Rational torsion on hyperelliptic jacobian varieties
Hamide Kuru, Mohammad Sadek
TL;DR
The paper advances Flynn's conjecture by constructing infinite families of genus $g$ hyperelliptic curves over ${\mathbb Q}$ whose Jacobians admit a ${\mathbb Q}$-rational torsion point of order $N$ for all $N$ in the range $3g\le N\le 4g+1$, under specific partition conditions. The central technique ties the torsion of the divisor at infinity to periodic continued fractions for $\sqrt{f(x)}$, and builds parameterized curves where $N$ equals $g+1+6\alpha+3\beta+\gamma$ with $2\alpha+2\beta+\gamma=g+1$, yielding the linear growth of torsion with genus. The authors obtain explicit infinite families realizing $N=4g+1$ for odd $g$ and $N=4g-1$ for even $g$, and showcase concrete examples for genus $3,4,5$ with torsion orders $13,15,17,18,21$, including infinitely many absolutely simple Jacobians. The work blends function-field continued fraction theory, Diophantine constructions, and Galois/irreducibility arguments to produce new torsion phenomena in higher-genus Jacobians, providing both theoretical and explicit constructive tools relevant to arithmetic geometry and number theory.
Abstract
It was conjectured by Flynn that there exists a constant $κ$ such that, for any integer $g \ge 2$, any $m \le κg$, 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 $κ=3$. 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.