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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 thestructure of the p-adic group ring Z_p[G], we prove many new cases of thep-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 otherconjectures concerning the vanishing of certain elements in relative algebraicK-groups. We then prove a conjecture of Burns concerning the annihilation ofclass groups as Galois modules for a wide class of interesting extensions,including cases in which the full ETNC in not known. Similarly, we constructannihilators of higher dimensional algebraic K-groups of the ring of integersin L.

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