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Cubic surfaces over finite fields

Published online by Cambridge University Press:  07 July 2010

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
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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:

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2010

References

REFERENCE

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