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

Banach Center Publications (2016)

- Volume: 108, Issue: 1, page 143-149
- ISSN: 0137-6934

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topTomasz 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 -

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