INTRODUCTION 9

CONJECTURE

C. For a e Aut(C), we have

= c(a)

Y[UL{™/2,\°y

^Am=i9K^(id,a-^-1Tiaia;{CjnJ€J,{i,7})ai

with a root of unity C(°")-

We can show that the validity of Conjectures B and C does not depend on the

choice of {Cj}jej and {aM}. We have defined X(c) carefully to get this consistency.

Conjecture C gives more precise information than Conjecture B and we can use it

for the numerical test. Since ^ ( i d , r; {Cj}jej, {aM}) has at most n "conjugates"

under the action of Aut(C), we call it an absolute CM-period. In chapter III, §4, we

will discuss several convincing examples. We will see that there is a close relation

between £(a) and V(c).

The building block to construct ^x(id, r) is the class invariant X(c). The

formula (19) shows that the invariant X(c) contains all information about the Stark-

Shintani units besides CM-periods. By the detailed investigation in chapter III, §5,

we get the following insight about the nature of X(c). Let F be a real quadratic

field and Cf be the ideal class group modulo fooicx)2 with an integral ideal f of

F. Take v\, v2 G OF SO that v\ = 1 mod f, z/2 = 1 mod f, i/^ 0, v[* 0,

z/ 2 0, i^2 0. Let Si G Cf be the class of (z/$), i = 1, 2. Then we can prove that

X(c) +X(cs2) = a(c) loge with a(c) £ Q where e is the fundamental unit of F. On

the other hand, exp(—2(X(c) + X(csi)) is closely related to a Stark-Shintani unit.

Assuming the Stark-Shintani conjecture, we see that the transcendantal degree of

the field generated over Q by four quantities

exp(X(c)), exp(X(csi)), exp(X(cs2)), exp(X(csis2))

is at most

l.7

We feel that the approach in chapter III, §5 gives a hint for a possible

proof of the Stark-Shintani conjecture.

In chapter IV, we will first study geometric construction of a cone decompo-

sition (10). For example, we will show that for a suitable index finite subgroup

E of Ep in place of Ep, we can take Cj =

C(VJI,VJ2,

. • • ,vjr(j)) with Vji G E,

1 ^ i ^ r(j) for all j . As an application of this result, we will prove that

(21) W(c) = -^\ogN(afij)CF(0,c),

m

(22) V(C) =

J2ai lQS

^

Wlth

^

G

^'

€* e Et

i=l

where F is the normal closure of F over Q. We expect that F can be replaced by

F in (22); (21) is equivalent to E j e j E*

6

*

(

c„c) C^A^^iz)) e Q (cf. (12)).

7 As similar phenomenon where transcendental and algebraic invariants appear simultane-

ously, we remind the reader that the values of the 77-function at CM-points can be described by

CM-periods while a suitable quotient of them gives a unit; cf. also chapter III, the ending remark

of §5 for a simple example.