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

TL;DR

This paper advances the understanding of the equivariant Tamagawa number conjecture for Tate motives by proving new cases and applying methods to annihilate class groups and higher algebraic K-groups in Galois extensions.

Contribution

It establishes new cases of the p-part of the ETNC for Tate motives and proves a conjecture on class group annihilation for various Galois extensions.

Findings

Proved many new cases of the p-part of the ETNC.

Confirmed a conjecture of Burns on class group annihilation.

Constructed annihilators for higher algebraic K-groups.

Abstract

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.