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Sufficient congruence conditions for the existence of rational points on certain cubic surfaces. (English) Zbl 0051.03202
L. J. Mordell [Publ. Math. 1, 1–6 (1949; Zbl 0033.16001)] has conjectured that the elementary congruence conditions are sufficient for the existence of rational points on cubic surfaces. The author proves this conjecture for the purely cubic equation \[ a_1x_1^3+a_2x_2^3+a_3x_3^3+a_4x_4^3=0,\qquad a_1a_2a_3a_4\neq 0, \tag{1} \] satisfying the additional condition that (for instance) (2) \(a_3a_4/a_1a_2= \) a rational cube. If (2) is satisfied, (1) can be transformed into \(x^3+my^3=n(u^3+mv^3)\), with integer, cubefree \(m\) and \(n\), and the last equation is treated in the field \(K\left(\root 3\of m\right)\). The class number of this field is significant. To cover all cases a result of H. Hasse [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1931, 64–69 (1931; Zbl 0003.19902)] is needed. Finally it is shown how the method can be extended to prove the conjecture for the more general equation \(f_3(x,y)=n\cdot f_3(u,v)\), where \(f_3\) is an arbitrary binary cubic form.
Reviewer: W. Ljunggren

MSC:
11D25 Cubic and quartic Diophantine equations
14G05 Rational points
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