Fine Selmer groups of congruent $p$-adic Galois representations

Abstract

We compare the Pontryagin duals of fine Selmer groups of two congruent $p$-adic Galois representations over admissible pro-$p$, $p$-adic Lie extensions $K_\infty$ of number fields $K$. We prove that in several natural settings the $\pi$-primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $\mu$-invariants. In the special case of a $\mathbb{Z}_p$-extension $K_\infty/K$, we also compare the Iwasawa $\lambda$-invariants of the fine Selmer groups, even in situations where the $\mu$-invariants are non-zero. Finally, we prove similar results for certain abelian non-$p$-extensions.