Roots of unity and higher ramification in iterated extensions

Abstract:

Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in \mathbb {P}^1(K)$, we study the extension $K(\phi ^{-\infty }(b))$ generated by the union over $n$ of all solutions to $\phi ^n(x) = b$, where $\phi ^n$ is the $n$th iterate of $\phi$. We ask when a finite extension of $K(\phi ^{-\infty }(b))$ can contain all $m$-power roots of unity for some $m \geq 2$, and prove that this occurs for several families of rational functions. 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 $\phi$, especially when $\phi$ 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.