6 INTRODUCTION

(A) As to the arithmeticity of automorphic forms, we stated only (0.6) as a basic

requirement. However, there are other natural questions about arithmeticity whose

answers become necessary in various applications. Let us mention here only a few

facts we shall prove in this connection: (i) all automorphic forms are spanned by the

Q-rational forms; (ii) the group action (defined relative to a fixed weight) preserves

Q-rationality; (iii) in these statements Q can be replaced by a smaller field such as

Q or Qab if the group and the weight are of special types.

(B) In Sections 19 through 25 we give a detailed treatment of Z{s, f, x)

a n (

i

E(z, s; f, x) in the symplectic case, as well as in the case G =

G77.

These cases

were mentioned but not discussed in detail in the previous book [S97]. Also, in

the symplectic case we can define Z and E even with respect to a half-integral

weight, and we believe that the subject acquires the status of a complete theory

only when that case is included. Therefore in this book we treat both integral and

half-integral weights, and present the main results for both, though at a few points

the details of the proof for a half-integral weight are referred to some papers of the

author.

(C) We have spoken of a CM-point, which is naturally related to an abelian

variety with complex multiplication. Thus it is necessary to view r \ ^ as a space

parametrizing a family of abelian varieties. This will be discussed in Sections 4 and

6. The topic was treated in [S98], but we prove here something which was not fully

explained in that book. Namely, in Section 9, we prove the reciprocity-law for the

value of an automorphic function at a CM-point, when J T \ 3 ^ is associated with a

PEL-type.

(D) In the elliptic modular case it is well-known that the space of all holomor-

phic modular forms is the direct sum of the space of cusp forms and the space

of Eisenstein series. In Section 27 we prove several results of the same nature for

symplectic and unitary groups. For example, we show that the orthogonal comple-

ment of the space of cusp forms in the space of all holomorphic automorphic forms

is spanned by certain Eisenstein series, and the direct sum decomposition can be

done Q-rationally. This will be proven for the weights with which the series are

defined beyond the line of convergence.

(E) Though we are mainly interested in the higher-dimensional cases, in Section

18 we give an elementary theory of Eisenstein series in the Hilbert modular case,

which leads to arithmeticity results on the critical values of an //-function of a CM-

field. Also, in the Appendix we include some material of expository nature such

as theta functions of a quadratic form and the estimate of the Fourier coefficients

of a modular form. Many of them are well-known when the group is 5Z/2(Q) or

even Sp(ny Q) for some statements, but the researchers have often had difficulties

in finding references for the results on a more advanced level. Therefore we have

expended conscious efforts in treating such standard topics in a rather general

setting.