Characterization of the torsion of the Jacobians of two families of hyperelliptic curves
Volume 161 / 2013
Abstract
Consider the families of curves $C^{n,A}:y^{2}=x^{n}+Ax$ and $C_{n,A}:y^{2}=x^{n}+A$ where $A$ is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}( \mathbb {Q}) $ and $C_{3,A}( \mathbb {Q}) $ are well known. We show that for any nonzero rational $A$ the torsion subgroup of $J^{7,A}( \mathbb {Q}) $ is a 2-group, and for $A\not =4a^{4},-1728,-1259712$ this subgroup is equal to $J^{7,A}( \mathbb {Q}) [ 2] $ (for a excluded values of $A$, with the possible exception of $A=-1728$, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ ($A\not =4$) and $J^{5,A}$. We also almost completely determine the $\mathbb {Q}$-rational torsion of $J_{p,A}$ for all odd primes $p$, and all $A\in \mathbb {Q}\setminus \{ 0\} $. We discuss the excluded case (i.e. $A\in ( -1) ^{( p-1) /2}p\mathbb {N}^{2}$).
