Grigor Grigorov, Jordan Rizov
Heights on elliptic curves and the diophantine equation x⁴ + y⁴ = cz⁴
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

11G30 Curves of arbitrary genus or genus $\ne 1$ over global fields, See also {14H25}

11D25 Cubic and quartic equations

Abstract: In this paper we give sharp explicit estimates for the difference of the Weil height and the Néron - Tate height on the elliptic curve v² = u³ - cu. We then apply this in the proof of the fact that if c > 2 is a fourth power free integer and the rank of v² = u³ - cu is 1 then the equation x⁴ + y⁴ = cz⁴ has no nonzero solutions in integers
Keywords: elliptic curves, canonical height, diophantine equation