On torsion of superelliptic Jacobians

Wojciech Wawrów

doi:10.5802/jtnb.1158

Wojciech Wawrów ¹

¹ Adam Mickiewicz University Faculty of Mathematics and Computer Science Uniwersytetu Poznańskiego 4 61-614 Poznań, Poland

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 223-235.

Abstract
Résumé

We prove a result describing the structure of a specific subgroup of the $m$ -torsion of the Jacobian of a general superelliptic curve $y^{m} = F (x)$ , generalizing the structure theorem for the $2$ -torsion of a hyperelliptic curve. We study existence of torsion on curves of the form $y^{q} = x^{p} - x + a$ over finite fields of characteristic $p$ . We apply those results to bound from below the Mordell–Weil ranks of Jacobians of certain superelliptic curves over $ℚ$ .

Nous démontrons un résultat décrivant la structure d’un sous- groupe de $m$ -torsion spécifique de la jacobienne d’une courbe superelliptique générale de la forme $y^{m} = F (x)$ , généralisant ainsi le théorème de structure pour la $2$ -torsion d’une courbe hyperelliptique. Nous étudions l’existence de points de torsion sur les courbes de la forme $y^{q} = x^{p} - x + a$ sur les corps finis de caractéristique $p$ . Nous appliquons ces résultats à la minoration du rank de Mordell–Weil des jacobiennes de certaines courbes superelliptiques sur $ℚ$ .

Received: 2020-07-01
Revised: 2020-10-19
Accepted: 2020-12-13
Published online: 2021-05-21

DOI: 10.5802/jtnb.1158

Classification: 14H40, 14G10, 14H45
Keywords: Jacobian variety, superelliptic curves, Mordell–Weil group

Author's affiliations:

Wojciech Wawrów ¹

¹ Adam Mickiewicz University Faculty of Mathematics and Computer Science Uniwersytetu Poznańskiego 4 61-614 Poznań, Poland

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On torsion of superelliptic {Jacobians}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Wojciech Wawrów. On torsion of superelliptic Jacobians. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 223-235. doi : 10.5802/jtnb.1158. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1158/

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