Fine Selmer groups of congruent Galois representations

Abstract

In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so is the other. Our results also compare the $\pi$-primary submodules of the two dual fine Selmer groups. We then apply our results to compare the structure of Galois group of the maximal abelian unramified pro-$p$ extension of an admissible $p$-adic Lie extension and the structure of the dual fine Selmer group over the said admissible $p$-adic Lie extension.

22 pages. This version supercedes and improves on the earlier one. Added new author (Ramdorai Sujatha)

References in corpus (5)

• Notes on the fine Selmer groups
• The growth of fine Selmer groups
• A remark on the ${\mathfrak{M}}_H(G)$-conjecture and Akashi series
• Comparing the $\pi$-primary submodules of the dual Selmer groups
• On the pseudo-nullity of the dual fine Selmer groups

Cited by in corpus (7)

• Iwasawa Invariants for Symmetric Square Representations
• Fine Selmer groups of congruent $p$-adic Galois representations
• Euler Characteristics and their Congruences in the Positive Rank Setting
• Structure of fine Selmer groups over ${\mathbb{Z}}_p$-extensions
• On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in ${\mathbb{Z}}_p$-extensions
• On the $\mu$ equals zero conjecture for the fine Selmer group in Iwasawa theory
• Estimating class numbers over metabelian extensions