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
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