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

This paper classifies quadratic rational maps over $\mathbb{Q}$ with nonabelian automorphism groups that possess rational periodic points of periods 1, 2, and 3, proves the nonexistence of such points for periods 4 and 5, and establishes that the total number of rational preperiodic points is at most 6.

The Gist

Imagine you have a magical machine that takes a number, does some math to it, and spits out a new number. You take that new number, feed it back into the machine, and keep going forever. This is called a dynamical system.

In this paper, mathematicians Hasan Bilgili and Mohammad Sadek are studying a specific type of machine: a quadratic rational map. Think of this as a machine with a very specific, slightly complicated formula involving squares and fractions.

Here is the breakdown of their adventure, explained simply:

1. The Special Machines (The "Nonabelian" Group)

Most of these machines are boring; they have a simple symmetry (like a circle that looks the same if you flip it). But the authors are only interested in a very rare, special breed of machines that have a complex, "non-abelian" symmetry.

• The Analogy: Imagine a standard Rubik's Cube. If you twist it one way, it looks the same. That's simple symmetry. Now imagine a machine that only works if you twist it in a very specific, chaotic dance of moves. If you change the order of the moves, the result is different. This "chaotic dance" is what mathematicians call a non-abelian group (specifically, the $S_3$ group).
• The authors found that these special machines can be described by a specific formula with two knobs, $k$ and $d$.

2. The Game of "Catch Me If You Can" (Periodic Points)

The main game they play is: If you feed a rational number (a fraction like 1/2 or 3/4) into the machine, will it ever come back to where it started?

• Period 1 (Fixed Point): You put in 5, it spits out 5. It's stuck in a loop of length 1.
• Period 2: You put in 5, it spits out 10, then 10 spits out 5. It's a loop of length 2.
• Period 3: 5 $\to$ 10 $\to$ 15 $\to$ 5. A loop of length 3.

The authors asked: How long can these loops get?

3. The Great Discovery: The "No-Go" Zones

For decades, mathematicians have wondered if these loops can get infinitely long. The authors proved some very strict rules for their special machines:

• Loops of length 1, 2, and 3: These are possible! They found exactly how to tune the knobs ($k$ and $d$) to make these loops happen.
• Loops of length 4 and 5: Impossible. They proved mathematically that no matter how you tune the machine, you can never create a loop of 4 or 5 steps using rational numbers.
• Loops of length 6: It's highly unlikely. They proved there are only a finite number of ways to even try, and in almost all cases, it fails.
• Loops of length 7 or more: They strongly suspect these are impossible too (though they didn't prove it 100% for every single case, they proved it for 4 and 5, and showed 6 is rare).

The Metaphor: Imagine a roller coaster. The authors proved that for this specific type of track, the coaster can go in a small circle (1, 2, or 3 cars), but if you try to build a track that loops 4 or 5 times before returning, the track physically cannot be built with these specific materials (rational numbers).

4. The "Pre-periodic" Points (The Run-up)

Sometimes, a number doesn't start in a loop immediately. It might take a few steps to get into the loop.

• Example: 10 $\to$ 5 $\to$ 10 $\to$ 5...
  • Here, 10 is "pre-periodic." It takes one step to get to the loop (5 $\to$ 10).

The authors counted how many of these "run-up" numbers can exist.

• The Limit: If the machine doesn't have any loops longer than 3 steps, then the total number of special rational numbers (both the ones in loops and the ones running up to them) is at most 6.

5. The Big Picture (The Conjecture)

This work is part of a massive, decades-old mystery in math called the Uniform Boundedness Conjecture.

• The Big Question: Is there a universal limit to how many "special numbers" any machine of this type can have?
• The Authors' Contribution: They solved this puzzle for this specific, complex type of machine. They showed that for these machines, the answer is "Yes, there is a limit," and that limit is very small (6).

Summary in a Nutshell

The authors studied a specific, complex type of mathematical machine. They discovered that:

1. These machines can have short loops (1, 2, or 3 steps).
2. They cannot have loops of 4 or 5 steps.
3. They likely cannot have loops of 6 or more.
4. The total number of interesting numbers these machines can interact with is very small (no more than 6).

It's like finding out that a specific type of lock can only be opened with keys of length 1, 2, or 3, and trying to make a key of length 4 or 5 is a physical impossibility. This helps mathematicians understand the fundamental "rules of the universe" for how numbers behave when they are repeatedly transformed.

Technical Summary

Here is a detailed technical summary of the paper "Rational Preperiodic Points of Quadratic Rational Maps Over $\mathbb{Q}$ with Nonabelian Automorphism Groups" by Hasan Bilgili and Mohammad Sadek.

1. Problem Statement

The paper addresses the Uniform Boundedness Conjecture in arithmetic dynamics, proposed by Morton and Silverman. This conjecture posits that for a fixed degree $d$, dimension $n$, and number field $K$, there exists a uniform bound $C$ on the number of $K$-rational preperiodic points for any morphism $f: \mathbb{P}^n \to \mathbb{P}^n$.

While significant progress has been made for quadratic polynomials over $\mathbb{Q}$ (e.g., Poonen's conjecture that the maximum number of preperiodic points is 9, and the non-existence of rational periodic points of exact period $\ge 6$), the case for quadratic rational maps (degree 2 maps on $\mathbb{P}^1$) is more complex.

The authors specifically investigate quadratic rational maps defined over $\mathbb{Q}$ that possess a nonabelian automorphism group.

• A quadratic rational map has a non-trivial automorphism group if and only if it is conjugate to $k(z + 1/z)$.
• If $k \neq -1/2$, the automorphism group is cyclic ($C_2$).
• If $k = -1/2$, the automorphism group is the symmetric group $S_3$ (nonabelian, order 6).

The paper focuses on the $S_3$ case, aiming to classify rational periodic and preperiodic points and determine bounds on their total number.

2. Methodology

The authors utilize a combination of algebraic geometry, dynamical systems theory, and computational number theory.

• Normal Form Parametrization:
  They utilize a specific normal form for maps with $S_3$ symmetry. Any such map $f$ over a field $K$ (char $\neq 2, 3$) is $K$-conjugate to:
  $$\theta_{d,k}(z) = \frac{kz^2 - 2dz + dk}{z^2 - 2kz + d}$$
  where $k, d \in K$, $d \neq 0$, and $k^2 \neq d$.

• Dynatomic Polynomials:
  To find periodic points of exact period $n$, the authors compute the $n$-th dynatomic polynomial $\Phi^*_{f,n}(z)$. The roots of this polynomial correspond to points of formal period $n$.

  • For periods 1, 2, and 3, they derive explicit algebraic conditions on parameters $d$ and $k$.
  • For periods 4, 5, and 6, the existence of rational periodic points corresponds to the existence of rational points on specific algebraic curves defined by these polynomials.

• Algebraic Curve Analysis:
  The core of the non-existence proofs involves analyzing the algebraic curves $C_n$ defined by the vanishing of the dynatomic polynomials.

  • The authors use Magma and Mathematica to compute these curves.
  • They demonstrate that these curves are not geometrically irreducible. Instead, they decompose into multiple irreducible components defined over number fields (extensions of $\mathbb{Q}$).
  • By analyzing the intersection of these components over $\mathbb{Q}$, they show that the only rational points are singular points that do not correspond to valid quadratic rational maps.
  • For period 6, they invoke Faltings' Theorem (Mordell's Conjecture) on curves of genus $>1$ to prove that the number of rational points is finite.

• Preperiodic Classification:
  They analyze preperiodic points (points that eventually map to a cycle) by studying generalized dynatomic polynomials $\Phi^*_{f, m, n}$. They classify the possible directed graphs (orbits) formed by these points.

3. Key Contributions and Results

A. Classification of Periodic Points

The authors completely classify maps with rational periodic points of periods 1, 2, and 3:

• Period 1 (Fixed Points): They provide a complete parametrization of maps with 1, 2, or 3 rational fixed points.
• Period 2: A map has a rational 2-cycle if and only if $d$ is a rational square ($d=c^2$).
• Period 3: A map has a rational 3-cycle if and only if $-3d$ is a rational square. They provide explicit parametrizations for the parameters $k$ and $d$ in these cases.

B. Non-Existence of Higher Periods

The paper proves the following non-existence results for rational periodic points:

• Period 4 & 5: There are no quadratic rational maps with $S_3$ symmetry over $\mathbb{Q}$ possessing a rational periodic point of exact period 4 or 5.
  • Proof Strategy: The associated curves $C_4$ and $C_5$ decompose into components. The rational points on these components are shown to be only singular points (e.g., $(0:0:1)$), which do not yield valid maps.
• Period 6: There are at most finitely many such maps with a rational periodic point of exact period 6.
  • Proof Strategy: The curve $C_6$ has components of high genus (e.g., genus 433). By Faltings' Theorem, these have finitely many rational points.

C. Conjecture and Preperiodic Bounds

Based on the results, the authors propose Conjecture 1.2:

Let $f$ be a quadratic rational map over $\mathbb{Q}$ with $\text{Aut}(f) \cong S_3$. Then $f$ has no rational periodic point of exact period $N > 3$.

Assuming this conjecture holds, they prove Theorem 1.3:

If $f$ has no rational periodic points of exact period $\ge 6$, then the number of rational preperiodic points is at most 6.
$$|\text{PrePer}(f, \mathbb{Q})| \le 6$$

D. Graph Enumeration

The authors provide a complete list of all possible directed graphs (orbit structures) for the set of rational preperiodic points, assuming the conjecture. They provide explicit examples of $(d, k)$ pairs that realize each possible graph structure (e.g., maps with one fixed point and a 2-cycle, maps with three fixed points, etc.).

4. Significance

• Extension of Poonen's Work: This paper extends the well-studied case of quadratic polynomials to quadratic rational maps with a specific symmetry group ($S_3$). It mirrors the results for polynomials (no period 4 or 5) but establishes new bounds for rational maps.
• Sharp Bound: The bound of 6 preperiodic points is conjectured to be sharp, providing a concrete target for the Uniform Boundedness Conjecture in this specific family.
• Methodological Advancement: The paper demonstrates the power of combining explicit algebraic parametrization with computational algebraic geometry (decomposing curves over number fields) to solve dynamical problems. It shows that even when curves are not geometrically irreducible, their rational points can be effectively enumerated.
• Computational Verification: The authors provide a GitHub repository with Magma code verifying all computations, ensuring reproducibility of the complex polynomial factorizations and curve analyses.

Summary Conclusion

Bilgili and Sadek provide a comprehensive arithmetic dynamical analysis of quadratic rational maps with nonabelian ($S_3$) automorphism groups. They prove the non-existence of rational periodic points of periods 4 and 5, establish a finite bound for period 6, and conjecture that periods $>3$ do not exist. Under this conjecture, they prove that the maximum number of rational preperiodic points is 6, offering a complete classification of the possible orbit structures for this class of maps.