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

Meng Fai Lim

Documenta mathematica, Tome 25 (2020), pp. 2445-2471

Résumé

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 Zp​-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 Zp​-extension has trivial Λ-corank. We then derive an asymptotic growth formula for the p-torsion subgroup of the dual fine Selmer group in a Zp​-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.

Détail

DOI : 10.4171/dm/803

Classification : 11G05, 11R23, 11S25
Mots-clés : fine Selmer groups, fine Tate-Shafarevich groups, fine Mordell-Weil groups, Zp​-extensions