Quadratic torsion orders on Jacobian varieties

Hamide Suluyer; Mohammad Sadek

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arXiv · 2410.14455

Quadratic torsion orders on Jacobian varieties

Hamide Suluyer·Mohammad Sadek

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Abstract

We establish the existence of hyperelliptic curves of genus $g\ge 2$ defined over $\mathbb{Q}$ whose Jacobians possess rational torsion points of order $N$ where $N=4g^2+2g-2$ or $4g^2+ 2g -4$. For $N = 2g^{2} + 7g + 1$, we introduce a 1-parameter family of polynomials $f_{t}(x)$ of degree $2g+1$. For all but finitely many rational values of $t$, if the discriminant of $f_{t}(x)$ is nonzero, then the hyperelliptic curve defined by $y^{2} = f_{t}(x)$ has a rational point of order $N$ on its Jacobian.

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Hamide Suluyer, Mohammad Sadek. 2024-10-18. Quadratic torsion orders on Jacobian varieties. https://arxiv.org/abs/2410.14455

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