On twists of modules over noncommutative Iwasawa algebras

It is well known that, for any finitely generated torsion module $M$ over the Iwasawa algebra $ℤ_{p} [[Γ]]$ , where $Γ$ is isomorphic to $ℤ_{p}$ , there exists a continuous $p$ -adic character $ρ$ of $Γ$ such that, for every open subgroup $U$ of $Γ$ , the group of $U$ -coinvariants $M {(ρ)}_{U}$ is finite; here $M (ρ)$ denotes the twist of $M$ by $ρ$ . This twisting lemma was already used to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a noncommutative generalization of this twisting lemma, replacing torsion modules over $ℤ_{p} [[Γ]]$ by certain torsion modules over $ℤ_{p} [[G]]$ with more general $p$ -adic Lie group $G$ . In a forthcoming article, this noncommutative twisting lemma will be used to prove the functional equation of Selmer groups of general $p$ -adic representations over certain $p$ -adic Lie extensions.

Keywords

Selmer group, noncommutative Iwasawa theory

Mathematical Subject Classification 2010

Primary: 11R23

Secondary: 16S50

Milestones

Received: 21 October 2015

Revised: 22 December 2015

Accepted: 1 February 2016

Published: 12 June 2016

Authors

Somnath Jha
Department of Mathematics and Statistics Indian Institute of Technology Kanpur Kanpur 208016 India

Tadashi Ochiai
Department of Mathematics Graduate School of Science Osaka University Machikaneyama 1-1 Toyonaka Osaka 5600043 Japan

Gergely Zábrádi
Department of Algebra and Number Theory Mathematical Institute, Eötvös Loránd University Bertalan Lajos utca 11 1111 Budapest Hungary

Vol. 10, No. 3, 2016

Somnath Jha, Tadashi Ochiai and Gergely Zábrádi

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors