6

INTRODUCTION

This is a finite set. To Cj, we associate an n x r(j)-matrix Aj by Aj = (vV) (the

(i, /)-component of Aj is vV). Then we have

(11) CF(S, C) = JV(aMf)- J2 E ^ ^ *(*))•

For 1 ^ i ^ n, let A^- denote the i-th row of Aj. Now we have

1

n

(12) CH0,c ) = - £ X ) E^O'^-^^eQ,

j£J z£R(Cj,c) i=l

CF(O,C ) = E E EC'M

J )

^) )

jeJ z£R(Cj:c)

L

2=l

(13)

+^ E {c'(o,(^)),^))-c'(o,Af,x(.))-c'(o,Af,^))

-log^a^CfCO.c).

The first formula gives a closed formula for CF(0, C) by (6) and the second formula

gives an expression of CF(0 C) by ^ n e multiple gamma function by (8).

Next let K be a CM-field of degree. 2n. Let p denote the complex conjugation.

We denote by JK the set of all isomorphisms of K into C and by IK the free abelian

group generated by JK- Shimura introduced the period symbol PK • IK X IK —•

C

x

/ Q which is linear with respect to both of the arguments; when $ is a CM-

type of K, PK{&, &), a £ & coincides with a geometric period of an abelian variety

with complex multiplication. The period symbol PK plays a fundamental role in

the analysis of the behavior of arithmetic automorphic forms at CM-points. For

the convenience of notation, we regard

PK(€IW)

a s a n element of C x well defined

modulo Q . For a, b G C, we write a ~ b if b ^ 0 and a/b G Q.

Assume that K is normal over Q. Put G = Gal(Jf/Q). A representation rj

of G is called odd (resp. even) if rj(p) = —id (resp. rj(p) — id). Let G (resp.

G-) denote the set of all equivalence classes of irreducible (resp. irreducible odd)

representations of G. When K is abelian over Q, we can show that

Here p(a) = 1 (resp. —1, resp. 0) if a — 1 (resp. a — p, resp. a ^ 1, p). This result

is essentially due to Anderson [An]. We give a proof in chapter III, §2. The idea is

to consider the factors of the jacobian variety of the Fermat curve

xn

+

yn

= 1. The

geometric periods of this factor can be expressed by the gamma function and we can

obtain sufficient information on p^(id, a) in terms of the gamma function6. When

6

This is the main reason of the success in the abelian case. For PK(&,T), K being a non-

abelian CM-field, we do not know any geometric objects which can supply a connection of CM-

periods to the multiple gamma function.