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On the Arithmetic of Bicritical Rational Functions

Vefa Goksel, Rafe Jones

TL;DR

This work develops arithmetic foundations for bicritical rational maps (maps with exactly two critical points), establishing a field-theoretic normal form and precise constraints on the field of definition of the critical points. It shows that after a finite extension, the arboreal Galois representation attached to a bicritical map embeds into an iterated wreath product of cyclic groups, with stronger embeddings when $K$ contains a primitive $d$th root of unity. For quadratic maps with $\phi(\\gamma_1)=\\gamma_2$, the authors adapt Odoni-Stoll techniques to prove arboreal surjectivity for an infinite subfamily, including explicit normal forms such as $ (z^2+a)/z^2 $. A key component combines reduction, ramification, and rigid-divisibility methods to study irreducibility and growth of Galois groups of iterates, drawing connections to the classical unicritical and quadratic polynomial cases and broadening them to rational dynamics.

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 Möbius 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 $φ\in \mathbb{Q}(x)$ with critical points $γ_1$ and $γ_2$ such that $φ(γ_1) = γ_2$. Adapting methods of Odoni-Stoll in the polynomial case to rational functions, we show that the arboreal representation is surjective for an infinite subfamily.

On the Arithmetic of Bicritical Rational Functions

TL;DR

This work develops arithmetic foundations for bicritical rational maps (maps with exactly two critical points), establishing a field-theoretic normal form and precise constraints on the field of definition of the critical points. It shows that after a finite extension, the arboreal Galois representation attached to a bicritical map embeds into an iterated wreath product of cyclic groups, with stronger embeddings when contains a primitive th root of unity. For quadratic maps with , the authors adapt Odoni-Stoll techniques to prove arboreal surjectivity for an infinite subfamily, including explicit normal forms such as . A key component combines reduction, ramification, and rigid-divisibility methods to study irreducibility and growth of Galois groups of iterates, drawing connections to the classical unicritical and quadratic polynomial cases and broadening them to rational dynamics.

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 Möbius 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 with critical points and such that . Adapting methods of Odoni-Stoll in the polynomial case to rational functions, we show that the arboreal representation is surjective for an infinite subfamily.
Paper Structure (6 sections, 10 theorems, 22 equations)

This paper contains 6 sections, 10 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Background on reduction and ramification for rational functions
  3. Defining polynomials for iterates of a rational function
  4. Normalized form and reduction with respect to a non-archimedean absolute value
  5. Ramification and tame Riemann-Hurwitz
  6. Conjugacy and normal forms for bicritical rational functions

Key Result

Theorem 1.1

Let $K$ be a field of characteristic $\ell \geq 0$, and let $\phi \in K(z)$ be bicritical of degree $d \geq 2$ with critical points $\gamma_1, \gamma_2 \in \mathbb{P}^1(\overline{K})$. Assume that $\ell = 0$ or $\ell > d$. If $\phi(\{\gamma_1, \gamma_2\}) \neq \{\gamma_1, \gamma_2\}$, then $\phi$ is for some $a, b \in K(\gamma_1, \gamma_2)$ with $a \neq b$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4: Tame Riemann-Hurwitz for $\mathbb{P}^1$
  • ...and 8 more