Glasnik Matematicki, Vol. 54, No. 2 (2019), 321-343.

GENERATORS AND INTEGRAL POINTS ON CERTAIN QUARTIC CURVES

Yasutsugu Fujita and Tadahisa Nara

College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan
e-mail: fujita.yasutsugu@nihon-u.ac.jp

Faculty of Engineering, Tohoku-Gakuin University, 1-13-1 Chuo, Tagajo, Miyagi 985-8537, Japan
e-mail: sa4m19@math.tohoku.ac.jp

Abstract.   In this paper, we study integral points and generators on quartic curves of the forms u2 ± v4=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.

2010 Mathematics Subject Classification.   11G05, 11D25, 11G50

Key words and phrases.   Elliptic curve, quartic curve, canonical height, integral points

Full text (PDF) (free access)

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

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