The equivariant local $\varepsilon $-constant conjecture for unramified twists of $\mathbb Z_p(1)$
Volume 178 / 2017
Abstract
Let $N/K$ be a finite Galois extension of $p$-adic number fields. We study the equivariant local $\varepsilon$-constant conjecture, denoted by $C^{\rm na}_{\rm EP}(N/K,V)$, as formulated in various forms by Kato–Benois–Berger, Fukaya–Kato and others, for certain $1$-dimensional twists $T = \mathbb{Z}_p(\chi^{\rm nr})(1)$ of $\mathbb{Z}(1)$ and $V=T\otimes_{\mathbb Z} \mathbb{Q}_p$. Following the ideas of recent work of Izychev and Venjakob we prove that for $T = \mathbb Z_p(1)$ a conjecture of Breuning is equivalent to $C^{\rm na}_{\rm EP}(N/K,V)$. As our main result we show the validity of $C^{\rm na}_{\rm EP}(N/K,V)$ for certain wildly and weakly ramified abelian extensions $N/K$. A crucial step in the proof is the construction of an explicit representative of $R\varGamma(N, T)$.
