INTRODUCTION
Let H be the Siegel upper-half space of degree n :
Hn = {Z 6 Mn(C)| Z =
tZ,
Im Z 0}.
Here M
n
(C) is tne ring of n x n matrices over C . The real
symplectic group of degree 2n , Sp(n,R), acts transtively on
H as a group of holomorphic automorphisms by the action,
'
M =
c
]
M(Z) =
(AZ+B)(CZ+D)"1"
, M =
|[
r
n|Din
Sp(n,R).
Let Sp(n,Z) = Sp(n,R) f i JV U (Z) be the discrete modular
subgroup of Sp(n,R). A holomorphic function f defined on
H is called a modular form of weight k and degree n with
respect to Sp(n,Z) if f satisfies the following condition:
( n 2)
1. f(M(Z)) =[det(CZ+D)]kf (Z) for all M = |„ | in
-Gn 3
Sp(n,Z).
The modular form f is called a cusp form if it satisfies the
further condition:
2. Suppose that I a (T)exp[2Tria (TZ)] is the Fourier
expansion of f ; then a(T) =0 if rank T n.
Here the summation is over all half integral matrices
T such that T _ 0 and a(TZ) = trace of TZ.
Denote by S(k;Sp(n,Z)) the vector space of holomorphic
cusp forms of weight k and degree n with respect to Sp(n,Z)
If k _ 2n+3 and n 2, the dimension of S (k;Sp(n,Z) ) over
C is given by Selberg's trace formula as follows [ 6 ]:
Received by the editor April 18, 1983 and, in revised form,December 5, 1983.
1
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