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

Tomasz Jędrzejak

Acta Arithmetica, Tome 161 (2013), p. 201-218 / Harvested from The Polish Digital Mathematics Library

Résumé

Consider the families of curves $C^{n,A} : y² = xⁿ + Ax$ and $C_{n,A} : y² = xⁿ + A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}(ℚ)$ and $C_{3,A}(ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}(ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}(ℚ)[2]$ (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 ≠ 4) and $J^{5,A}$. We also almost completely determine the ℚ-rational torsion of $J_{p,A}$ for all odd primes p, and all A ∈ ℚ∖0. We discuss the excluded case (i.e. $A ∈ (-1)^{(p-1)/2}pℕ²$).

Détail

Publié le : 2013-01-01

Zbl 1286.11094

EUDML-ID : urn:eudml:doc:279395