INTRODUCTION 11
If F = Q, (25) gives
exp(^M ) ~ ^(if
I^K)|4)1/,lK
~ *PK(id,
id)2
since
\r](wi)\4
~
p^(id,id)2,
1 ^ i ^ /i^; thus the Chowla-Selberg formula (15)
follows.
Comparing (25) with (24), we should be able to obtain arithmetical information
of h(z) at CM-points. In chapter V, we will investigate this problem in great detail
taking account of the Stark conjecture; we will see the source of difficulty in the
case n ^ 2.
In appendix I, we will give a general theory of Eisenstein series on GL(2) over
an arbitrary global field. In particular, we will prove analytic continuation and the
functional equation of Eisenstein series.
In view of the simple form of Conjecture A and the factorization process of
£^(0, c) employed to define X(c), it is tempting to make similar constructions start-
ing from the second derivative
CF(0C)-
^n fac^
by (9)
a n
d (H) we can express
CF(0
c)
using a natural generalization of the multiple gamma function. In appendix
II, we will investigate the problem to define the class invariant X^2\c) for c G Cf
so that EHi x ( 2 ) ( c 7 i ) =
CF(° C )
h o l d s i n analogy to (19). Then we will define
a symbol (7^'(id, r) by a similar procedure to that defines 7#-(kI, r). We will end
(2)
with a speculation on a property of the symbol
gKK'.
In appendix III, we assemble a few remarks on the transcendence property of
the periods, for the convenience of the reader.
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