On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in $\mathbb{Z}_p$-extensions

TL;DR

This paper establishes a control theorem for the fine Selmer group of an abelian variety over any -extension, showing finiteness and boundedness of kernels and cokernels, and analyzes the growth of the fine Tate-Shafarevich group and related structures.

Contribution

It proves a general control theorem for the fine Selmer group without restrictions on reduction or ramification, and derives growth formulas for the fine Tate-Shafarevich group in -extensions.

Findings

Kernel and cokernel of restriction maps are finite and bounded.

Fine Tate-Shafarevich group has trivial -corank over -extensions.

Provided conditions for precise asymptotic growth formulas.

Abstract

Let $A$ be an abelian variety defined over a number field $F$. We prove a control theorem for the fine Selmer group of the abelian variety $A$ which essentially says that the kernel and cokernel of the natural restriction maps in a given $\mathbb{Z}_p$-extension $F_\infty/F$ are finite and bounded. We emphasise that our result does not have any constraints on the reduction of $A$ and the ramification of $F_\infty/F$. As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary $\mathbb{Z}_p$-extension has trivial $\Lambda$-corank. We then derive an asymptotic growth formula for the $p$-torsion subgroup of the dual fine Selmer group in a $\mathbb{Z}_p$-extension. However, as the fine Mordell-Weil group needs not be $p$-divisible in general, the fine Tate-Shafarevich group needs not agree with the $p$-torsion of the dual fine Selmer group, and so the asymptotic growth formula for the dual fine Selmer groups do not carry over to the fine Tate-Shafarevich groups. Nevertheless, we do provide certain sufficient conditions, where one can obtain a precise asymptotic formula.