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, ϕ∈K(x) and α∈K then the extension of K generated by the abelian points (i.e. points that generate an abelian extension of K) in the backward orbit of α is ramified only at finitely many primes. This has the immediate strong consequence that if all points in the backward orbit of α are abelian then ϕ is post-critically finite. We use this result to prove two facts: on the one hand, if ϕ∈Q(x) is a quadratic rational function not conjugate over Qab to a power or a Chebyshev map and all preimages of α are abelian, we show that ϕ is 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 α to only contain finitely many cyclotomic preimages, extending previous results of the second author. Finally, we give necessary and sufficient conditions for a triple (ϕ,K,α), where ϕ is a K-Lattès map over a number field K and α∈K, for the whole backward orbit of α to only contain abelian points.