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Quadratic torsion orders on Jacobian varieties

This paper establishes the existence of hyperelliptic curves over $\mathbb{Q}$ with Jacobians possessing rational torsion points of specific high orders, including a constructive one-parameter family for the order $N = 2g^2 + 7g + 1$ .

Original authors: Hamide Kuru, Mohammad Sadek

Published 2026-07-10

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Hamide Kuru, Mohammad Sadek

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 mathematical world as a vast, bustling city of shapes called Jacobian varieties. These aren't just any shapes; they are the "shadow selves" of curved lines known as hyperelliptic curves. In this city, there are special travelers called torsion points. Think of these points as dancers who perform a routine: they take a step, then another, and eventually, after a specific number of steps, they land exactly back where they started. That number of steps is their order.

For a long time, mathematicians have been trying to build these curves so that their dancers have specific, predictable routine lengths. The big question was: Can we build a curve where the number of steps the dancer takes grows quadratically with the size of the curve? In other words, if the curve gets bigger (measured by a number called genus, let's call it $g$ ), can the dance routine length explode like $g^2$ instead of just growing slowly like $g$ ?

The Main Discovery: A New Dance Floor
Mohammad Sadek and Hamide Suluyer have built a brand-new set of dance floors (curves) where the dancers perform massive, quadratic routines. They proved that for any curve size $g$ (as long as $g \ge 2$ ), they can construct a hyperelliptic curve where the dancers have a routine length of exactly:

$4g^2 + 2g - 2$ steps, or
$4g^2 + 2g - 4$ steps.

To visualize this, imagine if a curve of size $g=4$ (a medium-sized shape) usually had dancers taking maybe 10 or 20 steps. With this new construction, the dancers on the $g=4$ curve take 70 steps before returning home. This is the first time anyone has found a curve of genus 4 with a rational torsion point of order 70. It's like discovering a new species of bird that can fly twice as high as anyone thought possible.

The "Magic" Formula
How did they do it? They didn't just guess; they built a machine. They created a special 1-parameter family of polynomials (think of these as blueprints for the curves). By tweaking a single dial (a variable called $t$ ), they can generate an infinite number of these curves.

For almost every setting of this dial (except for a few broken spots), if the blueprint doesn't collapse (the discriminant is nonzero), the resulting curve is guaranteed to have a dancer with a routine of $2g^2 + 7g + 1$ steps.

For a genus-3 curve, this formula gives a routine of 40 steps.
For a genus-5 curve, it gives 86 steps.

What They Didn't Do (and What They Ruled Out)
It's important to note what this paper doesn't do. They didn't find a magic number that works for every possible dance routine. They didn't prove that all quadratic numbers are possible. They specifically constructed curves for these two very specific quadratic formulas ( $4g^2 + 2g - 2$ and $4g^2 + 2g - 4$ ) and the one-parameter family ( $2g^2 + 7g + 1$ ).

They also didn't just simulate these curves on a computer and say, "It looks like they work." They proved it. They used rigorous mathematical arguments to show that the curves are indeed of the correct size (genus) and that the dancers must return home after exactly those specific numbers of steps. They even checked the "simple" nature of these curves (meaning they can't be broken down into smaller, simpler curves) using computer calculations for small examples, confirming they are unique, indivisible shapes.

The Confidence Level
The authors are extremely confident. They didn't just suggest these curves might exist; they established their existence.

They proved that for any integer $g \ge 2$ , these specific curves exist over the rational numbers (the fractions we use in everyday math).
They proved the order of the torsion points is exactly the numbers they claimed.
They verified with computer software (Magma) that for small values of $g$ (up to 1662 for one case and 1695 for another), the curves don't break down or become messy.

Why It Matters
Before this, the record for how fast the dance routine could grow was linear (like $3g$ or $4g$ ). This paper breaks that ceiling, showing that the routine can grow quadratically (like $g^2$ ). It's a significant step forward in understanding the limits of these mathematical shapes. They didn't solve the whole mystery of every possible dance routine, but they definitely opened a new door, proving that these massive, quadratic routines are not just a dream, but a reality we can build.

Technical Summary: Quadratic Torsion Orders on Jacobian Varieties

Problem Statement
The paper addresses the problem of constructing hyperelliptic curves of genus $g \ge 2$ defined over the rational field $\mathbb{Q}$ whose Jacobian varieties possess rational torsion points of specific orders. While the Mordell-Weil theorem guarantees that the group of rational points on an abelian variety is finitely generated, the structure of the torsion subgroup remains a central area of investigation. Previous research has established the existence of curves with torsion orders linear in $g$ (e.g., in the interval $[3g, 4g+1]$ ) and, more recently, quadratic orders in specific intervals such as $[g^2 + 2g + 1, g^2 + 3g + 1]$ and $[2g^2 + 2g + 1, 2g^2 + 5g + 5]$ . The authors pose the question of whether it is possible to construct abelian varieties of dimension $g$ possessing rational torsion points with orders that are quadratic in $g$ , specifically targeting orders larger than those previously constructed by Flynn and Leprévost.

Methodology
The authors employ a constructive approach based on the theory of divisors on hyperelliptic curves.

Curve Construction: They consider hyperelliptic curves $C$ defined by $y^2 = f(x)$ , where $f(x)$ is a polynomial of odd degree $2g+1$ . The polynomial is constructed in the form $f(x) = A(x)^2 - \lambda x^{g+1+d}(x-1)^{g-d}$ , where $A(x) \in \mathbb{Q}[x]$ has degree at most $g$ , and $\lambda, d$ are parameters.
Divisor Analysis: The study focuses on the torsion properties of divisors of the form $D = P - P_\infty$ , where $P$ is a rational point on the curve and $P_\infty$ is the point at infinity. The authors utilize Riemann-Roch spaces to identify rational functions $\phi$ with specific divisor properties, specifically $\text{div}(\phi) = (g+m)P'_0 + (g+m+2)P_1 - (2g+2m+2)P_\infty$ .
Algebraic Derivation: By analyzing the norm of these rational functions and imposing conditions on the coefficients of $A(x)$ , the authors derive explicit 1-parameter families of polynomials. They solve for parameters $\lambda$ and $\alpha$ (or a parameter $t$ in the infinite family) to ensure that the resulting curve is non-singular (square-free discriminant) and that the torsion order is exactly the target value.
Verification of Genus and Simplicity: For specific cases, the authors prove that the constructed curves have genus $g$ by demonstrating that the defining polynomials are square-free. This involves analyzing resultants of auxiliary polynomials and using height bounds (Smyth's theorem and Landau's inequality) to limit the range of $g$ requiring computational verification. They also verify that the Jacobians of low-genus examples are $\mathbb{Q}$ -simple by examining the irreducibility of their characteristic polynomials of Frobenius over finite fields.

Key Contributions and Results
The paper establishes the existence of hyperelliptic curves of genus $g \ge 2$ with rational torsion points of the following orders:

New Quadratic Orders: For any integer $g \ge 2$ $g \geq 2$ , the authors construct curves with rational torsion points of order $N = 4g^2 + 2g - 2$ $N = 4 g^{2} + 2 g - 2$ and $N = 4g^2 + 2g - 4$ $N = 4 g^{2} + 2 g - 4$ .
- Specifically, for $C_{g, g-1}$ , the torsion order is $4g^2 + 2g - 2$ .
- For $C_{g, g-2}$ , the torsion order is $4g^2 + 2g - 4$ .
- These results extend the known range of quadratic torsion orders beyond previous bounds established by Flynn and Leprévost.
Specific Examples: The authors provide the first known examples of a genus-4 hyperelliptic curve over $\mathbb{Q}$ with a rational torsion point of order 70 (where $g=4$ yields $4(16)+8-2 = 70$ ).
Infinite Families: For the order $N = 2g^2 + 7g + 1$ , the authors introduce a 1-parameter family of polynomials $f_t(x)$ of degree $2g+1$ . They prove that for all but finitely many rational values of the parameter $t$ , if the discriminant is non-zero, the resulting curve has a rational torsion point of this order. This family works for both odd and even $g$ (with specific parameter adjustments for even $g$ ).
Simplicity: The Jacobians of the constructed curves for low genera ( $2 \le g \le 5$ ) are shown to be $\mathbb{Q}$ -simple, meaning they are not isogenous to products of lower-dimensional abelian varieties.

Significance
The paper claims to extend the results of Flynn and Leprévost by producing the first examples of hyperelliptic curves with Jacobians possessing rational torsion points of orders $4g^2 + 2g - 2$ and $4g^2 + 2g - 4$ . These orders represent a significant increase over previously known quadratic bounds. Furthermore, the construction of the 1-parameter family for the order $2g^2 + 7g + 1$ demonstrates the existence of infinite families of such curves. The work contributes to the broader understanding of the uniform boundedness conjecture for torsion subgroups by expanding the set of realizable torsion orders for higher genus curves. The authors explicitly note that their genus-4 example with torsion order 70 is the first of its kind in the literature, distinguishing it from non-simple abelian varieties that might possess such torsion.

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