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

Abstract

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 fields with Galois group G, where K is a global number field and G is a p-adic Lie group of dimension 1 for an odd prime p. We attach to each finite 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 validity of the EIMC and the vanishing of the Iwasawa μ-invariant, we compute Fitting invariants of certain Iwasawa modules, and we show that this implies the minus part of the ETNC at p for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above p, provided that (an appropriate p-part of) the integrality conjecture holds.