We regard K as a subfield of Qc, the algebraic closure of Q in C. We also fix an algebraic
of % and a Q-embedding s :
~^ Qf.
Let F C
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
= {x{9)Y
£ ^ 5 g £ G.
Let F/ 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
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 ^ (
) = det( \p | Hom
(V^, C 0 A S ) )
where V^ is a CG-module with character x(g) =
the contragredient character to X-
Writing Ls{s, x)
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
) .
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 |
( 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)
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