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# Asian Journal of Mathematics

## Volume 25 (2021)

### Number 4

### Twisting lemma for $\lambda$-adic modules

Pages: 551 – 564

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n4.a5

#### Authors

#### Abstract

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.

#### Keywords

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