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

Tomasz Jędrzejak. "Characterization of the torsion of the Jacobians of two families of hyperelliptic curves." Acta Arithmetica 161.3 (2013): 201-218. <http://eudml.org/doc/279395>.

@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 -