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Rational torsion on hyperelliptic jacobian varieties

This paper proves the existence of parametric families of hyperelliptic Jacobian varieties over $\mathbb{Q}$ with rational torsion points of order $N$ for specific values in the interval $[3g, 4g+1]$ , thereby providing the first known infinite examples of such varieties for various torsion orders including 13, 15, 17, 18, and 21.

Original authors: Hamide Suluyer, Mohammad Sadek

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hamide Suluyer, Mohammad Sadek

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 the world of math as a giant, cosmic playground where shapes called "curves" dance around. Some of these curves are simple loops, but others are wild, twisting hyperelliptic curves that look like pretzels made of infinite possibilities. Hidden inside these curves are secret treasures called "torsion points." Think of these points as special keys. If you use a key to unlock a door (the curve's Jacobian, which is like the curve's internal engine room) and turn it exactly $N$ times, you end up back where you started, as if you never moved at all.

For a long time, mathematicians were trying to figure out how big these keys could get. They knew that for simple, one-humped curves (genus 1), the keys had a strict size limit. But for the more complex, multi-humped curves (genus $g \ge 2$ ), the rules were fuzzy. A mathematician named Flynn made a bold guess: he thought there was a magic constant, let's call it $\kappa$ , that acted like a ruler. He guessed that for any curve with $g$ humps, you could find a key of size $N$ as long as $N$ was smaller than $\kappa$ times $g$ . Another researcher, Leprévost, proved this was true, but only if the key was no bigger than $3g$ . It was like saying, "We can find keys up to size 30 for a 10-hump curve, but maybe not bigger."

But here's the twist: the space between $3g$ and the much larger, quadratic limits (where keys could be as big as $g^2$ ) was a dark, unexplored canyon. No one knew if keys existed in that gap.

Enter Mohammad Sadek and Hamide Suluyer. They decided to build a bridge across that canyon. They didn't just guess; they constructed a factory.

The Factory of Infinite Curves
The authors built a parametric family of curves. Imagine a giant 3D printer that can print out an infinite number of these hyperelliptic curves. By tweaking the dials (which are just variables like $s$ , $t$ , and $u$ ), they could print curves with specific properties.

Their main discovery is a new set of rules for the size of the keys. They proved that for any curve with at least 3 humps ( $g \ge 3$ ), they can create a family of curves that has a key of size $N$ , provided $N$ falls in the range between $3g$ and $4g + 1$ .

To make this concrete, they showed they could print:

Curves with 3 humps ( $g=3$ ) that have a key of size 13.
Curves with 4 humps ( $g=4$ ) that have a key of size 15.
Curves with 5 humps ( $g=5$ ) that have keys of size 17, 18, and 21.

These aren't just one-off lucky finds. The paper proves there are infinitely many different, non-identical curves for each of these sizes. It's like saying, "We can print an endless supply of 3-hump curves, and every single one of them will have a key of size 13."

How They Did It: The Magic of Fractions
How did they find these keys? They used a mathematical tool called "continued fractions." Imagine you are trying to describe a number by breaking it down into a chain of smaller and smaller fractions. Usually, this chain goes on forever. But for these special curves, the chain starts repeating itself, like a song with a catchy chorus.

The authors discovered a deep link: if the chain of fractions repeats in a specific, symmetrical pattern (which they call "skew symmetric"), then the curve has a torsion point. The size of the key ( $N$ ) is determined by adding up the lengths of the pieces in that repeating pattern. By carefully designing the curve so that the fraction chain repeats in just the right way, they forced the curve to have a key of the exact size they wanted.

What They Didn't Do (and What They Didn't Claim)
It's important to note what this paper doesn't say. They didn't prove that keys of every size exist, only those in that specific $3g$ to $4g+1$ range. They also didn't claim to have found the absolute maximum possible key size for these curves; they just pushed the boundary further than anyone else had before.

Furthermore, they didn't just simulate these curves on a computer and hope for the best. They used rigorous algebra to prove that these families exist. They even checked that many of these curves are "absolutely simple," meaning their internal engines are so complex they can't be broken down into smaller, simpler machines. This is a strong, proven fact, not a guess.

The Bottom Line
Sadek and Suluyer have filled a gap in the map of mathematical treasures. They showed that the "forbidden zone" between $3g$ and $4g+1$ is actually full of hidden keys. They built a machine that can generate an infinite supply of these curves, proving that for odd numbers of humps, you can get a key of size $4g+1$ , and for even numbers, you can get a key of size $4g-1$ .

So, the next time you hear about a 5-humped curve, remember: thanks to this work, we know for a fact that there are infinitely many of them hiding a secret key of size 21, waiting to be discovered.

Technical Summary: Rational Torsion on Hyperelliptic Jacobian Varieties

Problem Statement
The paper addresses the problem of determining which integers can be realized as the order of a rational torsion point on the Jacobian of a hyperelliptic curve of genus $g$ over $\mathbb{Q}$ . While Merel's theorem guarantees a uniform bound on torsion subgroups for elliptic curves (genus $g=1$ ) over number fields of fixed degree, no such uniform bound or complete classification exists for abelian varieties of dimension $d \ge 2$ .

Previous work by Flynn conjectured the existence of a constant $\kappa$ such that for any $g \ge 2$ and any $m \le \kappa g$ , there exists a hyperelliptic curve of genus $g$ with a rational torsion point of order $m$ . Leprévost proved this with $\kappa = 3$ , establishing the existence of torsion points of order up to $3g$ . However, a gap remained between these linear bounds ( $3g$ ) and previously known quadratic bounds ( $g^2 + \dots$ ). The authors aim to extend the range of realizable torsion orders beyond $3g$ and provide explicit parametric families for these new orders.

Methodology
The authors employ a connection between torsion divisors at infinity on hyperelliptic curves and the periodicity of continued fractions over function fields.

Theoretical Framework: Let $C$ be a hyperelliptic curve $y^2 = f(x)$ of genus $g$ . The divisor at infinity $D_\infty = \infty_+ - \infty_-$ is torsion of order $N$ if and only if the continued fraction expansion of $\sqrt{f(x)}$ is periodic. The order $N$ is determined by the degrees of the partial quotients in this expansion.
Construction Strategy: The authors focus on constructing curves where the continued fraction expansion of $\sqrt{f(x)}$ has a specific quasi-period length $m=6$ (and period length $n=6$ ). This corresponds to an expansion of the form $[a_0; a_1, a_2, a_3, a_2, a_1, 2a_0]$ .
Diophantine Analysis: To ensure the expansion yields a valid hyperelliptic curve, the authors derive a polynomial Diophantine equation relating the partial quotients $a_i(x)$ . They solve this equation using Lemma 3.1, expressing the coefficients in terms of arbitrary polynomials $a_1, r, u$ with specific degree constraints.
Parametric Families: By fixing the degrees of these polynomials ( $\deg a_1 = \alpha, \deg r = \beta, \deg u = \gamma$ ) and satisfying the partition condition $2\alpha + 2\beta + \gamma = g+1$ , they construct explicit families of curves. The order of the torsion point is calculated as $N = g + 1 + 6\alpha + 3\beta + \gamma$ .

Key Contributions and Results
The paper establishes the existence of parametric families of hyperelliptic Jacobian varieties with rational torsion points of orders $N$ in the interval $[3g, 4g+1]$ for $g \ge 3$ , provided $N$ satisfies specific partition conditions derived from the degrees of the polynomials in the construction.

Extended Bounds: The authors prove that for $g \ge 3$ , there exist infinite families of curves with torsion orders $N$ where $3g \le N \le 4g+1$ .
Specific Cases:
- If $g$ is odd, there exist families with torsion order $N = 4g + 1$ .
- If $g$ is even, there exist families with torsion order $N = 4g - 1$ .
Explicit Examples: The paper provides the first known infinite examples of hyperelliptic Jacobian varieties with specific torsion orders that were previously unlisted in the literature:
- Genus 3 ( $g=3$ ): Torsion order 13.
- Genus 4 ( $g=4$ ): Torsion order 15.
- Genus 5 ( $g=5$ ): Torsion orders 17, 18, and 21.
Absolute Simplicity: The authors demonstrate that for infinitely many parameters in these families, the resulting Jacobian varieties are absolutely simple (i.e., their endomorphism ring is $\mathbb{Z}$ ). This is established by analyzing the Galois groups of the defining polynomials, showing they are often the full symmetric group $S_{2g+1}$ or alternating group $A_{2g+1}$ .

Significance
The paper claims to address the gap between the linear bounds established by Leprévost ( $\kappa=3$ ) and higher-order bounds. By utilizing continued fractions over function fields, the authors successfully construct new infinite families of hyperelliptic curves with rational torsion points of orders significantly larger than $3g$ .

The work provides the first explicit infinite families for torsion orders 13, 15, 17, 18, and 21 for their respective genera. Furthermore, the result that these varieties are often absolutely simple adds significant value, as it ensures these are not merely products of lower-dimensional abelian varieties but represent distinct, irreducible structures. The methodology demonstrates that continued fractions, a tool historically linked to elliptic curves (genus 1), can be effectively extended to study torsion on higher-dimensional Jacobian varieties.

Technical Summary: Rational Torsion on Hyperelliptic Jacobian Varieties

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