Quadratic torsion orders on Jacobian varieties

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.