On the torsion of the Jacobians of the hyperelliptic curves $y^{2}=x^{n}+a$ and $y^{2}=x(x^{n}+a)$

Tomasz Jędrzejak¹

¹ Institute of Mathematics University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland

Acta Arithmetica, Tome 174 (2016) no. 2, pp. 99-120

Résumé

Consider two families of hyperelliptic curves (over $\mathbb{Q}$), $C^{n,a}:y^{2}=x^{n}+a$ and $C_{n,a}:y^{2}=x(x^{n}+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}( \mathbb{Q}) $ and $J_{n,a}( \mathbb{Q}) $. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of $n$ (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}( \mathbb{Q})$. Namely, we show that $J_{8,a}(\mathbb{Q})_{\rm tors} =J_{8,a}(\mathbb{Q})[2]$. In addition, we characterize the torsion parts of $J_{p,a}( \mathbb{Q}) $, where $p$ is an odd prime, and of $J^{n,a}( \mathbb{Q}) $, where $n=4,6,8$. The main ingredients in the proofs are explicit computations of zeta functions of the relevant curves, and applications of the Chebotarev Density Theorem.

Détail

DOI : 10.4064/aa8141-3-2016

Keywords: consider families hyperelliptic curves mathbb their respective jacobians partial characterization torsion part mathbb mathbb precisely only prime factors orders groups prime divisors upper bounds exponents moreover complete description torsion part mathbb namely mathbb tors mathbb addition characterize torsion parts mathbb where odd prime mathbb where main ingredients proofs explicit computations zeta functions relevant curves applications chebotarev density theorem