8 INTRODUCTION

jeJzeR(Cj,c)

V

fc=2 l^i,k^n,iy£k

7

" $ = C'(o, ( ^ J ,*(*)) - c'(o,^\z(z)) - c'(uf ,*(*)),

w ( c ) =

_IlogAT(aMf)^ £

C(0,A;

X )

,^)),

j€J*ei*(Cj,c)

XfcHGCcJ + VfcJ + W^c),

^(id,r ) = 7r-^)/

2

exp(-^X : £ £ E

w

(

c

»

f|f

w €

(G_)

f

' cecf

The class invariant X(c) depends on the choice of {Cj}jej and aM. We write it

as X(c; {Cj}jeJ»

a/u) a n

d #£r(id,r) as ^ ( i d , r; {Cj}

j e

j, {aM}) when we have to be

precise We have the decomposition

n

(19) ^ X ( c " ; { ^ }

j 6 J

, t t ? ) =

CF(0,C).

2 = 1

CONJECTURE B. pK(id,r) ~ gK{\&,r) for r G Gal(K/F).

We should like to stress the importance of the factorization process described

above; it suggests the possibility of further generalizations and also Conjecture B

is considerably deeper than Conjecture A.

We can test Conjecture B numerically using a theorem of Shimura. Let L be a

CM-field, 3 be a CM-type of L and q be an integral ideal of L. Let Iq(L) denote

the group of all fractional ideals of L relatively prime to q. Let A be a character of

7q (L) such that

\((a))=Y[(arp/\atr\)t~

if a = 1

modxq,

where t

a

, a G $ are non-negative integers; A is called a Grossencharacter of L. Let

m be an integer such that m = ta mod 2 and —ta m ^ ta for every a G $.

Then a theorem of Shimura states that

(20) L(m/2, A) ~

ir'2pL(£

ta • a, $),

where e = m[L : Q]/2 + E ^ G ^ **•

By virtue of this theorem, we can choose Grossencharacters Ai, A2, ... , Xq of

K and integers €1, €2, ... , eq, m so that

?

r

f [ L(m/2, A0Ci - TTA J J p ^ (id, a-V,^)^

2 = 1 2 = 1

holds with 1 G 2

_ 1

Z, ai, a2, •.., ar G Z, n , T2, ... , r

r

G Gal(if/F). Then we

expect