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Comparing the $π$-primary submodules of the dual Selmer groups

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 $π$-primary submodules of the Pontryagin dual of the Selmer groups of two Galois representations have the same elementary representations when the 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 $π$-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 $π$-primary modules over a noncommutative Iwasawa algebra which can be viewed as a generalization of a weak analog of the classical Iwasawa asymptotic formula.