The equivariant local $\epsilon$-constant conjecture for unramified twists of $\mathbb{Z}_p(1)$

TL;DR

This paper proves the equivariant local epsilon constant conjecture for certain ramified abelian extensions by constructing explicit cohomological representatives, advancing understanding of local epsilon constants in number theory.

Contribution

It establishes the conjecture's validity for specific wildly ramified extensions and links it to Breuning's conjecture, using explicit cohomological constructions.

Findings

Proves the conjecture for certain wildly ramified abelian extensions.

Shows equivalence between Breuning's conjecture and the epsilon constant conjecture.

Constructs explicit representatives of cohomology complexes.

Abstract

We study the equivariant local epsilon constant conjecture, denoted by $C_{EP}^{na}(N/K,V)$, as formulated in various forms by Kato, Benois and Berger, Fukaya and Kato and others, for certain 1-dimensional twists $T=\mathbb{Z}_p(\chi^{nr})(1)$ of $\mathbb{Z}_p(1)$. Following ideas of recent work of Izychev and Venjakob we prove that for $T=\mathbb{Z}_p(1)$ a conjecture of Breuning is equivalent to $C_{EP}^{na}(N/K,V)$. As our main result we show the validity of $C_{EP}^{na}(N/K,V)$ for certain wildly and weakly ramified abelian extensions $N/K$. A crucial step in the proof is the construction of an explicit representative of $R\Gamma(N,T)$.