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

TL;DR

This paper investigates the structure and corank bounds of the fine Selmer group of an elliptic curve over a $Z_p$-extension of a number field, providing new insights into its algebraic properties.

Contribution

It establishes conditions under which the fine Selmer group is cofinitely generated and derives an upper bound for its corank based on local and global invariants.

Findings

Fine Selmer group is cofinitely generated over $Z_p$ under certain conditions.

Derived an upper bound for the corank (lambda-invariant) of the fine Selmer group.

Provides relations between the corank and local/global invariants of the elliptic curve.

Abstract

Let $p$ be an odd prime and $F_\infty$ 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 $F_\infty$. 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.