On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)

Tomasz Jędrzejak

Acta Arithmetica, Tome 172 (2016), p. 99-120 / Harvested from The Polish Digital Mathematics Library

Résumé

Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}: y² = xⁿ+a$ and $C_{n,a}: y² = x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}(ℚ) $ and $J_{n,a}(ℚ)$. 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 $J_{8,a}(ℚ)$. Namely, we show that $J_{8,a}(ℚ)_{tors} = J_{8,a}(ℚ)[2]$. In addition, we characterize the torsion parts of $J_{p,a}(ℚ)$, where p is an odd prime, and of $J^{n,a}(ℚ)$, 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

Publié le : 2016-01-01

Zbl 06602749

EUDML-ID : urn:eudml:doc:286254