Cyclotomic and abelian points in backward orbits of rational functions

TL;DR

This paper investigates the properties of backward orbits of rational functions over number fields, focusing on abelian and cyclotomic points, and establishes finiteness and classification results related to these points.

Contribution

It proves finiteness of ramified primes in abelian backward orbits, characterizes quadratic functions with abelian preimages, and provides conditions for Lattès maps to have all abelian points in backward orbits.

Findings

Backward orbits with all abelian points generate extensions ramified at finitely many primes.

Quadratic rational functions not conjugate to powers or Chebyshev maps have specific forms if all preimages are abelian.

Conditions are given for Lattès maps to have all abelian points in backward orbits.

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.