Quadratic torsion orders on Jacobian varieties

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.