Hostname: page-component-6bb9c88b65-k72x6 Total loading time: 0 Render date: 2025-07-18T14:46:37.635Z Has data issue: false hasContentIssue false
  • English
  • Français

Cubic surfaces over finite fields

Published online by Cambridge University Press:  07 July 2010

PETER SWINNERTON–DYER
PETER SWINNERTON–DYER*
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Rd, Cambridge, CB3 0WB. e-mail: H.P.F.Swinnerton-Dyer@dpmms.cam.ac.uk
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.

Let V be a nonsingular cubic surface defined over the finite field Fq. It is well known that the number of points on V satisfies #V(Fq) = q2 + nq + 1 where −2 ≤ n ≤ 7 and that n = 6 is impossible; see for example [1], Table 1. Serre has asked if these bounds are best possible for each q. In this paper I shall show that this is so, with three exceptions:

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 3 , November 2010 , pp. 385 - 388
Copyright
Copyright © Cambridge Philosophical Society 2010

References

REFERENCE

[1]Manin, Yu. I.Cubic Forms: Algebra, Geometry, Arithmetic (Amsterdam, 1974).Google Scholar