Cyclotomic and abelian points in backward orbits of rational functions

Abstract

We prove several results on backward orbits of rational functions over number fields. First, we show that if $K$ is a number field, $\phi\in K(x)$ and $\alpha\in K$ then the extension of $K$ generated by the abelian points in the backward orbit of $\alpha$ is ramified only at finitely many primes. This has the immediate strong consequence that if all points in the backward orbit of $\alpha$ are abelian then $\phi$ is post-critically finite. We use this result to prove two facts: on the one hand, if $\phi\in \mathbb Q(x)$ is a quadratic rational function not conjugate over $\mathbb Q^{\text{ab}}$ to a power or a Chebyshev map and all preimages of $\alpha$ are abelian, we show that $\phi$ is $\mathbb Q$-conjugate to one of two specific quadratic functions, in the spirit of a recent conjecture of Andrews and Petsche. On the other hand we provide conditions on a quadratic rational function in $K(x)$ for the backward orbit of a point $\alpha$ to only contain finitely many cyclotomic preimages, extending previous results of the second author. Finally, we give necessary and sufficient conditions for a triple $(\phi,K,\alpha)$, where $\phi$ is a Latt\`es map over a number field $K$ and $\alpha\in K$ for the whole backward orbit of $\alpha$ to only contain abelian points.