# A dynamical Shafarevich theorem for rational maps over number fields and function fields

## TL;DR

This paper establishes a finiteness theorem for rational maps over number and function fields with fixed degeneracies and good reduction outside a finite set of places, extending classical results to a dynamical setting.

## Contribution

It proves a dynamical Shafarevich theorem for rational maps over number and function fields, including auxiliary results on divisors and models, broadening the scope of finiteness theorems.

## Key findings

- Finiteness of isomorphism classes of rational maps with fixed degeneracies.
- Finiteness of reduced effective divisors with good reduction outside S.
- Existence of global models for rational maps.

## Abstract

We prove a dynamical Shafarevich theorem on the finiteness of the set of isomorphism classes of rational maps with fixed degeneracies. More precisely, fix an integer d at least 2 and let K be either a number field or the function field of a curve X over a field k, where k is of characteristic zero or p>2d-2 that is either algebraically closed or finite. Let S be a finite set of places of K. We prove the finiteness of the set of isomorphism classes of rational maps over K with a natural kind of good reduction outside of S. We also prove auxiliary results on finiteness of reduced effective divisors in $\mathbb{P}^1_K$ with good reduction outside of S and on the existence of global models for rational maps.