Cyclotomic and abelian points in backward orbits of rational functions.(English)Zbl 1536.37089
Summary: 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 (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 \(\phi \in \mathbb{Q}(x)\) is a quadratic rational function not conjugate over \(\mathbb{Q}^{\mathrm{ab}}\) 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 ofJ. Andrews andC. Petsche [Algebra Number Theory 14, No. 7, 1981–1999 (2020;Zbl 1465.11218)]. 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 \(K\)-Lattès map over a number field \(K\) and \(\alpha \in K\), for the whole backward orbit of \(\alpha\) to only contain abelian points.
MSC:
| 37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |
| 37P15 | Dynamical systems over global ground fields |
| 11R18 | Cyclotomic extensions |
| 11R20 | Other abelian and metabelian extensions |
Citations:
Zbl 1465.11218Software:
MagmaCite
References:
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