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 Z p -extension F ∞ /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 ∞ /F . As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary Z p -extension has trivial Λcorank. We then derive an asymptotic growth formula for the ptorsion subgroup of the dual fine Selmer group in a 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 ptorsion 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.

Algebraic structure and characteristic ideals of fine Mordell–Weil groups and plus/minus Mordell–Weil groups

“…Note that unlike the case of Mordell-Weil groups studied in [Lee20], our control theorem is independent of the reduction type of E. Our proof relies on the corresponding control theorem for the fine Selmer groups proved by Lim [Lim20].…”

Structure of fine Selmer groups over -extensions

“…We begin reviewing the fine Selmer groups of abelian varieties following [3,4,12,28,31,[54][55][56]. Fix an odd prime p. Let A be an abelian variety defined over a number field F. Let S be a finite set of primes of F containing the primes above p, the bad reduction primes of A and the infinite primes.…”

“…It is not difficult to verify that the modules occurring in the exact sequence are finitely generated over Z p [[ ]] (for instance, see [28,Lemma 3•2]). In fact, one expects more (see [28,45,54]).…”

“…We record two results taken from [28] which serve as support for Conjecture Y, and will play some role in the subsequent discussion of the paper. THEOREM 3•3.…”

“…The aim of the paper is to consider the case of a -extension which is not the cyclotomic -extension. In this context, it has been conjectured that the fine Selmer group should be cotorsion over the ring , where denotes the Galois group of the -extension (see [ 28 , 45 , 54 ]). This will be called Conjecture Y in the paper.…”