This is a finite set. To Cj, we associate an n x r(j)-matrix Aj by Aj = (vV) (the
(i, /)-component of Aj is vV). Then we have
(11) CF(S, C) = JV(aMf)- J2 E ^ ^ *(*))•
For 1 ^ i ^ n, let A^- denote the i-th row of Aj. Now we have
(12) CH0,c ) = - £ X ) E^O'^-^^eQ,
j£J z£R(Cj,c) i=l
CF(O,C ) = E E EC'M
J )
^) )
jeJ z£R(Cj:c)
+^ E {c'(o,(^)),^))-c'(o,Af,x(.))-c'(o,Af,^))
The first formula gives a closed formula for CF(0, C) by (6) and the second formula
gives an expression of CF(0 C) by ^ n e multiple gamma function by (8).
Next let K be a CM-field of degree. 2n. Let p denote the complex conjugation.
We denote by JK the set of all isomorphisms of K into C and by IK the free abelian
group generated by JK- Shimura introduced the period symbol PK IK X IK —•
/ Q which is linear with respect to both of the arguments; when $ is a CM-
type of K, PK{&, &), a £ & coincides with a geometric period of an abelian variety
with complex multiplication. The period symbol PK plays a fundamental role in
the analysis of the behavior of arithmetic automorphic forms at CM-points. For
the convenience of notation, we regard
a s a n element of C x well defined
modulo Q . For a, b G C, we write a ~ b if b ^ 0 and a/b G Q.
Assume that K is normal over Q. Put G = Gal(Jf/Q). A representation rj
of G is called odd (resp. even) if rj(p) = —id (resp. rj(p) id). Let G (resp.
G-) denote the set of all equivalence classes of irreducible (resp. irreducible odd)
representations of G. When K is abelian over Q, we can show that
Here p(a) = 1 (resp. —1, resp. 0) if a 1 (resp. a p, resp. a ^ 1, p). This result
is essentially due to Anderson [An]. We give a proof in chapter III, §2. The idea is
to consider the factors of the jacobian variety of the Fermat curve
= 1. The
geometric periods of this factor can be expressed by the gamma function and we can
obtain sufficient information on p^(id, a) in terms of the gamma function6. When
This is the main reason of the success in the abelian case. For PK(&,T), K being a non-
abelian CM-field, we do not know any geometric objects which can supply a connection of CM-
periods to the multiple gamma function.
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