论文标题
统一的根和更高的迭代扩展分支
Roots of unity and higher ramification in iterated extensions
论文作者
论文摘要
给定一个字段$ k $，k（x）$中的有理函数$ ϕ \，以及一个$ b \ in \ mathbb {p}^1（k）$，我们研究扩展$ k（ϕ^{ - \ fty}（b）$，由$ n $ nit $ nit $ dis $ nit的$ n $ nit的$ n $ nit的$ n（ $ ϕ $。我们询问$ k（ϕ^{ - \ infty}（b））$的有限扩展可以包含所有$ m $ - 对某些$ m \ geq 2 $的Unity的根源，并证明几个有理功能的家族都这样做。一个激励的应用程序是了解$ k $是$ \ mathbb {q} _p $和$ p $的有限扩展时，较高的分支过滤会划分$ ϕ $的度，尤其是当$ ϕ $是$ ϕ $是严格的有限有限时（PCF）。我们表明，所有较高的分支群对于迭代扩展的新家族都是无限的，例如由具有周期性临界点的两位政治理性函数给出的。我们还提供了迭代扩展的新示例，其延伸范围满足了更强的分支理论条件，称为算术分配性。我们推测，由PCF映射引起的每个迭代扩展都应具有这种更强的属性的范围，这将为pcf地图提供SEN定理的动态类似物。
Given a field $K$, a rational function $ϕ\in K(x)$, and a point $b \in \mathbb{P}^1(K)$, we study the extension $K(ϕ^{-\infty}(b))$ generated by the union over $n$ of all solutions to $ϕ^n(x) = b$, where $ϕ^n$ is the $n$th iterate of $ϕ$. We ask when a finite extension of $K(ϕ^{-\infty}(b))$ can contain all $m$-power roots of unity for some $m \geq 2$, and prove that several families of rational functions do so. A motivating application is to understand the higher ramification filtration when $K$ is a finite extension of $\mathbb{Q}_p$ and $p$ divides the degree of $ϕ$, especially when $ϕ$ is post-critically finite (PCF). We show that all higher ramification groups are infinite for new families of iterated extensions, for example those given by bicritical rational functions with periodic critical points. We also give new examples of iterated extensions with subextensions satisfying an even stronger ramification-theoretic condition called arithmetic profiniteness. We conjecture that every iterated extension arising from a PCF map should have a subextension with this stronger property, which would give a dynamical analogue of Sen's theorem for PCF maps.
