletreihen,
Math. Ann. 122 (1950), 90-108.
[8] H. Maass: Uber die Darstellung der Modulformen n-ten grades
durch Poincaresche Reihen, Math, Ann. 123 (1951),121-151.
[9] H.Maass: Die Primzahlen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 124 (1951), 87-122.
[10] H. Maass: Die Differentialgleichungen in der Theorie der
Siegelschen Modulfunktionen, Math. Ann. 126 (1953), 44-68.
281
282
312
BIBLIOGRAPHY
[11] H. Maass: Die Bestimming der Dirichletreihen mit Grossencharakteren zu den Modulformen n-ten Grades, J. Indian Math.
Soc. (in print).
[12] C.L.Siegel: Uber die analytische Theorie der quadratischen Formen, Ann, of Math. 36 (1935), 527-606.
[13] C.L. Siegel: Einfuhrung in die Theorie der Mokulfunktionen n-ten
Grades, Math. Ann. 116 (1939), 617-657.
[14] C.L.Siegel: Einheiten quadratischer Formen, Abh. a. d. Seminar d.
Hamburger Univ. 13 (1940), 209-239.
[15] C.L.Siegel: Symplectic geometry, Amer. J. Math. 65 (1943), 1-86.
[16] C.L. Siegel: Indefinite quadratische Formen und Funktionentheorie. I. Math. Ann. 124 (1952),17-54.
[17] C.L.Siegel: Indefinite quadratische Formen und funktionen - theorie. II. Math. Ann. 124 (1952), 364-387.
[18] E. Witt: Eine Identitate zwischen Modulformen zweiten Grades,
Abh.a.d.Seminar d. Hamburger Univ. 14 (1941), 321-327.

On Siegel`s Modular Functions - School of Mathematics, TIFR

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