A note on the torsion of the Jacobians of superelliptic curves $y^q=x^p+a$

Tomasz Jędrzejak. "A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$." Banach Center Publications 108.1 (2016): 143-149. <http://eudml.org/doc/286101>.

@article{TomaszJędrzejak2016,
abstract = {This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_\{q,p,a\}: y^\{q\} = x^\{p\} + a$, and its Jacobians $J_\{q,p,a\}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_\{3,5,a\}(ℚ)$ (resp. $J_\{q,p,a\}(ℚ)$). The main tools are computations of the zeta function of $C_\{3,5,a\}$ (resp. $C_\{q,p,a\}$) over $_\{l\}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp)) and applications of the Chebotarev Density Theorem.},
author = {Tomasz Jędrzejak},
journal = {Banach Center Publications},
keywords = {diagonal curve; superelliptic curve; Jacobian; Jacobi sum; torsion part; zeta function},
language = {eng},
number = {1},
pages = {143-149},
title = {A note on the torsion of the Jacobians of superelliptic curves $y^\{q\} = x^\{p\} + a$},
url = {http://eudml.org/doc/286101},
volume = {108},
year = {2016},
}

TY - JOUR
AU - Tomasz Jędrzejak
TI - A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$
JO - Banach Center Publications
PY - 2016
VL - 108
IS - 1
SP - 143
EP - 149
AB - This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}: y^{q} = x^{p} + a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}(ℚ)$ (resp. $J_{q,p,a}(ℚ)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over $_{l}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp)) and applications of the Chebotarev Density Theorem.
LA - eng
KW - diagonal curve; superelliptic curve; Jacobian; Jacobi sum; torsion part; zeta function
UR - http://eudml.org/doc/286101
ER -