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

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£(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

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^ ( 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

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L(o^y

= n 7 + g _

2

l o g |D*'

- 2 ^ ^ ;

-r-DM^O-iogll^O)-