Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups

Abstract

Let $f:\mathbb{P}^1 \rightarrow \mathbb{P}^1$ be a quadratic rational map defined over the rational field $\mathbb Q$ with nonabelian automorphism group. We completely classify such maps that have $\mathbb Q$-rational periodic points of period $1$, $2$, and $3$. We then prove that no such map has a $\mathbb Q$-rational periodic point of exact period $4$ or $5$. We also show that if such a map has no $\mathbb Q$-rational periodic points of exact period exceeding $3$, then the number of its $\mathbb Q$-rational preperiodic points is at most $6$.

Theorems & Definitions (24)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Proposition 3.2
• Proposition 3.3