论文标题
在$ \ mathbb {z} _p $ - extensions中的精细selmer组的控制定理和精美的泰特 - 萨法尔维奇组的增长
On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in $\mathbb{Z}_p$-extensions
论文作者
论文摘要
让$ a $是在数字字段$ f $上定义的Abelian品种。我们证明了Abelian品种的精细Selmer组的控制定理$ A $，从本质上说，在给定的$ \ Mathbb {z} _p $ extension $ f_ \ f_ \ infty/f $中，自然限制地图的内核和cokernel是有限且有限的。我们强调的是，我们的结果对减少$ a $和$ f_ \ infty/f $的后果没有任何限制。作为控制定理的第一个结果，我们表明，在任意$ \ mathbb {z} _p $ - extension上的精细泰特 - 萨法维奇组具有微不足道的$λ$ -Corank。然后，我们在$ \ Mathbb {z} _p $ - extension中为$ p $ torsion子组得出一个渐近增长公式。但是，由于一般不需要$ p $ - tate-shafarevich组不需要$ p $ - 不需要与双重塞尔默组的$ p $ torsion一致，因此，双重精美的selmer群体的渐近增长公式并不能够延续到精美的Tate-Shafarevich组。然而，我们确实提供了某些足够的条件，可以获得精确的渐近公式。
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.
