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

• Autorzy: Tomasz Jędrzejak
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11G10: Abelian varieties of dimension > 1
• 11G20: Curves over finite and local fields
• 11L10: Jacobsthal and Brewer sums; other complete character sums
• 11L05: Gauss and Kloosterman sums; generalizations
• 11G25: Varieties over finite and local fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 174
• Numer: 2
• Strony: 99-120

Daty

wydano

2016

Twórcy

autor

Tomasz Jędrzejak

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

Identyfikatory

DOI

10.4064/aa8141-3-2016