Original scientific paper

https://doi.org/10.3336/gm.54.2.04

Generators and integral points on certain quartic curves

Yasutsugu Fujita orcid.org/0000-0001-7985-9667 ; College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan
Tadahisa Nara ; Faculty of Engineering, Tohoku-Gakuin University, 1-13-1 Chuo, Tagajo, Miyagi 985-8537, Japan

Abstract

In this paper, we study integral points and generators on quartic curves of the forms \(u^2 \pm v^4=m\) for a nonzero integer $m$. The main results assert that certain integral points on the curves can be extended to bases for the Mordell-Weil groups of the elliptic curves attached to the quartic curves in the cases where the Mordell-Weil ranks are at most two. As corollaries, we explicitly describe the integral points on the quartic curves in each case where the ranks are one and two.

Keywords

Elliptic curve; quartic curve; canonical height; integral points

Hrčak ID:

229601

URI

https://hrcak.srce.hr/229601

Publication date:

11.12.2019.

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