Doubly isogenous curves of genus two with a rational action of $D_6$

This paper investigates an unexpected abundance of doubly isogenous genus 2 curves with $D_6$ automorphism groups over finite fields, leading to the surprising construction of a global example over a number field, a theoretical limit on such instances via the Zilber–Pink conjecture, and potential applications in deterministic polynomial-time polynomial factorization.

The Gist

Imagine you are a detective trying to solve a mystery in a vast, mathematical city called Finite Field. In this city, there are special shapes called Curves. Some of these curves are like simple loops (genus 1), but the ones in this story are more complex, shaped like pretzels with two holes (genus 2).

The paper you asked about is a report on a strange phenomenon the authors discovered while investigating these "pretzel" curves. Here is the story broken down into simple concepts, analogies, and metaphors.

1. The Setup: The "Double-Isogenous" Mystery

In this city, every curve has a hidden "shadow" called a Jacobian. Think of the Jacobian as the curve's DNA or its fingerprint. Usually, if two curves have different shapes, their DNA (Jacobian) is different.

However, the authors were looking for a very specific, rare event: Doubly Isogenous Curves.

• Level 1: Two different curves have the exact same DNA (their Jacobians are "isogenous," meaning they are mathematically related like cousins).
• Level 2: Not only are the curves related, but if you take a specific "zoom-in" view of them (called a double cover), the DNA of those zoomed-in views is also the same.

Usually, finding two curves that match on Level 1 is rare. Finding two that match on both Level 1 and Level 2 is like finding two strangers who not only have the same face but also the same fingerprints, voice, and handwriting. It should be almost impossible.

2. The "D6" Club: A Special Group of Curves

The authors focused on a specific club of curves called D6 Curves.

• The Analogy: Imagine a group of people who all wear a specific type of hat (the D6 symmetry). These hats have 12 distinct ways to be rotated or flipped while still looking the same.
• The authors expected that within this club, finding "Double-Isogenous" pairs would be extremely rare, based on standard mathematical probability (heuristics).

3. The Glitch: Too Many Matches!

When the authors ran computer simulations to count these pairs, they found a glitch.

• The Expectation: They expected to find almost zero matches.
• The Reality: They found way too many matches. It was as if they were looking for a needle in a haystack, but the haystack was full of needles.

Why? They discovered two reasons for this "overabundance":

Reason A: The "Twins" (Twists)

Some of the matches were actually just twins. In math, you can take a curve and "twist" it (like flipping a pancake). Sometimes, a curve and its twisted version look different but are secretly the same family.

• The Fix: The authors realized that for certain conditions (like specific prime numbers), these twins automatically match the "Double-Isogenous" test. Once they filtered these out, the numbers dropped, but...

Reason B: The "Extraordinary Coincidence" (The Global Pair)

Even after removing the twins, there were still too many matches. The authors found a Global Pair of curves.

• The Metaphor: Imagine two people, Alice and Bob, who live in a different country (a "Number Field," specifically involving $\sqrt{29}$). They are not related by a simple twist. They are completely different people.
• The Miracle: However, when Alice and Bob travel to the "Finite Field" city (by reducing their coordinates modulo a prime number), they happen to look exactly alike.
• The Surprise: This is a "miracle" in mathematics. It's like two people from different universes meeting in a specific city and realizing they have the exact same fingerprints, DNA, and voice. The authors call this the "Extraordinary Curves."

4. The Detective Work: Zilber-Pink and the "Finiteness" Proof

The existence of this "Extraordinary Pair" was so shocking it seemed like a fluke. Could there be infinite such pairs?

• The Zilber-Pink Conjecture: This is a famous, unproven mathematical rule (like a "Law of the Universe") that says: "If you have a complex geometric shape, you can't have too many points where different symmetries accidentally line up."
• The Conclusion: The authors used this conjecture to prove that while this "Extraordinary Pair" exists, it is a one-time miracle. There are only a finite number of such pairs in the entire universe of math. Once you account for this one pair and the "twins," the rest of the data fits the original "rare event" theory perfectly.

5. The Superpower: Factoring Polynomials

Why does this matter? The authors show that this family of curves can be used as a super-tool for cracking codes.

• The Problem: Factoring large polynomials (breaking down a complex equation into simple pieces) is a hard problem. There are fast probabilistic ways to do it (like guessing), but no fast deterministic way (guaranteed to work every time without guessing).
• The Solution: The authors propose using their "D6 Curves" to build a machine.
  • They take 15 different "views" (elliptic curves) of their main curve.
  • They shift the parameters slightly to ensure no two inputs look the same.
  • The Magic: If you feed a polynomial into this machine, the 15 views will produce 15 different "signatures" (Frobenius traces). Because the signatures are so unique, the machine can deterministically (guaranteed) find the factors of the polynomial in a reasonable amount of time.

Summary

1. The Mystery: The authors looked for rare mathematical "twins" (doubly isogenous curves) in a specific family.
2. The Glitch: They found too many twins, which confused their probability models.
3. The Explanation: They found that some twins were just "flipped versions" of each other, and one was a "Global Miracle" (a pair of curves from a different mathematical world that coincidentally matched).
4. The Proof: Using a high-level math conjecture, they proved that this miracle is a one-off event and won't happen infinitely often.
5. The Application: They turned this family of curves into a new, highly efficient, and guaranteed method for factoring polynomials, which is useful for cryptography and computer science.

In short: They found a mathematical anomaly, explained why it happened, proved it's rare, and then turned it into a super-efficient tool for solving hard problems.

Technical Summary

1. Problem Statement and Context

The paper investigates the phenomenon of doubly isogenous curves. Two genus-$g$ curves $(C, \iota)$ and $(C', \iota')$ over a field $k$ are defined as doubly isogenous if:

1. Their Jacobian varieties, $\text{Jac}(C)$ and $\text{Jac}(C')$, are isogenous over $k$.
2. The Jacobians of their "double covers" $C(2)$ and $C'(2)$ (defined as the pullbacks of the multiplication-by-2 map on the Jacobians via the embeddings $\iota, \iota'$) are also isogenous over $k$.

The authors focus on a specific one-parameter family of genus-2 curves, known as D6 curves, whose geometric automorphism groups contain the dihedral group $D_6$ of order 12. The central problem is to understand the frequency of non-isomorphic, doubly isogenous pairs within this family over finite fields.

The Paradox:

• Heuristic Expectation: Based on random heuristics regarding isogenies between elliptic curves, the probability of finding such pairs should be extremely low (decaying as $q^{-1/2}$ or faster), suggesting they should be rare.
• Empirical Observation: Computational data over finite fields reveals a significant "overabundance" of such pairs, far exceeding heuristic predictions.
• Goal: The authors aim to explain this discrepancy, construct global examples (over number fields) that influence local behavior, and determine if these curves can be distinguished using higher-order covers.

2. Methodology

A. Parametrization and Geometry

The authors utilize a specific parametrization of D6 curves $C_{r,c}$ defined by the equation:
$$cy^2 = x^6 + (r - 18)x^4 + (81 - 2r)x^2 + r$$
where $r, c \in K^\times$. They analyze the decomposition of the Jacobian of the double cover $C(2)$ into simple abelian varieties (specifically, products of elliptic curves).

• Decomposition: The Jacobian of $C(2)$ decomposes into a product of elliptic curves with specific $j$-invariants ($j_6, j_{3a}, j_{3b}, j_{3c}$) derived from the parameter $u$ (related to $r$).
• Prym Varieties: They explicitly compute the Prym varieties of unramified cyclic covers of degree 3 and 4. These covers provide additional invariants (L-functions) to test for isogeny.

B. Computational Analysis

The authors performed extensive computations over finite fields $\mathbb{F}_q$ (for primes near $2^n$ up to $n=30$). They enumerated pairs of D6 curves and checked:

1. If their Jacobians are isogenous.
2. If the Jacobians of their double covers are isogenous.
3. If the Prym varieties of their degree-3 and degree-4 unramified covers are isogenous.

C. Theoretical Tools

• Zilber–Pink Conjecture: Used to argue that the number of "exceptional" global examples (pairs of curves over number fields that are doubly isogenous) must be finite.
• Buium's Criterion: An algorithmic method using differential algebra (specifically the invariant $\chi(y)$) to detect isogeny correspondences between families of elliptic curves. This allows the authors to rule out the existence of infinite families of simultaneous isogenies.
• Kayal–Poonen Factoring Framework: The paper explores the application of these curve families to deterministic polynomial-time polynomial factorization over finite fields.

3. Key Contributions and Results

A. Explanation of the "Overabundance"

The discrepancy between heuristics and data is explained by two distinct phenomena:

1. Doubly Isogenous Twists: The authors prove (Theorem 7.1) that for certain primes $p \equiv 3 \pmod 4$, a D6 curve and its quadratic twist are automatically doubly isogenous under specific supersingularity conditions. This accounts for a constant number of "twist" pairs in the data.
2. The "Extraordinary" Global Pair: The most significant finding is the construction of a pair of non-isomorphic D6 curves, $C_1$ and $C_2$, defined over the number field $K = \mathbb{Q}(\sqrt{29})$.
   • Theorem 7.4: These curves are doubly isogenous over a degree-6 extension $L$ (the Hilbert class field of $\mathbb{Q}(\sqrt{-29})$).
   • Impact: For primes $p$ that split completely in $L$, the reductions of $C_1$ and $C_2$ modulo $p$ appear as doubly isogenous pairs over $\mathbb{F}_p$. This single global example explains a large portion of the "nontwist" pairs observed in finite fields, violating the assumption of independence in the random heuristics.

B. Finiteness of Global Examples

Using the Zilber–Pink conjecture, the authors prove (Theorem 8.7) that there can only be finitely many such extraordinary pairs of doubly isogenous D6 curves in characteristic zero.

• They developed an algorithm (Section 9) based on Buium's work to search for simultaneous isogeny correspondences between the elliptic factors of the Jacobians.
• Their computation ruled out the existence of any families of such correspondences, confirming that the observed pairs are isolated "unlikely intersections."

C. Distinguishing Curves via Higher Covers

The paper addresses whether doubly isogenous curves can be distinguished by their L-functions:

• Twists: The Pryms of degree-4 covers fail to distinguish a curve from its quadratic twist in the "extraordinary" cases.
• Nontwists: However, the authors show that for non-twist pairs, the isogeny classes of the Prym varieties of unramified degree-4 covers (and specific degree-3 covers) are sufficient to distinguish non-isomorphic, doubly isogenous curves.
• Result: When the "extraordinary" global pair and the twist cases are excluded, the remaining data aligns perfectly with the refined heuristics.

D. Application to Polynomial Factoring

The authors propose a new candidate for a deterministic polynomial-time algorithm to factor univariate polynomials over finite fields, based on the work of Kayal and Poonen.

• Mechanism: Instead of using a high-genus curve (as in previous candidates), they use a product of 15 elliptic curves derived from the D6 family (5 base curves + 10 shifted by parameter changes $u \to u-22$ and $u \to u-30$).
• Advantage: Computing Frobenius traces for elliptic curves is significantly faster ($O(\log^5 p)$) than computing zeta functions for high-genus curves.
• Conjecture: They conjecture that for generic parameters, these 15-tuples of traces are distinct for all primes, which would guarantee the algorithm's correctness.

4. Significance

1. Resolution of a Statistical Anomaly: The paper provides a rigorous explanation for why "unlikely" isogenies occur more frequently than random heuristics predict in specific arithmetic families, linking local finite field behavior to global number field geometry.
2. Explicit Construction of Unlikely Intersections: The construction of the $C_1, C_2$ pair over $\mathbb{Q}(\sqrt{29})$ is a concrete example of an "unlikely intersection" in the moduli space of abelian varieties, a central theme in modern arithmetic geometry.
3. Algorithmic Advances: The development of the Buium-based algorithm to rule out simultaneous isogeny correspondences provides a powerful tool for studying the distribution of isogenies in families of abelian varieties.
4. Cryptographic/Computational Implications: The proposed factoring algorithm offers a potentially more efficient deterministic alternative to existing methods, bridging the gap between theoretical number theory and practical computation. It highlights the utility of using products of elliptic curves rather than single high-genus curves for such applications.

In summary, the paper combines deep arithmetic geometry, computational number theory, and algorithm design to resolve a counter-intuitive statistical phenomenon in the distribution of isogenous curves, while simultaneously proposing a novel application for polynomial factorization.