@article{TomaszJędrzejak2013,
abstract = {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. $A ∈ (-1)^\{(p-1)/2\}pℕ²$).},
author = {Tomasz Jędrzejak},
journal = {Acta Arithmetica},
keywords = {hyperelliptic curves; Jacobian varieties},
language = {eng},
number = {3},
pages = {201-218},
title = {Characterization of the torsion of the Jacobians of two families of hyperelliptic curves},
url = {http://eudml.org/doc/279395},
volume = {161},
year = {2013},
}
TY - JOUR
AU - Tomasz Jędrzejak
TI - Characterization of the torsion of the Jacobians of two families of hyperelliptic curves
JO - Acta Arithmetica
PY - 2013
VL - 161
IS - 3
SP - 201
EP - 218
AB - 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. $A ∈ (-1)^{(p-1)/2}pℕ²$).
LA - eng
KW - hyperelliptic curves; Jacobian varieties
UR - http://eudml.org/doc/279395
ER -
