2 INTRODUCTIO N

modular functions of two variables. This work of Hecke was

generalized3

to the

theory of complex multiplication of abelian varieties by Shimura-Taniyama-Weil

around 1955. By the work of Shimura during 1960-1980, we have now a good

perspective on the zeta functions and periods of abelian varieties with complex

multiplication together with their relations to modular forms (cf. [Shl2]). As a

related but a different approach to Hilbert's 12th problem, Hecke tried to generalize

Kronecker's limit formula to algebraic number fields of higher degree. He was

interested in the special function which appears in the constant term of the Laurent

expansion of the partial zeta function of an algebraic number field. He expected

that such a function would give an analogue of the 77-function, hence should play

a key role in constructing abelian extensions of a number field. He investigated

this problem from various points of view. We refer the reader to papers No. 3,

10, 15, 17, 20, 21 in his Werke. A part of Hecke's work is presented together with

Kronecker's work by the master hand of Siegel [Si2]. In 1970, Asai [As] published a

generalization of Kronecker's limit formula to an arbitrary algebraic number field.

Also in this year, Stark [Stl] published a conjecture on the general nature of L(l, %).

This conjecture was made very precise by the work of Stark-Shintani in 1970's.

Let us explain their research. Using the functional equation, we see that (1) is

equivalent to

(1') CF(S) ~ - ^ + " - \

s

^ 0 .

Wp

Let K be a Galois extension of F of finite degree. Put G = Gal(K/F) and let G

denote the set of all irreducible characters of G. We have

(4)

(K(S)=

Y[L(s,x,K/F)d™*

xec

where L(s, x K/F) denotes the Artin //-function associated to x« We set

L(s,X,K/F) =

c(x)sr^+0(sr^+1),

s^0

with c(x) £ C

x

and 0 ^ r(x) £ Z Stark defined a generalized regulator R(x) and

conjectured that

{

' \R(x)J R{x°)

for every a G Aut(C). Here Aut(C) denotes the automorphism group of C. The

generalized regulator gives a decomposition of RK\ hence (5) can be regarded as

a factorization of the right-hand side of

(1;)

according to (4). The generalized

regulator R(x)

*s

the determinant of a matrix of size r(x) whose entries are linear

combinations of the logarithms of the absolute values of units of K with algebraic

coefficients. Thus it was expected that the investigation of 1/(0, x) when r(x) = 1

would yield very interesting results on the units of K.

To explain the conjecture more explicitly, let F be a totally real algebraic

number field of degree n. Let O^ be the ring of all integers of F and Ep — Op be

the group of units of F. Let 001, 002, ... , oon denote all the infinite places of F.

Let K be an abelian extension of finite degree of F such that 001 is unramified and

3 This word is not completely precise. See the introduction of [Shi2] for comments on this

work of Hecke.