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# Asian Journal of Mathematics

## Volume 21 (2017)

### Number 6

### Comparing the $\pi$-primary submodules of the dual Selmer groups

Pages: 1153 – 1182

DOI: https://dx.doi.org/10.4310/AJM.2017.v21.n6.a7

#### Author

#### Abstract

In this paper, we compare the structure of Selmer groups of certain classes of Galois representations over an admissible $p$-adic Lie extension. Namely, we show that the $\pi$-primary submodules of the Pontryagin dual of the Selmer groups of two Galois representations have the same elementary representations when two Galois representations in question are either Tate dual to each other or are congruent to each other. In the first situation, our result gives a partial answer to the question of Greenberg on whether the Pontryagin dual of the Selmer groups of two Galois representations that are Tate dual to each other are pseudo-isomorphic (up to a twist of the Iwasawa algebra). In the second situation, our result will be applied to study the variation of the $\pi$-primary submodules of the dual Selmer groups of certain specialization of a big Galois representation. One of the important ingredient in our proofs is an asymptotic formula for $\pi$-primary modules over a noncommutative Iwasawa algebra which can be viewed as a generalization of a weak analog of the classical Iwasawa asymptotic formula.

#### Keywords

Selmer groups, $\pi$-primary submodules, elementary representations, admissible $p$-adic Lie extensions, asymptotic formula

#### 2010 Mathematics Subject Classification

11F80, 11R23, 11R34

Received 25 May 2016

Accepted 3 October 2016

Published 6 March 2018