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

Abstract

Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a prime and let r ≤ 0 be an integer. By examining the structure of the p-adic group ring Zp[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h0(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 large class of interesting extensions, including cases in which the full ETNC is not known. Similarly, we construct annihilators of higher dimensional algebraic K-groups of the ring of integers in L.