zbMATH — the first resource for mathematics

Some quartic curves with no points in any cubic field. (English) Zbl 0602.14020
Proc. Lond. Math. Soc., III. Ser. 52, 193-214 (1986); corrigendum ibid. 116, No. 4, 1028 (2018).
It is shown that for certain values of the rational number D the equation \(x^ 4+y^ 4=Dz^ 4\) has no (non-zero) solution, even if (x,y,z) are allowed to lie in a cubic extension of the rationals. The essential condition on D is that the elliptic curve \(x^ 2+y^ 4=Dz^ 4\) has rank \(\leq 1\) over \({\mathbb{Q}}.\)
The method of proof uses an idea of J. W. S. Cassels [Proc. R. Soc. Edinb., Sect. A 100, 201-218 (1985; Zbl 0589.14029)]. The author also provides some very interesting examples.
Reviewer: G.Faltings

14G05 Rational points
11D25 Cubic and quartic Diophantine equations
14H25 Arithmetic ground fields for curves
Full Text: DOI