A dynamical Shafarevich theorem for twists of rational morphisms

TL;DR

This paper proves that only finitely many twists of a given rational morphism over a number field have good reduction outside a finite set of places, confirming a conjecture of Silverman.

Contribution

It establishes a dynamical analogue of the Shafarevich theorem for twists of rational morphisms, showing finiteness under certain reduction conditions.

Findings

Finiteness of twists with good reduction outside S

Answers Silverman's question affirmatively

Extends Shafarevich-type results to dynamical systems

Abstract

Let $K$ denote a number field and a finite set $S$ of places of $K$ and $\phi:\PP^n\rightarrow\PP^n$ be rational morphism defined over $K$. The main result of this paper proves that there are only finitely many twists of $\phi$ defined over $K$ which have good reduction at all places outside $S$. This answers a question of Silverman in the affirmative.