Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak¹

¹ Institute of Mathematics University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland

Acta Arithmetica, Tome 161 (2013) no. 3, pp. 201-218.

Résumé

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. $A\in ( -1) ^{( p-1) /2}p\mathbb {N}^{2}$).

Détail

DOI : 10.4064/aa161-3-1

Keywords: consider families curves where nonzero rational denote their respective jacobian varieties torsion points mathbb mathbb known nonzero rational torsion subgroup mathbb group subgroup equal mathbb excluded values possible exception group has point order variant corresponding results almost completely determine mathbb rational torsion odd primes mathbb setminus discuss excluded p mathbb