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.