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 )
'
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