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

摘要

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.