Fine Selmer groups of congruent p -adic Galois representations

Abstract

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 \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 \infty /K$ , we also compare the Iwasawa $\lambda $ -invariants of the fine Selmer groups, even in situations where the $\mu $ -invariants are nonzero. Finally, we prove similar results for certain abelian non- p -extensions.

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 \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 \infty /K$ , we also compare the Iwasawa $\lambda $ -invariants of the fine Selmer groups, even in situations where the $\mu $ -invariants are nonzero. Finally, we prove similar results for certain abelian non- p -extensions.