Fields of definition of dynamical systems on $\mathbb{P}^{1}$. Improvements on a result of Silverman

TL;DR

This paper characterizes precisely when dynamical systems on the projective line descend to their field of moduli, extending Silverman's results to include non-polynomial systems of odd degree.

Contribution

It provides a complete characterization of descent conditions for all dynamical systems on P^1, improving Silverman's partial results.

Findings

Complete characterization of descent for all dynamical systems on P^1

Extension of Silverman's results to odd degree, non-polynomial cases

Clarification of the field of moduli descent criteria

Abstract

J. Silverman proved that a dynamical system on $\mathbb{P}^{1}$ 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 $\mathbb{P}^{1}$ descend to the field of moduli.