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
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