Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups
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| Format: | Preprint |
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2026
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| _version_ | 1866914399313199104 |
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| author | Bilgili, Hasan Sadek, Mohammad |
| author_facet | Bilgili, Hasan Sadek, Mohammad |
| contents | 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 prove that no such map has a $\mathbb{Q}$-rational periodic point with exact period $N\ge 4$. We also give an explicit parametrization of such maps that have $\mathbb{Q}$-rational periodic points of period $1$, $2$, and $3$. Consequently, we show that the number of $\mathbb{Q}$-rational preperiodic points of such a map is at most $6$; establishing Morton-Silverman Uniform Boundedness Conjecture for this family of quadratic rational maps. As a result, we completely classify all portraits of $\mathbb{Q}$-rational preperiodic points for such $f$ showing that there are exactly $5$ such portraits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06203 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups Bilgili, Hasan Sadek, Mohammad Number Theory Dynamical Systems 37P05, 37P15 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 prove that no such map has a $\mathbb{Q}$-rational periodic point with exact period $N\ge 4$. We also give an explicit parametrization of such maps that have $\mathbb{Q}$-rational periodic points of period $1$, $2$, and $3$. Consequently, we show that the number of $\mathbb{Q}$-rational preperiodic points of such a map is at most $6$; establishing Morton-Silverman Uniform Boundedness Conjecture for this family of quadratic rational maps. As a result, we completely classify all portraits of $\mathbb{Q}$-rational preperiodic points for such $f$ showing that there are exactly $5$ such portraits. |
| title | Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups |
| topic | Number Theory Dynamical Systems 37P05, 37P15 |
| url | https://arxiv.org/abs/2603.06203 |