On torsion of superelliptic Jacobians

Wawrów, Wojciech

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

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 $y^m=F\left(x\right)$, 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 $\mathbb{Q}$.

We prove a result describing the structure of a specific subgroup of the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F\left(x\right)$, 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 $\mathbb{Q}$.

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

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