Cyclotomic and abelian points in backward orbits of rational functions

We prove several results on backward orbits of rational functions over number fields. First, we show that if K is a number field, phi E K(x) and alpha E 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 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 E Q(x) is a quadratic rational function not conjugate over Qab to a power or a Chebyshev map and all preimages of alpha are abelian, we show that phi 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 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 K-Lattes map over a number field K and alpha E K, for the whole backward orbit of alpha to only contain abelian points. (c) 2023 Elsevier Inc. All rights reserved.