Abstract. 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 [C-S] 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.

In this paper, we study the fine Selmer group of p-adic Galois representations and their deformations. We show that for an infinite family of elliptic cuspforms, if the μ-invariant of the dual fine Selmer group is zero for one member of the family, then the same holds for all the other members. Further the λ-invariants are equal for all but finitely many members in the family.The fine Selmer group for elliptic curves has been studied for many years by various authors. In recent times, there has been a renewed interest with several conjectures being proposed (see [C-S, Wu,Ha]). In this article, we study the fine Selmer group for ordinary Hida deformations and extend results of [C-S] in this context. In particular, we formulate an analogue of Conjecture A of [C-S] for the fine Selmer group in this setting. We then study its relation to the corresponding conjecture for the ordinary representations associated to modular forms that arise from specializations. We prove the equivalence of these conjectures and investigate its invariance in a Hida family.Throughout the article, p will denote an odd prime integer and N a natural number prime to p.We will fix an embedding of an algebraic closureQ of the field Q of rational numbers, into C and also an embedding ofQ into a fixed algebraic closureQ p of the field Q p of the p-adic numbers. Let Σ be a finite set consisting of primes of Q dividing Np and denote by Q Σ the maximal algebraic extension of Q unramified outside Σ . The cyclotomic Z p -extension of Q is denoted by Q ∞ , and Γ is the Galois group Gal(Q ∞ /Q). For any Galois extension of fields L/F and a discrete module M over the Galois group Gal(L/F ), the Galois cohomology groups are denoted by H i (L/F , M).The article consists of five sections. Section 1 is preliminary in nature and we define the fine Selmer group for p-adic Galois representations and Hida deformations in Section 2, where we also formulate conjectures on its structure as an Iwasawa module. In Section 3, we prove a control theorem and apply it in Section 4 to specializations in order to study the relationship between these conjectures. Finally, in Section 5, we illustrate our results through concrete numerical examples. We

We establish a duality result proving the 'functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic Z p extension of a totally real number field. Further, we use this result to establish a duality or algebraic 'functional equation' for the 'big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the 'big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the Z 2 p extension of an imaginary quadratic field.

We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯ℱ/F cyc ) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤp extension of a general number field F, under the assumption that there is at least one arithmetic specialization whose Selmer group is torsion over its Iwasawa algebra. For a general number field, the two-variable cyclotomic Iwasawa main conjecture for ordinary Hida family is not proved and this can be thought of as an evidence to the validity of the Iwasawa main conjecture. The central idea of the proof is to prove a variant of the result of Perrin-Riou [Groupes de Selmer et accouplements; cas particulier des courbes elliptiques, Doc. Math.2003 (2003) 725–760, Extra Volume: Kazuya Kato's fiftieth birthday] by constructing a generalized pairing on the individual Selmer groups corresponding to the arithmetic points and make use of the appropriate specialization techniques of Ochiai [Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble)55(1) (2005) 113–146].

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