Quadratic torsion orders on Jacobian varieties

Official URL: https://dx.doi.org/10.1007/s00229-026-01738-y

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 ft(x) of degree 2g+1. For all but finitely many rational values of t, if the discriminant of ft(x) is nonzero, then the hyperelliptic curve defined by y2=ft(x) has a rational point of order N on its Jacobian.