NOTATION

Z,Q,R,C : ring of rational integers and the fields of rational

numbers, real numbers and complex numbers respectively.

M (Z),Mn(R),Mn(C) : rings of n x n matrices over Z,R,C

respectively.

GL (Z)/ GL (R) : general linear groups over Z,R respectively.

SL (Z),SL (R) : special linear groups over Z/R respectively.

U(n) : group of n x n unitary matrices;

U(n) = {u € Mn(C) | if1 = fcU }.

Sp(n,R) : the real symplectic matrices of degree n ; specifically,

Sp(n,R) = {M 6 M2n(R)| ^JM = J, J = _E

Q

n } .

Here E is the identity of matrices ring M (C).

Sp(n,Z) = Sp(n,R) 0 M2 (Z) : the discrete modular subgroup of

degree n.

r (N) : the principal congruence subgroup of level N of Sp(n,z)/

specifically,

Tn(N) = { M £ Sp(n,Z) | M = E2n( mod N ) } .

H : Siegel upper-half space of degree n ; specifically,

Hn = { Z 6 Mn(C)| tZ = Z, Im Z 0 } .

D : the generalized disc of degree n ; specifically,

Dn = { W € Mn(C) | fcW = W, E - W ^ 0}

[ S , U ] : element of Sp(n,R) of the form

L°EJ

U 0

o

V11

diag [ a,,a2,... ,a ] or [ a^a^ .. . ,aR] : the diagonal matrix

[ai-] with ai:L = ai (i= 1,2, ... ,n) and ai- = 0 (i ^ j).

VI