Rational torsion on hyperelliptic jacobian varieties

TL;DR

This paper constructs infinite families of hyperelliptic Jacobian varieties with rational torsion points of specified orders, extending known results and providing explicit examples for certain torsion values.

Contribution

It proves the existence of hyperelliptic Jacobians with rational torsion points of orders in [3g, 4g+1], including explicit examples for new torsion values such as 13, 15, 17, 18, and 21.

Findings

Existence of hyperelliptic Jacobians with rational torsion of order N in [3g, 4g+1]

Explicit examples of torsion points for specific genera and orders

Many of these Jacobians are proven to be absolutely simple

Abstract

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its Jacobian. Lepr\'{e}vost proved this conjecture with $\kappa=3$. In this work we prove that given an integer $N$ in the interval $[3g,4g+1]$, $g\ge 3$, satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order $N$. In particular, we establish the existence of such varieties for $N=4g+1$ when $g$ is odd and for $N=4g-1$ when $g$ is even. A few explicit applications of this result produce the first known infinite examples of torsion $13$ when $g=3$, torsion $15$ when $g=4$, and torsion $17,18,21$ when $g=5$. In fact, we show that infinitely many of the latter abelian varieties are absolutely simple.