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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

##### MSC:
 14G05 Rational points 11D25 Cubic and quartic Diophantine equations 14H25 Arithmetic ground fields for curves
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