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The Lang–Weil Estimate for Cubic Hypersurfaces

Published online by Cambridge University Press:  20 November 2018

T. D. Browning
T. D. Browning*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom e-mail: t.d.browning@bristol.ac.uk
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Abstract

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An improved estimate is provided for the number of ${{\text{F}}_{q}}$-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

Keywords

11G2514G15cubic hypersurfacerational pointsfinite fields
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 3 , 01 September 2013 , pp. 500 - 502
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Bombieri, E., On exponential sums in finite fields. II. Invent. Math. 47 (1978, no. 1, 29–39. http://dx.doi.org/10.1007/BF01609477 Google Scholar
[2] Bruce, J.W. and Wall, C. T. C., On the classification of cubic surfaces. J. London Math. Soc. 19 (1979, no. 2, 245–256. http://dx.doi.org/10.1112/jlms/s2-19.2.245 Google Scholar
[3] Coray, D. F. and Tsfasman, M. A., Arithmetic on singular Del Pezzo surfaces. Proc. London Math. Soc. 57 (1988, no. 1, 25–87. http://dx.doi.org/10.1112/plms/s3-57.1.25 Google Scholar
[4] Davenport, H. and Lewis, D. J., Exponential sums in many variables. Amer. J. Math. 84 (1962, 649–665. http://dx.doi.org/10.2307/2372871 Google Scholar
[5] Davenport, H. and Lewis, D. J., Notes on congruences. II. Quart. J. Math. Oxford Ser. (2) 14 (1963, 153–159. http://dx.doi.org/10.1093/qmath/14.1.153 Google Scholar
[6] Deligne, P., La conjecture de Weil. II. Inst. Hautes E´ tudes Sci. Publ. Math. 52 (1980, 137–252.Google Scholar
[7] Hooley, C., On the number of points on a complete intersection over a finite field. J. Number Theory 38 (1991, no. 3, 338–358. http://dx.doi.org/10.1016/0022-314X(91)90023-5 Google Scholar
[8] Lang, S. and Weil, A., Number of points of varieties in finite fields. Amer. J. Math. 76 (1954, 819–827. http://dx.doi.org/10.2307/2372655 Google Scholar
[9] Swinnerton-Dyer, H. P. F., The zeta function of a cubic surface over a finite field. Proc. Cambridge Philos. Soc. 63 (1967, 55–71. http://dx.doi.org/10.1017/S0305004100040895 Google Scholar