Potential good reduction of degree 2 rational maps

TL;DR

This paper characterizes when degree two rational maps have potential good reduction over local fields, linking it to integral points in the moduli space, and provides algorithms for conjugation to maps with unit resultant.

Contribution

It offers a complete characterization of potential good reduction for degree two rational maps and introduces an algorithm for conjugation to maps with unit resultant.

Findings

Potential good reduction occurs exactly at integral points in the moduli space

An explicit algorithm for conjugating maps to have unit resultant

Results extend to fields over principal ideal domains and include specific cases over the rationals

Abstract

We give a complete characterization of degree two rational maps with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space. We detail an algorithm by which to conjugate any degree two rational map corresponding to an integral point in the moduli space into a map with unit resultant. The local fields result is used to solve the same problem for fields over a principal ideal domain. Some additional results are given for degree 2 rational maps over the rationals.