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

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 $R\Gamma (N,T)$.

63 pages

Cited by in corpus (2)

• The epsilon constant conjecture for higher dimensional unramified twists of ${\mathbb{Z}}_p^r(1)$
• A representative of $R\Gamma (N,T)$ for higher dimensional twists of ${\mathbb{Z}}_p^r(1)$