Annihilating Wild Kernels

Abstract

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin $L$-series at $s=r$ to the compact support cohomology of the étale $p$-adic sheaf $\mathbb{Z}_p(r)$. We show that our conjecture is essentially equivalent to the $p$-part of the equivariant Tamagawa number conjecture for the pair $(h^0(\text{Spec}(L))(r),\mathbb{Z}[G])$. We derive from this explicit constraints on the Galois module structure of Banaszak's $p$-adic wild kernels.