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

Yasutsugu Fujita, and Nobuhiro Terai. "Generators and integer points on the elliptic curve y² = x³ - nx." Acta Arithmetica 160.4 (2013): 333-348. <http://eudml.org/doc/286335>.

@article{YasutsuguFujita2013,
abstract = {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.},
author = {Yasutsugu Fujita, Nobuhiro Terai},
journal = {Acta Arithmetica},
keywords = {elliptic curves; generators; integer points; canonical heights},
language = {eng},
number = {4},
pages = {333-348},
title = {Generators and integer points on the elliptic curve y² = x³ - nx},
url = {http://eudml.org/doc/286335},
volume = {160},
year = {2013},
}

TY - JOUR
AU - Yasutsugu Fujita
AU - Nobuhiro Terai
TI - Generators and integer points on the elliptic curve y² = x³ - nx
JO - Acta Arithmetica
PY - 2013
VL - 160
IS - 4
SP - 333
EP - 348
AB - 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.
LA - eng
KW - elliptic curves; generators; integer points; canonical heights
UR - http://eudml.org/doc/286335
ER -