Galois theory of quadratic rational functions

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 $\alpha \in K$ under a degree-two rational function $\varphi \in K(x)$, with particular attention to the case when $\varphi$ commutes with a non-trivial Möbius transformation. In a sense this is a dynamical systems analogue to the $\ell$-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 $\varphi$, 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 $\varphi$. In the case where $\varphi$ commutes with a non-trivial Möbius transformation, 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.