C. For a e Aut(C), we have
= c(a)
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
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 =
. ,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),
(22) V(C) =
J2ai lQS
€* e Et
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*
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.
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