Equivariant Tamagawa Numbers, Fitting Ideals and Iwasawa Theory

Abstract

Let L/K be a finite Galois extension of number fields of group G. In [4] the second named author used complexes arising from étale cohomology of the constant sheaf $\mathbb Z$ to define a canonical element TΩ(L/K) of the relative algebraic K-group K₀($\mathbb Z$[G],$\mathbb R$). It was shown that the Stark and Strong Stark Conjectures for L/K can be reinterpreted in terms of TΩ(L/K), and that the Equivariant Tamagawa Number Conjecture for the $\mathbb Q$[G]-equivariant motive h⁰(Spec L) is equivalent to the vanishing of TΩ(L/K). In this paper we give a natural description of TΩ(L/K) in terms of finite G-modules and also, when G is Abelian, in terms of (first) Fitting ideals. By combining this description with techniques of Iwasawa theory we prove that TΩ(L/$\mathbb Q$) vanishes for an interesting class of Abelian extensions L/$\mathbb Q$.