Asian Journal of Mathematics

Volume 25 (2021)

Number 4

Twisting lemma for $\lambda$-adic modules

Pages: 551 – 564



Sohan Ghosh (Department of Mathematics and Statistics, IIT Kanpur, India)

Somnath Jha (Department of Mathematics and Statistics, IIT Kanpur, India)

Sudhanshu Shekhar (School of Mathematics and Computer Science, IIT Goa, India)


A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb{Z}_p[[Γ]]$ with $\Gamma \cong \mathbb{Z}_p$, there exists a continuous character $\theta : \Gamma \to \mathbb{Z}^\times_p$ such that, the $\Gamma^{p^n}$‑Euler characteristic of the twist $M(\theta)$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra of a general compact $p$‑adic Lie group $G$. In this article, we consider a further generalization of the twisting lemma to $\mathcal{T}[[G]]$ modules, where $G$ is a compact $p$‑adic Lie group and $\mathcal{T}$ is a finite extension of $\mathbb{Z}_p[[X]]$. Such modules naturally occur in Hida theory. We also indicate arithmetic applications by considering the ‘big’ Selmer (respectively fine Selmer) group of a $\Lambda$‑adic form over a $p$‑adic Lie extension.


Iwasawa theory, Selmer groups, $\lambda$-adic form, $G$-Euler characteristic

2010 Mathematics Subject Classification

11G05, 11R23, 14Fxx

S. Jha acknowledges the support of a SERB MATRICS grant and of a SERB ECR grant.

S. Shekhar is supported by a DST INSPIRE faculty award grant.

Received 24 August 2020

Accepted 4 January 2021

Published 25 April 2022