On the torsion of the Jacobians of the hyperelliptic curves $y^{2}=x^{n}+a$ and $y^{2}=x(x^{n}+a)$
Volume 174 / 2016
Abstract
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.
