Rational torsion on hyperelliptic jacobian varieties

Abstract

It was conjectured by Flynn that there exists a constant $\hat{I}º$ such that, for any integer $g\ge 2$, any $m\le \hat{I}ºg$, there exists a hyperelliptic curve of genus $g$ over $\mathbb{Q}$ with a rational $m$-torsion point on its Jacobian. Leprévost proved this conjecture with $\hat{I}º=3$. 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.