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 $\overline{f}$.

16 pages; to appear in The Ramanujan Journal