Fields of definition of dynamical systems on $\mathbb{P}^{1}$. Improvements on a result of Silverman
Number Theory
2024-05-13 v2 Algebraic Geometry
Dynamical Systems
Abstract
J. Silverman proved that a dynamical system on descends to the field of moduli if it is polynomial or it has even degree, but for non-polynomial ones of odd degree the picture is less clear. We give a complete characterization of which dynamical systems over descend to the field of moduli.
Keywords
Cite
@article{arxiv.2405.03612,
title = {Fields of definition of dynamical systems on $\mathbb{P}^{1}$. Improvements on a result of Silverman},
author = {Giulio Bresciani},
journal= {arXiv preprint arXiv:2405.03612},
year = {2024}
}
Comments
There is a mistake in the proof of the main theorem. In the middle of page 5, we state that \phi has equal orders of vanishing in 0 and \infty. However, the argument only proves the weaker statement that the ramification degrees are equal. This is not sufficient to conclude