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

Abstract

Let \({\mathcal {O}}\) be the ring of integers of a finite extension of \({\mathbb {Q}}_p\). We prove two control theorems for fine Selmer groups of general cofinitely generated modules over \({\mathcal {O}}\). We apply these control theorems to compare the fine Selmer group attached to a modular form f over the cyclotomic \({\mathbb {Z}}_p\)-extension of \({\mathbb {Q}}\) to its counterpart attached to the conjugate modular form \({\overline{f}}\).