# Annihilating wild kernels

## TL;DR

This paper proposes a conjecture linking special values of equivariant Artin L-series to étale cohomology, connecting it to the equivariant Tamagawa number conjecture and deriving constraints on wild kernels in number theory.

## Contribution

It formulates a new conjecture relating L-series values to cohomology and shows its equivalence to the p-part of the equivariant Tamagawa number conjecture, providing insights into wild kernels.

## Key findings

- Conjecture relates L-series values to étale cohomology.
- Equivalence established with the p-part of the Tamagawa number conjecture.
- Constraints on Galois module structure of wild kernels derived.

## 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 \'etale $p$-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 $(h^0(\mathrm{Spec}(L))(r), \mathbb Z[G])$. We derive from this explicit constraints on the Galois module structure of Banaszak's $p$-adic wild kernels.