On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)

Tomasz Jędrzejak. "On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)." Acta Arithmetica 174.2 (2016): 99-120. <http://eudml.org/doc/286254>.

@article{TomaszJędrzejak2016,
abstract = {Consider two families of hyperelliptic curves (over ℚ), $C^\{n,a\}: y² = xⁿ+a$ and $C_\{n,a\}: y² = x(xⁿ+a)$, and their respective Jacobians $J^\{n,a\}$, $J_\{n,a\}$. We give a partial characterization of the torsion part of $J^\{n,a\}(ℚ) $ and $J_\{n,a\}(ℚ)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_\{8,a\}(ℚ)$. Namely, we show that $J_\{8,a\}(ℚ)_\{tors\} = J_\{8,a\}(ℚ)[2]$. In addition, we characterize the torsion parts of $J_\{p,a\}(ℚ)$, where p is an odd prime, and of $J^\{n,a\}(ℚ)$, where n = 4,6,8. The main ingredients in the proofs are explicit computations of zeta functions of the relevant curves, and applications of the Chebotarev Density Theorem.},
author = {Tomasz Jędrzejak},
journal = {Acta Arithmetica},
keywords = {hyperelliptic curve; Jacobian; Gauss sum; Jacobi sum; jacobstahl sum; torsion part; zeta function},
language = {eng},
number = {2},
pages = {99-120},
title = {On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)},
url = {http://eudml.org/doc/286254},
volume = {174},
year = {2016},
}

TY - JOUR
AU - Tomasz Jędrzejak
TI - On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)
JO - Acta Arithmetica
PY - 2016
VL - 174
IS - 2
SP - 99
EP - 120
AB - Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}: y² = xⁿ+a$ and $C_{n,a}: y² = x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}(ℚ) $ and $J_{n,a}(ℚ)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}(ℚ)$. Namely, we show that $J_{8,a}(ℚ)_{tors} = J_{8,a}(ℚ)[2]$. In addition, we characterize the torsion parts of $J_{p,a}(ℚ)$, where p is an odd prime, and of $J^{n,a}(ℚ)$, where n = 4,6,8. The main ingredients in the proofs are explicit computations of zeta functions of the relevant curves, and applications of the Chebotarev Density Theorem.
LA - eng
KW - hyperelliptic curve; Jacobian; Gauss sum; Jacobi sum; jacobstahl sum; torsion part; zeta function
UR - http://eudml.org/doc/286254
ER -