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

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 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 $\pi$-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.

36 pages. Some further minor corrections. The work of this paper generalizes the results in a previous article of the author in arXiv:1405.5289

References in corpus (3)

• Algebraic functional equations and completely faithful Selmer groups
• Comparing the Selmer group of a $p$-adic representation and the Selmer group of the Tate dual of the representation
• Pontryagin duality for Iwasawa modules and abelian varieties

Cited by in corpus (8)

• Fine Selmer groups of congruent Galois representations
• A note on asymptotic class number upper bounds in $p$-adic Lie extensions
• On the growth of even $K$-groups of rings of integers in $p$-adic Lie extensions
• Fine Selmer groups of congruent $p$-adic Galois representations
• Some remarks on Kida's formula when $\mu \ne 0$
• ${\mathfrak{M}}_H(G)$-property and congruence of Galois representations
• On the algebraic functional equation of the eigenspaces of mixed signed Selmer groups of elliptic curves with good reduction at primes above $p$
• On order of vanishing of characteristic elements