Rational Torsion on Hyperelliptic Jacobian Varieties

Abstract

It was conjectured by Flynn that there exists a constant such that, for any integer , any , there exists a hyperelliptic curve of genus over with a rational -torsion point on its Jacobian. Lepr & eacute;vost proved this conjecture with . In this work, we prove that given an integer in the interval , , satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order . In particular, we establish the existence of such varieties for when is odd and for when is even. A few explicit applications of this result produce the first known infinite examples of torsion 13 when , torsion 15 when , and torsion 17,18,21 when . In fact, we show that infinitely many of the latter abelian varieties are absolutely simple.