Contemporary Mathematics

Volume 358, 2004

Rubin's Integral Refinement of

the Abelian Stark Conjecture

Cristian D. Popescu

To Harold Stark, on his 65th birthday.

ABSTRACT.

This paper is a survey of results obtained by the present author

towards proving Rubin's integral version of Stark's conjecture for abelian £-

functions of arbitrary order of vanishing at the origin. Rubin's conjecture is

stated and its links to the classical integral Stark conjecture for £-functions

of order of vanishing 1 are discussed. A weaker version of Rubin's conjecture

formulated by the author in [P4] is also stated and its links to Rubin's conjec-

ture are discussed. Evidence in support of the validity of Rubin's conjecture is

provided. A series of applications of Rubin's conjecture to the theory of Euler

Systems, groups of special units and Gras-type conjectures are given.

Introduction

In the 1970s and early 1980s, Stark [St] developed a remarkable Galois-equi-

variant conjectural link between the values at s

=

0 of the first non-vanishing

derivatives of the Artin £-functions LKjk(p, s) associated to a Galois extension

K/k

of number fields and a certain Q[Gal(K/k)]-module invariant associated to

the group of global units of

K.

Stark's Main Conjecture should be viewed as a vast

Galois-equivariant generalization of the unrefined, rational version of Dirichlet's

class-number formula

lim

~(k(s)

E Qx · Rk,

s-+0

sr

in which the zeta function (k is replaced by a Galois-equivariant £-function

eK/k,s(s) =

L

LK/k,s(p,

s).

ep'

pEG

with values in the center of the group-ring Z(C[Gal(K/k)]), the regulator Rk is

replaced by a Galois-equivariant regulator with values in Z(C[Gal(K/k)]), and the

1991 Mathematics Subject Classification. 11R42, 11R58, 11R27.

Key words and phmses. global £-functions, Units, Class Groups, Euler Systems.

The author was supported in part by NSF Grants DMS-0200543 and DMS-0350441.

©

2004 American Mathematical Society

http://dx.doi.org/10.1090/conm/358/06534