Hostname: page-component-76d6cb85b7-s74w7 Total loading time: 0 Render date: 2026-07-20T04:09:35.374Z Has data issue: false hasContentIssue false
  • English
  • Français

The Brauer group of cubic surfaces

Published online by Cambridge University Press:  24 October 2008

Sir Peter Swinnerton-Dyer
Sir Peter Swinnerton-Dyer
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection

is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 3 , May 1993 , pp. 449 - 460
Copyright
Copyright © Cambridge Philosophical Society 1993

References

REFERENCES

[1]Colliot-Thélène, J.-L.. Kanevsky, D. and Sansuc, J. J.. Arithmétique des surfaces cubiques diagonales. In Diophantine Approximation and Transcendence Theory, Lecture Notes in Math. vol. 1290 (Springer-Verlag, 1987), pp. 1–108.CrossRefGoogle Scholar
[2]Colliot-Thélène, J.-L., Sansuc, J. J. and Swinnerton-Dyer, Sir Peter. Intersections of two quadrics and Châtelet surfaces. J. Reine Angew. Math. 373 (1987), 37–107Google Scholar
Colliot-Thélène, J.-L., Sansuc, J. J. and Swinnerton-Dyer, Sir Peter. Intersections of two quadrics and Châtelet surfaces. J. Reine Angew. Math. and 374 (1987), 72–168.Google Scholar
[3]Henderson, A.. The Twenty-seven Lines upon the Cubic Surface (Cambridge University Press, 1911).Google Scholar
[4]Iskovskih, V. A.. Rational surfaces with a pencil of rational curves. Mat. Sbornik 74 (116) (1967), 608–638.Google Scholar
[5]Manin, Yu. I.. Cubic Forms: Algebra, Geometry, Arithmetic (North Holland, 1974).Google Scholar
[6]Segre, B.. The Nonsingular Cubic Surfaces (Oxford University Press, 1942).Google Scholar
[7]Swinnerton-Dyer, H. P. F.. The zeta function of a cubic surface over a finite field. Proc. Cambridge Philos. Soc. 63 (1967), 55–71.CrossRefGoogle Scholar
[8]Urabe, T. A remark on a conjecture of Manin. Preprint.Google Scholar