On the equivariant Tamagawa number conjecture in tame CM-extensions, II

We use the notion of non-commutative Fitting invariants to give a reformulation of the equivariant Iwasawa main conjecture (EIMC) attached to an extension F/K of totally real elds with Galois group G, where K is a global number eld and G is a p-adic Lie group of dimension 1 for an odd prime p. We attach to each nite Galois CM-extension L/K with Galois group G a module SKu(L/K) over the center of the group ring ZG which coincides with the Sinnott-Kurihara ideal if G is abelian. We state a conjecture on the integrality of SKu(L/K) which follows from the equivariant Tamagawa number conjecture (ETNC) in many cases, and is a theorem for abelian G. Assuming the vanishing of the Iwasawa μ-invariant, we compute Fitting invariants of certain Iwasawa modules via the EIMC, and we show that this implies the minus part of the ETNC at p for an in nite class of (non-abelian) Galois CM-extensions of number elds which are at most tamely rami ed above p, provided that (an appropriate p-part of) the integrality conjecture holds.