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),
Previous Page Next Page