Generators and integer points on the elliptic curve y² = x³ - nx

• Autorzy: Yasutsugu Fujita , Nobuhiro Terai
• Języki publikacji: EN

Abstrakty

EN

Let E be an elliptic curve over the rationals ℚ given by y² = x³ - nx with a positive integer n. We consider first the case where n = N² for a square-free integer N. Then we show that if the Mordell-Weil group E(ℚ ) has rank one, there exist at most 17 integer points on E. Moreover, we show that for some parameterized N a certain point P can be in a system of generators for E(ℚ ), and we determine the integer points in the group generated by the point P and the torsion points. Secondly, we consider the case where n = s⁴ + t⁴ for distinct positive integers s and t. We then show that if n is fourth-power-free, the points P₁ = (-t²,s²t) and P₂ = (-s²,st²) can be in a system of generators for E(ℚ ). Furthermore, we prove that if n is square-free, then there exist at most nine integer points in the group Γ generated by the points P₁, P₂ and the torsion point (0,0). In particular, in case n = s⁴ + 1 the group Γ has exactly seven integer points.

Kategorie tematyczne

• 11D25: Cubic and quartic equations
• 11G50: Heights
• 14G05: Rational points
• 11G05: Elliptic curves over global fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 160
• Numer: 4
• Strony: 333-348

Daty

wydano

2013

Twórcy

autor

Yasutsugu Fujita

• College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan

autor

Nobuhiro Terai

• Division of Information System Design, Ashikaga Institute of Technology, 268-1 Omae, Ashikaga, Tochigi 326-8558, Japan

Identyfikatory

DOI

10.4064/aa160-4-3