Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

• Autorzy: Tomasz Jędrzejak
• Języki publikacji: EN

Abstrakty

EN

Consider the families of curves $C^{n,A} : y² = xⁿ + Ax$ and $C_{n,A} : y² = xⁿ + A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}(ℚ)$ and $C_{3,A}(ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}(ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}(ℚ)[2]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ (A ≠ 4) and $J^{5,A}$. We also almost completely determine the ℚ-rational torsion of $J_{p,A}$ for all odd primes p, and all A ∈ ℚ∖{0}. We discuss the excluded case (i.e. $A ∈ (-1)^{(p-1)/2}pℕ²$).

Kategorie tematyczne

• 11G10: Abelian varieties of dimension > 1
• 11G20: Curves over finite and local fields
• 11L10: Jacobsthal and Brewer sums; other complete character sums
• 11G30: Curves of arbitrary genus or genus ≠ 1 over global fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 161
• Numer: 3
• Strony: 201-218

Daty

wydano

2013

Twórcy

autor

Tomasz Jędrzejak

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

Identyfikatory

DOI

10.4064/aa161-3-1