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.

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