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 etale cohomology of the constant sheaf ℤ to define a canonical element TΩ(L/K) of the relative 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 TΩ(L/K), and that the Equivariant Tamagawa Number Conjecture for the ℚ[G]-equivariant motive h 0(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/ℚ) vanishes for an interesting class of Abelian extensions L/ℚ.