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

Tomasz Jędrzejak

Banach Center Publications, Tome 108 (2016), p. 143-149 / Harvested from The Polish Digital Mathematics Library

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Résumé

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.

Publié le : 2016-01-01

Zbl 06622291

EUDML-ID : urn:eudml:doc:286101

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     author = {Tomasz J\k edrzejak},
     title = {A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$
            },
     journal = {Banach Center Publications},
     volume = {108},
     year = {2016},
     pages = {143-149},
     zbl = {06622291},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-11}
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Tomasz Jędrzejak. A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$
            . Banach Center Publications, Tome 108 (2016) pp. 143-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-11/