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Some cubic surfaces with no rational points

Published online by Cambridge University Press:  24 October 2008

Andrew Bremner
Andrew Bremner
Affiliation:
Emmanuel College, Cambridge
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Extract

Selmer(1) conjectured that the Hasse principle holds for all cubic surfaces of the type

that is, such a surface has a rational point whenever it has points defined over every p-adic field Qp; and he proved this assertion in the case that ab = cd.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 2 , September 1978 , pp. 219 - 223
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Selmer, E. S.Sufficient congruence conditions for the existence of rational points on certain cubic surfaces. Math. Scand. 1 (1953), 113–119.CrossRefGoogle Scholar
(2)Cassels, J. W. S. and Guy, M. J. T.On the Hasse principle for cubic surfaces. Mathematika 13 (1966) 111–120.CrossRefGoogle Scholar
(3)Manin, Y. I.Cubic forms (North Holland).Google Scholar
(4)Selmer, E. S.Tables for the purely cubic field K (), Avh. Norske Vid. Akad. Oslo I (1955), no. 5.Google Scholar
(5)Lang, S.Algebraic number theory (Addison Wesley).Google Scholar
(6)Jacobi, C. G. J.Canon arithmeticus (nach Berechnungen von W. Patz neu herausgegeben von H. Brandt) (Akademie–Verlag, 1956).Google Scholar
(7)Skolem, Th.Unlösbarkeit von Gleichungen, deren entsprechende Kongruenz für jeden Modul lösbar ist. Avh. Norske Vid. Akad. Oslo I (1955). no. 4.Google Scholar