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Arithmetic exceptionality of Lattès maps

This paper proves Odabaş's conjecture that the $k$ -th Lattès map attached to an elliptic curve $E/\mathbb{Q}$ is arithmetically exceptional if and only if $E$ lacks a rational $x$ -coordinate $k$ -torsion point for curves with complex multiplication by imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-11})$ , while demonstrating that the conjecture fails when $E$ has CM by $\mathbb{Q}(\sqrt{-11})$ and $6$ divides $k$ .

Original authors: Chatchawan Panraksa, Detchat Samart, Songpon Sriwongsa

Published 2026-03-27

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CC BY 4.0

Original authors: Chatchawan Panraksa, Detchat Samart, Songpon Sriwongsa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical machine called a Lattès Map. You feed it a number, and it spits out a new number. Now, imagine you take this machine and shrink it down to work inside a tiny, finite world made of just a few numbers (like a clock with only 5 hours, or a calendar with only 7 days).

The big question mathematicians are asking is: Does this machine shuffle the numbers perfectly?

If you put in every number from 0 to 6, does the machine give you back every number from 0 to 6 exactly once, with no duplicates and no missing numbers? If it does this for infinitely many different-sized worlds (different "clocks"), we call the machine "Arithmetically Exceptional." It's a special, rare machine that always plays fair.

The Big Guess (The Conjecture)

A mathematician named Odabaş made a bold guess about what makes these machines special. He said:

"A Lattès machine is 'Exceptional' (plays fair) if and only if the underlying blueprint (an Elliptic Curve) doesn't have any 'special points' with rational coordinates that get stuck at a specific spot."

Think of the Elliptic Curve as the engine that powers the machine. Odabaş guessed that if the engine has a specific type of "knot" (a torsion point) that is easy to describe with simple fractions, the machine will jam up and fail to shuffle the numbers perfectly. If the engine is "clean" of these knots, the machine works perfectly.

What This Paper Does

The authors of this paper, Chatchawan, Detchat, and Songpon, decided to test Odabaş's guess. They focused on a very specific, fancy type of engine called Complex Multiplication (CM). These engines have extra symmetry, like a snowflake or a perfect crystal.

Here is what they found, broken down simply:

1. The General Rule (Most Engines)

For almost all of these fancy crystal engines, Odabaş was right!

If the engine has no "knots" (rational torsion points), the machine shuffles perfectly.
If the engine does have a knot, the machine jams.
They proved this by looking at how the machine behaves on different "clocks" (prime numbers) and using deep number theory tools (like Chebotarev's Density Theorem, which is like a cosmic lottery that guarantees certain patterns will appear infinitely often).

2. The Big Exception (The "11" Engine)

However, they found one specific engine that breaks the rules. This is the engine associated with the number $\sqrt{-11}$ .

The Problem: For this specific engine, even if there are no "knots" (no rational points), the machine still jams if you try to shuffle it in groups of 6 (or multiples of 6).
Why? It's a hidden trap. The number 3 splits in a weird way in this engine's world, and the number 2 is always there. When you combine them (making 6), the machine always produces duplicates, no matter how clean the engine looks.
The Result: Odabaş's guess fails here. The machine is not exceptional for multiples of 6, even though the engine looks "clean."

3. The "Non-CM" Engines

The authors also looked at engines that don't have that fancy crystal symmetry (non-CM). They found a few more examples where the machine jams for multiples of 6, even when the engine looks clean. This suggests that the rule might need a small update: "The machine works perfectly for any group size except multiples of 6, unless the engine has a specific knot."

The Analogy of the "Jamming Machine"

Imagine you are a DJ with a playlist of songs (numbers).

The Goal: You want to play every song exactly once in a loop.
The Engine (Elliptic Curve): This is your music library.
The Knot (Torsion Point): This is a song that is "broken" or "stuck." If your library has a stuck song, your DJ machine will skip it or play it twice, ruining the shuffle.
The Exception (The $\sqrt{-11}$ case): Even if your library is perfect and has no broken songs, if you try to play the playlist in a loop of 6 songs, the machine's internal gears (the math of the number 11) will always cause a skip. It's a mechanical flaw in the universe for that specific number.

Why Does This Matter?

This isn't just about shuffling numbers. It's about understanding the deep connection between geometry (the shape of the curve), algebra (the numbers), and dynamics (how things move and change).

For Cryptography: These maps are used to build secure codes. Knowing exactly when they shuffle perfectly (and when they don't) is crucial for safety.
For Math Theory: It helps mathematicians understand the "DNA" of numbers. The fact that the number 6 is a special "obstruction" for the number 11 is a beautiful, surprising discovery that adds a new piece to the puzzle of how numbers behave.

The Bottom Line

The authors proved that for most fancy mathematical engines, a simple rule predicts if a shuffling machine will work. But they also discovered a unique "bug" in the system for the number 11 when dealing with groups of 6. They didn't just say "Odabaş was right"; they said, "Odabaş was right, except for this one tricky case, and here is exactly why."

They even proposed a new, slightly more careful guess (Conjecture 20) that accounts for this "6" problem, inviting other mathematicians to solve the rest of the puzzle.

1. Problem Statement

The paper investigates the arithmetic exceptionality of Lattès maps.

Definition: A rational function $r(x) \in \mathbb{Q}(x)$ is arithmetically exceptional if it induces a permutation (bijection) on the projective line $\mathbb{P}^1(\mathbb{F}_p)$ for infinitely many primes $p$ .
Context: Lattès maps $L_k$ are rational functions derived from an elliptic curve $E/\mathbb{Q}$ and an integer $k \geq 1$ . They satisfy the commutative diagram $\text{proj} \circ [k] = L_k \circ \text{proj}$ , where $[k]$ is the multiplication-by- $k$ map on $E$ and $\text{proj}$ is the projection to the $x$ -coordinate.
The Conjecture: Based on computational evidence, Odabaş conjectured that for any $k \in \mathbb{N}$ , the $k$ -th Lattès map $L_k$ attached to $E/\mathbb{Q}$ is arithmetically exceptional if and only if $E$ possesses no $k$ -torsion point whose $x$ -coordinate is rational.
Goal: The authors aim to prove or disprove this conjecture, specifically focusing on elliptic curves with Complex Multiplication (CM) and identifying potential counterexamples.

2. Methodology

The authors employ a combination of arithmetic geometry, algebraic number theory, and computational verification.

Permutation Criterion: They utilize a known criterion (Proposition 3) stating that $L_k$ permutes $\mathbb{P}^1(\mathbb{F}_p)$ if and only if $\gcd(A_p(E), k) = 1$ , where $A_p(E) = (p+1)^2 - a_p^2 = |E(\mathbb{F}_p)| \cdot |E'(\mathbb{F}_p)|$ (with $E'$ being a quadratic twist).
CM Theory & Frobenius Traces: For CM curves, they leverage Deuring's results on Frobenius traces $a_p$ $a_{p}$ . Specifically, $a_p$ $a_{p}$ depends on whether $p$ $p$ splits or is inert in the CM field $K$ $K$ .
- If $p$ is inert, $a_p = 0$ , so $A_p(E) = (p+1)^2$ .
- If $p$ splits, $a_p = \pi + \bar{\pi}$ , where $\pi$ is a prime in the CM order.
Reciprocity Laws: To construct primes $p$ $p$ satisfying specific congruence conditions (ensuring $\gcd(A_p(E), k) = 1$ $g cd (A_{p} (E), k) = 1$ ), the authors use:
- Dirichlet's Theorem on arithmetic progressions.
- Chebotarev Density Theorem to find splitting primes with specific residue properties.
- Cubic and Sextic Reciprocity Laws (specifically for the field $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{-11})$ ) to control the divisibility of $A_p(E)$ by small primes like 2, 3, and 7.
Case-by-Case Analysis: The proof is structured by the discriminant $D$ of the CM order, handling class number 1 fields ( $D \in \{-3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163\}$ ) individually.

3. Key Contributions and Results

A. Proof for Most CM Fields (Theorem 2)

The authors prove Odabaş's conjecture is true for all CM elliptic curves $E/\mathbb{Q}$ with CM by an imaginary quadratic field $K$ , except when $K = \mathbb{Q}(\sqrt{-11})$ .

Strategy: For a given $k$ (assuming no rational $k$ -torsion $x$ -coordinate), they construct an infinite set of primes $p$ such that $\gcd(A_p(E), k) = 1$ .
Mechanism:
- For most discriminants, they choose inert primes where $A_p(E) = (p+1)^2$ and select $p$ such that $p \equiv -1 \pmod{\ell}$ for prime factors $\ell$ of $k$ .
- For discriminants involving $\mathbb{Q}(\sqrt{-3})$ (e.g., $D=-3, -12, -27$ ), where 2 and 3 divide $A_p(E)$ for inert primes, they use splitting primes and sextic reciprocity to ensure $A_p(E)$ is coprime to $k$ .
- They explicitly handle cases where $E$ has rational torsion (e.g., $C_2, C_6$ ), showing the conjecture holds only when $k$ is coprime to the order of the torsion subgroup.

B. The Exceptional Case: $K = \mathbb{Q}(\sqrt{-11})$

The paper identifies a genuine obstruction for the field $K = \mathbb{Q}(\sqrt{-11})$ when $6 \mid k$ .

Proposition 11: If $E$ has CM by $\mathbb{Q}(\sqrt{-11})$ and $6 \mid k$ , then $L_k$ is never arithmetically exceptional, regardless of whether $E$ has rational $k$ -torsion points.
Reasoning:
- For inert primes $p$ , $A_p(E) = (p+1)^2$ is even (divisible by 2).
- For split primes $p$ , Lemma 10 proves that $3 \mid A_p(E)$ because 3 splits in $\mathbb{Q}(\sqrt{-11})$ and forces a specific residue structure on the Frobenius element.
- Since $6 \mid k$ , $\gcd(A_p(E), k)$ is always at least 2 or 3. Thus, $L_k$ fails to permute $\mathbb{P}^1(\mathbb{F}_p)$ for all but finitely many primes.
Proposition 12: If $6 \nmid k$ , the conjecture holds for this field.

C. Counterexamples for Non-CM Curves

The authors provide explicit counterexamples to the original conjecture for non-CM curves, suggesting the conjecture needs refinement.

Proposition 18: They construct a family of non-CM curves $E_u: y^2 = x^3 - 9u^2x + 12u^3$ $E_{u} : y^{2} = x^{3} - 9 u^{2} x + 12 u^{3}$ .
- These curves have no rational $k$ -torsion points with rational $x$ -coordinates for $6 \mid k$ .
- However, for every prime $p$ of good reduction, $\gcd(A_p(E_u), k) \neq 1$ (specifically, 2 divides it if 3 is a cubic residue mod $p$ , and 3 divides it otherwise).
- Thus, $L_k$ is not arithmetically exceptional, disproving the "if and only if" nature of Odabaş's conjecture for $6 \mid k$ in the non-CM case.

D. Reformulated Conjecture (Conjecture 20)

Based on these findings, the authors propose a refined conjecture:

For any positive integer $k$ such that $6 \nmid k$ , the $k$ -th Lattès map $L_k$ is arithmetically exceptional if and only if $E$ has no $k$ -torsion point with a rational $x$ -coordinate.

4. Significance

Resolution of a Specific Conjecture: The paper largely resolves Odabaş's conjecture for the class of CM elliptic curves, providing a complete classification except for the specific obstruction at $K=\mathbb{Q}(\sqrt{-11})$ .
Discovery of New Obstructions: It reveals that the arithmetic exceptionality of Lattès maps is not solely determined by the existence of rational torsion points. The splitting behavior of small primes (specifically 2 and 3) in the CM field can create "global" obstructions that prevent permutation for all primes, even when the torsion condition is met.
Galois Representation Perspective: The paper connects the problem to the image of Galois representations. It shows that $L_k$ is arithmetically exceptional if and only if the image of the mod- $k$ Galois representation intersects a specific set $C_k$ (matrices with determinant $\neq 0$ ). The counterexamples correspond to cases where this intersection is empty.
Future Directions: The work opens new avenues for investigating arithmetic exceptionality in non-CM settings and over general number fields, utilizing the density of primes provided by the Chebotarev Density Theorem.

5. Conclusion

The authors successfully prove Odabaş's conjecture for almost all CM elliptic curves. They demonstrate that the conjecture fails specifically when $6 \mid k$ and the CM field is $\mathbb{Q}(\sqrt{-11})$ , or for certain non-CM curves, due to unavoidable divisibility of the group order $A_p(E)$ by 2 or 3. This leads to a more nuanced understanding of the relationship between torsion structures and permutation properties of Lattès maps, suggesting the conjecture holds true only when $k$ is not divisible by 6.