Equivariant Tamagawa numbers, fitting ideals and Iwasawa theory

Let L/K be a finite Galois extension of number fields of group G. In [4] th e second named author used complexes arising from etale cohomology of the c onstant sheaf bb Z to define a canonical element T Omega (L/K) of the relat ive algebraic K-group K-0(Z[G], R). It was shown that the Stark and Strong Stark Conjectures for L/K can be reinterpreted in terms of T Omega (L/K), a nd that the Equivariant Tamagawa Number Conjecture for the bb Q[G]-equivari ant motive h(0)(Spec L) is equivalent to the vanishing of T Omega (L/K). In this paper we give a natural description of T Omega (L/K) in terms of fini te G-modules and also, when G is Abelian, in terms of (first) Fitting ideal s. By combining this description with techniques of Iwasawa theory we prove that T Omega (L/Q) vanishes for an interesting class of Abelian extensions L/Q.