A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$

• Autorzy: Tomasz Jędrzejak
• Języki publikacji: EN

Abstrakty

EN

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}: y^{q} = x^{p} + a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}(ℚ)$ (resp. $J_{q,p,a}(ℚ)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over $𝔽_{l}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp)) and applications of the Chebotarev Density Theorem.

Kategorie tematyczne

• 11G10: Abelian varieties of dimension > 1
• 11G20: Curves over finite and local fields
• 11L05: Gauss and Kloosterman sums; generalizations
• 11G30: Curves of arbitrary genus or genus ≠ 1 over global fields
• 11G25: Varieties over finite and local fields
• 11G15: Complex multiplication and moduli of abelian varieties
• 11R45: Density theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2016
• Tom: 108
• Numer: 1
• Strony: 143-149

Daty

wydano

2016

Twórcy

autor

Tomasz Jędrzejak

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

Identyfikatory

DOI

10.4064/bc108-0-11