On the Fine Selmer Group

TL;DR

This paper proves a conjecture that the fine p-infinity Selmer group of an elliptic curve over an anticyclotomic Z_p-extension is finitely generated, under certain hypotheses on the curve and prime.

Contribution

It establishes the finite generation of the fine Selmer group in a specific Iwasawa-theoretic setting, confirming a previous conjecture.

Findings

Proves the finite generation of the fine Selmer group under certain conditions.

Supports the conjecture with specific hypotheses on E and p.

Advances understanding of Selmer groups in Iwasawa theory.

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 $p$.