Structure of fine Selmer groups over $\mathbb{Z}_p$-extensions

TL;DR

This paper investigates the conjectured torsion property of fine Selmer groups over various $Z_p$-extensions, providing evidence and connections to existing conjectures, and extends the study to modular forms and broader $p$-adic Lie extensions.

Contribution

It offers new evidence supporting the torsion conjecture for fine Selmer groups over non-cyclotomic $Z_p$-extensions and relates these properties to other conjectures in Iwasawa theory.

Findings

Conjectural torsionness aligns with the pseudo-nullity conjecture of Coates-Sujatha.

If true for cyclotomic extensions, then it holds for almost all $Z_p$-extensions.

Torsionness of fine Selmer groups for modular forms relates to growth number conjecture.

Abstract

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_p$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_p$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates-Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_p$-extension, then it holds for almost all $\mathbb{Z}_p$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary $p$-adic Lie extension.