Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

Let O be the ring of integers of a finite extension of Q p . We prove two control theorems for fine Selmer groups of general cofinitely generated modules over O . We apply these control theorems to compare the fine Selmer group attached to a modular form f over the cyclotomic Z p -extension of Q to its counterpart attached to the conjugate modular form f ¯ .