Quadratic torsion orders on Jacobian varieties

2024/10/18 by Suluyer, Hamide, Kuru, Hamide, Mohammad Sadek +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Nonlinear Waves and Solitons #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2410.14455

Abstract

We establish the existence of hyperelliptic curves of genus g≥ 2 defined over ℚ whose Jacobians possess rational torsion points of order N where N=4g²+2g-2 or 4g²+ 2g -4. For N = 2g² + 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 y² = f_t(x) has a rational point of order N on its Jacobian.

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