Rational periodic points of quadratic rational maps with S3 automorphism group

### What is the conjecture

Let $f: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $2$ defined over $\mathbb{Q}$ with automorphism group $\mathrm{Aut}(f) \cong \mathfrak{S}_3$. Then $f$ has no rational periodic point of exact period $N > 3$.

Equivalently, every such map is conjugate over $\mathbb{Q}$ to a map of the normal form

$$f(z) = \frac{k z^2 - 2dz + dk}{z^2 - 2kz + d}$$

for some $k \in \mathbb{Q}$, $d \in \mathbb{Q} \setminus \{0\}$ with $k^2 \ne d$, and the conjecture states that no such map admits a $\mathbb{Q}$-rational periodic point of exact period $N \ge 4$.

**Context:** This is an instance of the Morton–Silverman Uniform Boundedness Conjecture in arithmetic dynamics (1994). It is the nonabelian-automorphism analogue of Poonen's conjecture (1998) for quadratic polynomials and Manes' conjecture (2008) for quadratic rational maps with $\mathrm{Aut}(f) \cong \mathbb{Z}/2\mathbb{Z}$. In the source paper, Bilgili and Sadek prove that periods $N = 4$ and $N = 5$ do not occur (Theorem 1.1(i)), and that at most finitely many such maps have periodic points of period $N = 6$ (Theorem 1.1(ii)), but the general case $N \ge 6$ remains open.

* **Source Paper:** *Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups* ([arXiv:2603.06203v1](https://arxiv.org/abs/2603.06203v1), [HTML](https://arxiv.org/html/2603.06203v1), [PDF](https://arxiv.org/pdf/2603.06203v1))
* **Authors:** Hasan Bilgili, Mohammad Sadek
* **In-Paper Location:** [Conjecture 1.2 (Page 2)](https://arxiv.org/pdf/2603.06203v1#page=2) in Section "Introduction"
* **OpenConjecture ID:** 980 ([OpenConjecture](https://github.com/davisrbr/conjectures-arxiv))

### Prerequisites needed

* **Rational functions:** [`Mathlib.FieldTheory.RatFunc.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/Basic.html)
* **Polynomials:** [`Mathlib.Algebra.Polynomial.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Basic.html)
* **Dynamical systems and periodic points:** [`Mathlib.Dynamics.PeriodicPts.Defs`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/PeriodicPts/Defs.html), [`Mathlib.Dynamics.FixedPoints.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/FixedPoints/Basic.html)
* **Permutations / symmetric group:** [`Mathlib.GroupTheory.Perm.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Basic.html)

### Choose either option
- [ ] I plan on adding this conjecture to the repository
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

This issue was generated by an AI agent and reviewed by me.