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 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

Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

Let L/K be a finite Galois extension of number fields with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well known Coates-Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen-Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h0(Spec(L))(r), where r is a strictly negative integer. In particular, this implies the ETNC for the pair (h0(Spec(L))(r),M), where L is totally real, r < 0 is odd and M is a maximal order containing Z[ 1/2 ]G, and will also provide some evidence for our conjecture

On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a wide class of interesting extensions, including cases in which the full ETNC in not known. Similarly, we construct annihilators of higher dimensional algebraic K-groups of the ring of integers in L.Comment: 33 pages, error in section 3.4 corrected. To appear in Transactions of the AM