Fine Selmer group of Hida deformations over non-commutative $p$-adic Lie extensions

We study the Selmer group and the fine Selmer group of $p$-adic Galois representations defined over a non-commutative $p$-adic Lie extension and their Hida deformations. For the fine Selmer group, we generalize the pseudonullity conjecture of J. Coates and R. Sujatha, "Fine Selmer group of elliptic curves over $p$-adic Lie extensions," in this context and discuss its invariance in a branch of a Hida family. We relate the structure of the ‘big’ Selmer (resp. fine Selmer) group with the specialized individual Selmer (resp. fine Selmer) groups.

Keywords

Selmer group, congruences of modular forms, Hida theory, $p$-adic Galois representation, non-commutative Iwasawa theory

2010 Mathematics Subject Classification

Primary 11F33, 11F80, 11R23. Secondary 11G05, 14G05, 16E40.

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Published 6 April 2012