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

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.