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arXiv 2006.16447math.NT
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关于细Selmer群的控制定理和细Tate-Shafarevich群在$\mathbb{Z}_p$ -扩张中的增长

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

Meng Fai Lim

备注：
20 pages; some minor changes and added one extra paragraph in the intro with some new references
发表信息：
Doc. Math. 25, 2445-2471 (2020)
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摘要：

设$A$是定义在数域$F$上的阿贝尔簇。本文证明了交换阿贝尔簇$A$的精细Selmer群的一个控制定理，它实质上是说在给定的$\mathbb{Z}_p$ -扩张$F_\infty/F$中自然限制映射的核和上核是有限有界的.我们强调，我们的结果没有任何限制的减少$A$和衍生物的$F_\infty/F$。作为控制定理的第一个结果，我们证明了任意$\mathbb{Z}_p$ -扩张上的精细Tate-Shafarevich群具有平凡的$Λ$ -corank。然后我们得到了对偶精细Selmer群在$\mathbb{Z}_p$ -扩张中的$p$ -扭子群的渐近增长公式.然而，由于精细Mordell-Weil群一般不需要是$p$ -可分的，精细Tate-Shafarevich群不需要与对偶精细Selmer群的$p$ -挠率一致，因此对偶精细Selmer群的渐近增长公式不适用于精细Tate-Shafarevich群。然而，我们确实提供了某些充分条件，在那里人们可以得到一个精确的渐近公式。

英文摘要：

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 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 $Λ$-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.

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