Galois theory of quadratic rational functions

TL;DR

This paper investigates the Galois action on quadratic rational functions over number fields, providing criteria for maximal Galois image size based on critical orbit arithmetic and extending results to functions commuting with Mobius transformations.

Contribution

It introduces a new criterion for the Galois image to be maximal in quadratic rational dynamics, especially when the function commutes with a Mobius transformation, and proves finite-index results.

Findings

Criterion for maximal Galois image based on critical orbit discriminants

Extension of maximality criteria to functions commuting with Mobius transformations

Finite-index Galois image results in specific cases

Abstract

For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a non-trivial Mobius transfomation. In a sense this is a dynamical systems analogue to the l-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of phi, we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of phi. In the case where phi commutes with a non-trivial Mobius transfomation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.