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Counting Rational Points on Cubic Hypersurfaces

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  21 December 2009

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

Let X ⊂ ℙN be a geometrically integral cubic hypersurface defined over ℚ, with singular locus of dimension at most dim X − 4. The main result in this paper is a proof of the fact that X(ℚ) contains OɛX (BdimX + ɛ) points of height at most B.

MSC classification

Secondary: 11G35: Varieties over global fields 14G05: Rational points
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 93 - 112
Copyright
Copyright © University College London 2007

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References

1Browning, T. D. and Heath-Brown, D. R., Counting rational points on hypersurfaces. J. reine angew. Math. 584 (2005), 83–115.CrossRefGoogle Scholar
2Browning, T. D. and Heath-Brown, D. R., Counting rational points on smooth hypersurfaces, II. Proc. London Math. Soc. 93 (2006), 273–303.CrossRefGoogle Scholar
3Browning, T. D. and Heath-Brown, D. R., Rational points on quartic hypersurfaces. J. reine angew. Math. (to appear).Google Scholar
4Browning, T. D. and Heath-Brown, D. R., Integral points on cubic hypersurfaces. Analytic Number Theory: Essays in honour of Klaus Roth, Cambridge University Press (to appear).Google Scholar
5Davenport, H., Cubic forms in sixteen variables. Proc. Roy. Soc. Ser. A 272 (1963), 285–303.Google Scholar
6Heath-Brown, D. R., Cubic forms in ten variables. Proc. London Math. Soc. 47 (1983), 225–257.Google Scholar
7Heath-Brown, D. R., The density of rational points on non-singular hypersurfaces. Proc. Indian Acad. Sci. 104 (1994), 13–29.Google Scholar
8Heath-Brown, D. R., The density of rational points on curves and surfaces. Annals of Math. 155 (2002), 553–595.CrossRefGoogle Scholar
9Heath-Brown, D. R., Rational points on cubic hypersurfaces. Invent. Math. 170 (2007), 199–230.Google Scholar
10Pila, J., Density of integral and rational points on varieties. Astérisque 228 (1995), 183–187.Google Scholar
11Salberger, P., Counting rational points on projective varieties (in preparation).Google Scholar
12Salberger, P., Integral points on hypersurfaces of degree at least 3 (in preparation).Google Scholar