Cyclotomic and abelian points in backward orbits of rational functions

2022/03/18 by Ferraguti, Andrea, Ostafe, Alina, Zannier, Umberto · 5 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2203.10034

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 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∈ \mathbb Q(x) is a quadratic rational function not conjugate over \mathbb Q^ab to a power or a Chebyshev map and all preimages of α are abelian, we show that ϕ 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 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 Lattès map over a number field K and α∈ K for the whole backward orbit of α to only contain abelian points.

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