On the Fine Selmer Group

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ an odd prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$. In a previous article the author conjectured that the fine $p^{\infty}$-Selmer group $R_{p^{\infty}}(E/K_{\infty})$ is confinitely generated over $\mathbb{Z}_p$. In this note we prove this conjecture assuming some hypotheses on $E$ and $p$.