Quadratic torsion orders on Jacobian varieties

TL;DR

This paper constructs hyperelliptic curves over rationals with Jacobians having rational torsion points of specific quadratic orders, expanding understanding of torsion structures in algebraic geometry.

Contribution

It introduces explicit families of hyperelliptic curves with Jacobians containing rational torsion points of new quadratic orders, including infinite families for certain orders.

Findings

Existence of hyperelliptic curves with Jacobians having torsion points of orders 4g^2+2g-2 and 4g^2+2g-4.

Construction of a 1-parameter family with torsion points of order 2g^2+7g+1.

Demonstration of explicit examples for various genera.

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 hyperelliptic curves of genus $g$ over $\mathbb{Q}$ with a rational torsion point of order $N$ on their Jacobians.