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On the Hasse principle for cubic surfaces

Published online by Cambridge University Press:  26 February 2010

J. W. S. Cassels  and
M. J. T. Guy
J. W. S. Cassels
Affiliation:
University of Cambridge.
M. J. T. Guy
Affiliation:
University of Cambridge.
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It was conjectured by Mordell [6] that the Hasse principle holds for cubic surfaces in 3-dimensional projective space other than cones†: i.e., that such a surface defined over the rational field 0 has a rational point whenever it has points defined over every p-adic field Qp. This conjecture was verified for singular cubic surfaces by Skolem [11” and for surfaces

with

by Selmer [9]: but it was disproved for cubic surfaces in general by Swinnerton-Dyer [12] (see also Mordell [7]). It therefore becomes of interest to specify fairly wide classes of cubic surfaces for which the Hasse principle does hold. It was shown independently by F. Châtєlet and by Swinnerton-Dyer (both, apparently, unpublished) that this is the case when it contains a set of either 3 or 6 mutually skew lines which are rational as a whole (and trivially true when there is a rational pair of lines, since then there are always rational points). Selmer [9] conjectures on the basis of numerical evidence that the Hasse principle is also true for all surfaces of the type (1). It is the object of this note to disprove this by showing that the Hasse principle fails for

Type
Research Article
Information
Mathematika , Volume 13 , Issue 2 , December 1966 , pp. 111 - 120
Copyright
Copyright © University College London 1966

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References

1. Brauer, R., “Beziehungen zwischen Klassenzahlen von Teilkörpern eines G-aloisschen Körpers”, Math. Nachr., 4 (1950), 158–174.CrossRefGoogle Scholar
2. Dirichlet, P. G.Lejeune, Vorlesungen über Zahlentheorie. Mit Zusdtzen versehen von B. Dedekind (Braunschweig, 4te Auflage, 1894).Google Scholar
3. Hilbert, D., “Die Theorie der algebraischen Zahlkörper”, Jahresbericht der DMV, 4 (1897), 175–546.Google Scholar
4. Jacobi, C. G. J., Canon Arithmeticus (nach Berechnungen von W. Patz neu herausgegeben von H. Brandt) (Akademie-Verlag, 1956).Google Scholar
5. Kuroda, S., “Über die Klassenzahlen algebraischer Zahlkörper”, Nagoya Math. J., 1 (1950), 1–10.CrossRefGoogle Scholar
6. Mordell, L. J., “Rational points on cubic surfaces”, Pub. Math. Debrecen, 1 (1949), 1–6.Google Scholar
7. Mordell, L. J., “On the conjectures for rational points on a cubic surface”, Journal London Math. Soc, 40 (1965), 149–158.CrossRefGoogle Scholar
8. Nehrkorn, H., Über absolute Idealklassengruppen und Einheiten in algebraischen Zahlkörpern”, Abh. Math. Sem. Hamburg. Univ., 9 (1933), 318–334.CrossRefGoogle Scholar
9. Selmer, E. S., “Sufficient congruence conditions for the existence of rational points on certain cubic surfaces”, Math. Scand., 1 (1953), 113“119.CrossRefGoogle Scholar
10. Selmer, E. S. “Tables for the purely cubic field ”, Avh. Norske Vid. Akad. Oslo I, 1955 No. 5, 38 pp.Google Scholar
11. Skolom, Th., “Einige Bemerkungen über die Auffindung von rationalen Punkten auf gewissen algebraischen Gebilden”, Math. Zeitschrift, 63 (1955), 295–312.CrossRefGoogle Scholar
12. Swinnerton-Dyer, H. P. F., “Two special cubic surfaces”, Mathematika, 9 (1962), 54–56.CrossRefGoogle Scholar
13. Berwick, W. E. H., Integral Bases. Cambridge Tracts in Mathematics and Mathematical Physics, No. 22 (Cambridge, 1927).Google Scholar