Rational Torsion on Hyperelliptic Jacobian Varieties

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