INTRODUCTION 7

K is an imaginary quadratic field of discriminant —d, (14) gives the Chowla-Selberg

formula:

d-i

£'(o,x),

(15) W ( i d , i d )

2

~ J J r ( ^ r ^ ) /

2

^ = d e x p ( ^ ^ ) ,

a=l d L ( 0 ' X )

where x

1S

the Dirichlet character corresponding to K. As a natural generalization

of (14), we conjecture

CONJECTURE

A. Let K be a CM-field normal over Q. Let c be a conjugacy

class of G. Then

a. )

n»^—"2n_«pf^m-

#ere /i(c) = 1 (resp. —1, resp. 0) if c = {1} (resp. c = {p}, resp. c ^ {1}, {p});

Xv is the character of n and we denote L(s,Xr),K/Q) as L(s,rj).

This conjecture is essentially equivalent to a conjecture of Colmez [Co]; the

present formulation is due to the author [Y3]. Conjecture A provides an expression

of r i a E c ^ O d '

a

)

m

terms of the multiple gamma function but not of the individual

£x(id, a). To get a reasonable formula on p^(id, cr), we must factorize the right-

hand side of (16) suitably.

Now let F be a totally real algebraic number field of degeree n and K be an

abelian extension of F of finite degree. We assume that K is a CM-field. Set

G = Gdl(K/F). Let Jp = {tf"i, 02,... , o~n} as before and we use the same letter o~i

for any of its extensions to JK- From Conjecture A, we can derive a relation

Hi ) nB(,,™,)~,-*»- n ' ^ f f | )

i=1

coed-

for r G G. By the properties of the period symbol PK- the left hand side mod Q

does not depend on the choice of c^. We also have

PK{O~I, TGI)

~

pxai

(id,

a^1rai).

Let fooioo2 • • • oon be the conductor of K as a class field over F , f being an integral

ideal of F. For an integral ideal f of F, let (G_)f denote the set of all characters of

G- whose conductors are equal to fooi(X)2 • • • oon. Then (17) can be written as

n

Y\pK"i (id,

o~XTOi)

(18)

i = 1

! , x

— ^ e

X

p

(

^ E E ^ £ ^ ( 0 , c ) ) .

flf "€(G-)

f

v cec,

The formula (13) for

£'F(0,C) n a s t n e

f °

r m a s

the sum of "conjugate terms"; in view

of this, we decompose £F(0 C) naturally. Regard F as a subfield of R and choose

o"i = id. For c G Q , we put

G(c)

= E E c'MVw),