On the corank of the fine Selmer group of an elliptic curve over a ${\mathbb{Z}}_p$-extension

Let $p$ be an odd prime and $F_{\mathrm{\infty }}$ be a ${\mathbb{Z}}_p$-extension of a number field $F$. Given an elliptic curve $E$ over $F$, we study the structure of the fine Selmer group over $F_{\mathrm{\infty }}$. It is shown that under certain conditions, the fine Selmer group is a cofinitely generated module over ${\mathbb{Z}}_p$ and furthermore, we obtain an upper bound for its corank (i.e., the $\lambda$-invariant), in terms of various local and global invariants.