2

JURGEN RITTER AND ALFRED WEISS

We regard K as a subfield of Qc, the algebraic closure of Q in C. We also fix an algebraic

closure

%c

of % and a Q-embedding s :

Qc

~^ Qf.

Let F C

Qc

be a finite Galois extension of Q with Galois group 0 = G(F/Q) and large enough

that every complex representation of G is realizable over F. Then 0 acts on the ring R(G)

of all complex characters % of (2 by

xa{9)

= {x{9)Y •

J

£ ^ 5 g £ G.

Let F/ C

Q/c

be a finite Galois extension of Qi with Galois group 0/ = G(Fi/Qi), containing

s(F) and large enough that every (^-representation of G is realizable over F[. Then 0/ acts

on the ring Ri(G) of all l-adic characters \ of G. Note that s : R(G) — Ri(G) is bijective.

We tend to regard this setting as fixed and, when there is no danger of confusion, we even

suppress I and S from the notation.

1.2 Stark's conjecture

Define the Dirichlet map

A : E - K g) A S , w t- - ] T log |tz|p 0 p

pes

as in [GRW2, §7). It induces an isomorphism R 0 £ - ^ M 0 AS of EG-modules, from which

the existence of G-monomorphisms (p : AS — • E follows. To such a ip we associate the

Stark-Tate regulator (see [Ta2])

R^ : R(G) - C , i ^ (

X

) = det( \p | Hom

cG

(V^, C 0 A S ) )

where V^ is a CG-module with character x(g) =

x(0~l)

• the contragredient character to X-

Writing Ls{s, x) f°

r

the Artin //-function with Euler factors at primes of k below S deleted,

we define cs(x) to be the leading coefficient in the Taylor series expansion of Ls(s,x) a t

s — 0, and then set

Ap(x) = Rp(x)/cs(x)-

C O N J E C T U R E (Stark). The homomorphism A^ is in Hom

0

(R(G),F

X

) .

Eventually we will assume that Stark's conjecture is true for K/k and all its intermediate

extensions. This is the case e.g. when K/Q is abelian.

1.3 Chinburg's Root Number Conjecture

If S is large, meaning that S contains all primes of K which are ramified with respect to k

and enough primes to generate the ideal class group of K, then there exists an exact sequence

of finitely generated ZG-modules, a Tate sequence,

E - A - B -» AS

in which A and B have finite projective dimension over TLG and B is Z-torsionfree. The

corresponding extension class r € E x t |

G

( A 5 , E) is called the Tate canonical class and has

been arithmetically constructed by Tate [Tal].

Chinburg [Chi] attaches to r the element

n = [A] - [B] e K0(ZG)