**Somnath Jha**

TIFR

September 29, 2011

**Fine Selmer group of Hida deformations**:
Fine Selmer group of an elliptic curve is an arithmetic module which
is studied in Iwasawa theory. In this talk, we will study the fine
Selmer groups associated to modular forms and $\Lambda$-adic forms.
These modules are defined over a $p$-adic Lie extension of a number
field.
Inspired by some deep classical conjectures of Iwasawa and Greenberg,
Coates and Sujatha have proposed certain conjectures regarding the
structure of the fine Selmer group. We will formulate analogues of these conjectures in the setting of modular forms and also for $\Lambda$-adic forms. We will relate the structure of the `big' fine Selmer group of a $\Lambda$-adic form to the fine Selmer groups associated to the individual modular forms which are specializations of the $\Lambda$-adic form. We will also compare the usual Greenberg Selmer groups (resp. fine Selmer group) in a family of congruent modular forms associated to a $\Lambda$-adic form.