On the Control Theorem for Fine Selmer Groups and the Growth of Fine Tate-Shafarevich Groups in ${\mathbb{Z}}_p$-Extensions

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 an arbitrarily 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 need not be $p$-divisible in general, the fine Tate-Shafarevich group need 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.