8 INTRODUCTION

forms mentioned at the end of II.

IV. We take our setting to be (V, p) with V = U 0 FL © FK over a totally real

F under the condition that p has signature (1, n — 3) on U at every archimedean

prime. As explained above,

G+(V)

acts on a hermitian tube domain $), and so

we can define holomorphic automorphic forms and even can consider the Fourier

expansion of such a form similar to that of a Siegel modular form. Thus we take

a holomorphic Hecke eigenform f, for which Z(s, f, x)

c a n

be defined in the same

manner. Then we shall prove that this can be expressed as a finite linear combina-

tion of the form

(11) Z(s, f, X) = 5 ^ Z ( s - 1, 9i, X)

id

Lc(2s + 2 - n , ^x 2 ) ifne2Z ,

1 ifn^2Z .

Here the symbols are as follows: h is a fixed element of U; gi is a Hecke eigenform

on G + (iy), where W is the orthogonal complement of Fh in U; a^ £ C, 0

bij G R; Cj ( ) denotes the Fourier coefficients of a holomorphic automorphic form

associated with f; £ runs over the totally positive elements of F modulo a group

of units. For the precise definition of

^-k~su

the reader is referred to (25.22). Our

result of (G) is applicable to Z(s, gi, x), an essential point. We can then derive

meromorphic continuation of Z(s, f, x) from (11) (Theorem 25.10). It should be

noted that if n — 5 and F = Q, it can happen that G^(V) is the group of

similitudes of an alternating form of degree 4 which has 5p(2, Q) as a subgroup,

as already mentioned in connection with (6). In such a case, the Cj become the

Fourier coefficients of a Siegel modular form of degree 2. When the form belongs to

Sp(2, Z) Andrianov proved a formula of type (11) in his well known paper [And].

Our methods are far less computational than his.

Though the book is primarily intended for the treatment of these new theories

thus far explained, we expended considerable efforts in presenting relevant exposi-

tory material in clear and readable forms; we also included some basic results which

may be essentially known, but cannot easily be found in the existing literature.

• $ ( ^ "

f c

~

( f l + 2 - n )

'