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On diagonal cubic surfaces. (English) Zbl 0652.14019

The author studies rational curves on the surface \(ax^3+by^3+cz^3=dt^3\) for cube-free rational numbers \(a,b,c,d\) extending work done by B. Segre [Math. Notae 11, 1–68 (1951; Zbl 0043.27501)] and H. P. F. Swinnerton-Dyer [1969 Number Theory Inst., Proc. Symp. Pure Math. 20, 1–52 (1971; Zbl 0228.14001)]. The author shows that there are no curves of arithmetic genus 0 defined over \(\mathbb Q\) unless (after permuting coordinates) \(a=b=1\) and \(c=d\). For the surface \(x^3+y^3- cz^3=ct^3\), he shows that there are exactly two such curves if \(c\neq 2\), and four if \(c=2\); and he gives explicit equations for these curves. Finally he generalizes an example of D. H. Lehmer [J. Lond. Math. Soc. 31, 275–280 (1956; Zbl 0072.26804)] to show that the surfaces \(x^3+y^3+cz^3=ct^3\) contain infinitely many curves defined over \(\mathbb Q\) of geometric genus 0 and arithmetic genus greater than 0.

MSC:

11D25 Cubic and quartic Diophantine equations
11D72 Diophantine equations in many variables
11G35 Varieties over global fields
14G25 Global ground fields in algebraic geometry
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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References:

[1] LEHMER, D.H., On the Diophantine equationx 3+y 3+z 3=1, J. London Math. Soc., 31, 275-280, (1956). · Zbl 0072.26804 · doi:10.1112/jlms/s1-31.3.275
[2] SEGRE, B., On the rational solutions of homogeneous cubic equations in four variables, Math. Notae (Rosario), 11, 1-68, (1951). · Zbl 0043.27501
[3] SWINNERTON-DYER, H. P. F., Applications of algebraic geometry to number theory, 1969 Institute on Number Theory, Amer. Math. Soc. Proceedings XX, 1-52.
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