Preface

Between 1971 and 1980, Harold Stark published a series of four papers in

Advances in Mathematics concerning the Artin £-functions L(s,

x)

associated to a

Galois extension of number fields

K / k.

In the general case, he conjectured that their

leading terms at s

=

0 (or, by the functional equation, at s

=

1) should be equal

to a certain 'x-regulator' of S-units of K times an algebraic constant depending

Gal(Ql/Q)-equivariantly on X· Under special conditions he made conjectures with

stronger consequences, e.g. a solution of Hilbert's 12th Problem in certain instances

where the conjecture is 'first order'.

These conjectures were rapidly taken up and developed by other number theo-

rists, including Chinburg, Deligne, Gross, Hayes, Serre, and Tate. The last-named

also gave a course on the subject at Orsay in 1980/81. Lecture notes written up

by Bernardi and Schappacher gave rise to Tate's 1984 book, Les conjectures de

Stark sur les fonctions L d'Artin en s

=

0, which collected together most of the

developments up to that date. In the years that followed, this became the standard

reference for researchers on Stark's Conjectures. Steady progress was made in the

area alongside- and often interacting with- advances on certain other '£-function

conjectures' then being actively investigated (to cite but three: those due to Bloch

and Kato and to Fontaine and Perrin-Riou, as well as the Main Conjecture of lwa-

sawa Theory). By the turn of the millennium, there was a clear need to take stock of

the 'state-of-the-art' in the area. With this in mind, the editors organised a confer-

ence on Stark's Conjectures and Related Topics, held at Johns Hopkins University

in Baltimore from 5 to 9 August 2002. The more than 60 participants included

almost all the experts in the field. At the conclusion of a very successful week there

was general enthusiasm for creating a permanent record of its proceedings.

This volume is the result.

It

does not claim to replace Tate's book as an

introduction to the Stark Conjectures. Nor, coming almost exactly twenty years

later, can it pretend to catalogue all the intervening work in the area. The ten

articles which it contains cannot even cover the great thematic diversity of the

more than thirty talks given at the conference. Nevertheless, almost all of the

authors were among the invited speakers, and most of their contributions expand

on the subject matter of their plenary talks.

It

is therefore the editors' belief that

the first four, 'survey-type' contributions provide an introduction to a good part

of the recent work relating to Stark's Conjectures. The remaining six should give

a taste of some major themes currently being explored in the area (non-abelian

and p-adic aspects of the conjectures, abelian refinements, etc.). At the same time,

we are acutely conscious of some major omissions, and of two in particular: we

have been unable to include Gross's p-adic refinement of the 'first order' abelian

Stark Conjecture and its recent generalizations to the 'arbitrary order' abelian Stark

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