10
INTRODUCTION
In chapter V, we will study the relation of Conjecture B with the limit formula
of Kronecker's type. For simplicity, we assume, in this introduction, that the class
number of F is 1. Let S) denote the complex upper half plane. Let a denote the set
of all archimedean places of F. For x [x\, x2,... , xn) G R+ and s G C, we put
xsa
= nr=i
xt- P u t
V = {QF® D
F
)\{(0,0)}. For z={zuz2,... zn) G £
n
, s G C,
we define a standard Eisenstein series E(z, s) by
E{z,s)
1^ y
(c,d)ev/EF
\cz + d\
-2sa
Here cz + d = (c^z, + d^\c^z2 + d2),... ,
c
W*
n
+ *»), y - (yi,y
2
,... ,2/n),
i/fc = 9(zjt). As a function of 5, E(z,s) can be continued meromorphically to the
whole s-plane, holomorphic except for the simple pole at s 1. In the usual
manner, SL(2,F) acts on S)n. We have E(iz,s) = E(z,s) for 7 G SL(2,D
F
). The
limit formula of Kronecker's type is
(23)
£(z,s) =
•yn 2__nni?f
7T
^
1
n l / 2
1
+ /i(z)-logy
a + 0 ( * - l ) ,
Here -fp is generalized Euler's constant denned by
CF(S)
=
2n~1RF 1
D
1/2
- + 7 F + 0 ( 5 - 1 ) ,
and h(z) is a certain real analytic function on
S)n
with the automorphic property
such that h(z)
logya
is invariant under SL(2,DF); V y{z) is the imaginary
part of z as above. The function h(z) is closely related to a holomorphic function
studied by Hecke which has a very complicated automorphic property (Werke No.
20, p. 398). If n - 1, we have
h(z) = -log\r,(z)\4
where rj(z) is the Dedekind //-function. Let K be a CM-field such that [K : F) 2.
Let x be the Hecke character of F which corresponds to the quadratic extension
K/F. Let L(s, x) denote the L-function attached to x- As a special case of Con-
jecture A, we have
(24)
^ ( T T M ) - ^ " EI PK{?,°)-
L(0,x)
CJ€JK
Let 2li, 2(.25 •, 21/IK be integral ideals of K representing ideal classes of K. We
write % =
OFUJ[I)
©
OFJ^
and put w{ =
LJ^/U^.
Choose a CM-type $; of K
so that 3(wf) 0 for every o G I2. Then we have
(25)
L(o^y
= n 7 + g _
2
l o g |D*'
- 2 ^ ^ ;
-r-DM^O-iogll^O)-
Previous Page Next Page