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Examples of Quadratic Polynomials over $\mathbb{Q}$ with Surjective Arboreal Galois Representations

Luck Henderson, Jamie Juul, Brenner Lattin, Enrique Mercado, Mia Schaefer

Abstract

We explore families of pairs of quadratic polynomials $f(x)=x^2+c\in \mathbb{Q}$ and $a\in \mathbb{Q}$ with $a$ being a strictly preperiodic point of $f$ to provide infinitely many new examples for which the associated arboreal Galois representations are surjective.

Examples of Quadratic Polynomials over $\mathbb{Q}$ with Surjective Arboreal Galois Representations

Abstract

We explore families of pairs of quadratic polynomials and with being a strictly preperiodic point of to provide infinitely many new examples for which the associated arboreal Galois representations are surjective.

Paper Structure

This paper contains 7 sections, 13 theorems, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Families of quadratic polynomials with preperiodic points over $\mathbb{Q}$
  3. Repeated prime divisors in adjusted critical orbits
  4. Evaluating the signs of elements in the adjusted critical orbits
  5. Proofs of Main Theorems
  6. Proof of Theorem \ref{['thm:main1']}
  7. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1

Let $a\in \mathbb{Q}$, let $c=-a-a^2$, and let $f(x)=x^2+c$. Write $a=\frac{r}{s}$ where $r,s\in \mathbb{Z}$, $s>0$, and $\gcd(r,s)=1$. Define where $\beta =\frac{1}{3} (-2 + (19 - 3\sqrt{33})^\frac{1}{3} + (19 + 3\sqrt{33})^\frac{1}{3}) \approx 0.839$. Also define Suppose $a-c$ is not a square in $\mathbb{Q}$ and at least one of the following hold: Then the associated arboreal Galois represent

Figures (4)

  • Figure 1: Tree diagram for backward orbit of $a$.
  • Figure 2: Julia set of $f(x)=x^2-\frac{3}{4}$ approximated by backward orbit of $a=\frac{1}{2}$.
  • Figure 3: Orbit of $a$ for $f(x)=x^2+(-a-a^2)$.
  • Figure 4: Orbit of $a$ for $f(x)=x^2+(-1+a-a^2)$.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Example 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 17 more