On the Arithmetic of Bicritical Rational Functions

TL;DR

This paper develops foundational arithmetic and Galois-theoretic results for bicritical rational functions, including normal forms and properties of their arboreal Galois representations, with applications to specific quadratic families.

Contribution

It introduces new Galois-theoretic results for bicritical rational functions, including normal forms and injectivity of arboreal representations, extending polynomial methods to rational functions.

Findings

Arboreal Galois representation injects into an iterated wreath product of cyclic groups after finite extension.

Normal form for bicritical rational functions over their field of definition.

Surjectivity of arboreal representation for an infinite subfamily of quadratic rational functions.

Abstract

Bicritical rational functions -- those with precisely two critical points -- include the well-studied families of unicritical polynomials and quadratic rational functions. In this article we lay out general foundations for studying arithmetic dynamical properties of bicritical rational functions, and prove new Galois-theoretic results for a family with special properties. We study the field of definition of the critical points, and give a normal form up to M\"obius conjugacy over this field. As a corollary, we show that after a finite extension of the ground field, the arboreal Galois representation attached to a bicritical rational function injects into an iterated wreath product of cyclic groups. We then examine the family of quadratic $\phi \in \mathbb{Q}(x)$ with critical points $\gamma_1$ and $\gamma_2$ such that $\phi(\gamma_1) = \gamma_2$. Adapting methods of Odoni-Stoll in the polynomial case to rational functions, we show that the arboreal representation is surjective for an infinite subfamily.