The equivariant local $\varepsilon $-constant conjecture for unramified twists of $\mathbb Z_p(1)$

Werner Bley¹ ; Alessandro Cobbe²

¹ Mathematisches Institut der Universität München Theresienstr. 39 D-80333 München, Germany
² Institut für Theoretische Informatik, Mathematik und Operations Research Fakultät für Informatik Universität der Bundeswehr München Werner-Heisenberg-Weg 39 D-85577 Neubiberg, Germany

Acta Arithmetica, Tome 178 (2017) no. 4, pp. 313-383.

Résumé

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 $R\varGamma(N, T)$.

Détail

DOI : 10.4064/aa8567-10-2016

Keywords: finite galois extension p adic number fields study equivariant local varepsilon constant conjecture denoted formulated various forms kato benois berger fukaya kato others certain dimensional twists mathbb chi mathbb otimes mathbb mathbb following ideas recent work izychev venjakob prove mathbb conjecture breuning equivalent main result validity certain wildly weakly ramified abelian extensions crucial step proof construction explicit representative vargamma