INTRODUCTION

In number theory, various kinds of "class invariants" were introduced and stud-

ied. One of the most remarkable is the invariant studied by Stark-Shintani, which

is the exponential of the derivative at s — 0 of the partial zeta function of a class

c. This invariant is conjectured to give a unit of an abelian extension of a num-

ber field and should give an answer to Hilbert's 12th problem. The aim of this

book is to describe a more enlarged and coherent picture by introducing Shimura's

CM-periods into our consideration of class invariants.

To explain the contents of this book in a proper perspective, we start with

a brief historical survey on the subjects. Let F be an algebraic number field of

finite degree. One of the most famous formula in number theory, due to Dirichlet-

Dedekind, is the analytic class number

formula1:

s-»l WF\DF\ '

Here CF(S) is the Dedekind zeta function of F, r\ (resp. r2) denotes the number of

real (resp. imaginary) infinite places of F; hp, RF, Dp denote the class number,

the regulator, the discriminant of F respectively; wp denotes the number of roots of

unity contained in F. When K is an abelian extension of F, the calculation of the

relative class number hx/hp times

RK/RF

can be reduced to that of L(l,x) for

the non-trivial characters of Gal(K/F). In fact, Kummer performed the calculation

for the cyclotomic fields and derived many interesting results on class numbers and

units.

On the other hand, Kronecker applied his limit formula to the study of algebraic

number fields. Let F be an imaginary quadratic field, c be an ideal class of F and

(F(S, C) be the partial zeta function of the class c. His formula

gives2

(2)

C^ C)

= ^ P (^1

+

^

+

°

( S

"

1})

'

8^

(3) 7(c) = log *

| 1 / 2

+ 2

7

- log$X(w) - log

\rj(w)\4.

Here we take an ideal 21 in the class c and write 21 —

ZUJI

+ Za;2, 3(^1/^2) 0?

w = UJ1/U2; rj is the Dedekind eta function and 7 is Euler's constant.

Next let us briefly recall Hecke's work. In his dissertation, Hecke generalized

classical theory of complex multiplication to CM-fields of degree 4 using Hilbert

xcf. [Ded], III, p. 176.

2cf. [Kr], IV, p. 494-495 for his limit formula. In the form (2), (3), see [Ded], II, p. 227.

1