Rational torsion on hyperelliptic jacobian varieties

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.