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Manin's conjecture for a cubic surface with 2A2 + A1 singularity type

Published online by Cambridge University Press:  14 August 2012

PIERRE LE BOUDEC
PIERRE LE BOUDEC*
Affiliation:
Université Denis Diderot (Paris VII), Institut de Mathématiques de Jussieu, UMR 7586, Case 7012 - Bâtiment Chevaleret, Bureau 7C14, 75205 Paris Cedex 13, France. e-mail: pleboude@math.jussieu.fr
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Abstract

We establish Manin's conjecture for a cubic surface split over ℚ and whose singularity type is 2A2 + A1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath–Brown) and draws on the work of Deligne.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 3 , November 2012 , pp. 419 - 455
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[BB10]Baier, S. and Browning, T. D. Inhomogeneous cubic congruences and rational points on del Pezzo surfaces. J. Reine Angew. Math. To appear, arXiv:1011.3434v2 (2010).Google Scholar
[BBD07]Bretèche, R. de la, Browning, T. D. and Derenthal, U.On Manin's conjecture for a certain singular cubic surface. Ann. Sci. École Norm. Sup. (4) 40 (1) (2007), 1–50.Google Scholar
[BD09a]Browning, T. D. and Derenthal, U. Manin's conjecture for a cubic surface with D 5 singularity. Int. Math. Res. Not. IMRN (14) (2009), 2620–2647.CrossRefGoogle Scholar
[BD09b]Browning, T. D. and Derenthal, U.Manin's conjecture for a quartic del Pezzo surface with A 4 singularity. Ann. Inst. Fourier (Grenoble) 59 (3) (2009), 1231–1265.CrossRefGoogle Scholar
[Bre98]Bretèche, R. de la Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière. Astérisque (251) (1998), 51–77. Nombre et répartition de points de hauteur bornée (Paris, 1996).Google Scholar
[Bro06]Browning, T. D.The density of rational points on a certain singular cubic surface. J. Number Theory 119 (2) (2006), 242–283.CrossRefGoogle Scholar
[BSD07]Bretèche, R. de la and Swinnerton–Dyer, P.Fonction zêta des hauteurs associée à une certaine surface cubique. Bull. Soc. Math. France 135 (1) (2007), 65–92.CrossRefGoogle Scholar
[BT98]Batyrev, V. V. and Tschinkel, Y.Manin's conjecture for toric varieties. J. Algebraic Geom. 7 (1) (1998), 15–53.Google Scholar
[Bur62]Burgess, D. A.On character sums and primitive roots. Proc. London Math. Soc. (3) 12 (1962), 179–192.CrossRefGoogle Scholar
[BW79]Bruce, J. W. and Wall, C. T. C.On the classification of cubic surfaces. J. London Math. Soc. (2) 19 (2) (1979), 245–256.CrossRefGoogle Scholar
[Cay69]Cayley, A.A memoir on cubic surfaces. Phil. Trans. Roy. Soc. 159 (1869), 231–326.Google Scholar
[CLT02]Chambert-Loir, A. and Tschinkel, Y.On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2) (2002) 421–452.CrossRefGoogle Scholar
[Del74]Deligne, P.La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. (43) (1974), 273–307.CrossRefGoogle Scholar
[Der06a]Derenthal, U. Geometry of universal torsors. PhD. thesis. Georg-August-Universität Göttingen (2006).Google Scholar
[Der06b]Derenthal, U. Singular del Pezzo surfaces whose universal torsors are hypersurfaces. arXiv:math/0604194v1. (2006).CrossRefGoogle Scholar
[Der07]Derenthal, U.On a constant arising in Manin's conjecture for del Pezzo surfaces. Math. Res. Lett. 14 (3) (2007), 481–489.CrossRefGoogle Scholar
[DJT08]Derenthal, U., Joyce, M. and Teitler, Z.The nef cone volume of generalized del Pezzo surfaces. Algebra Number Theory 2 (2) (2008), 157–182.CrossRefGoogle Scholar
[DL10]Derenthal, U. and Loughran, D.Singular del Pezzo surfaces that are equivariant compactifications. Zapiski Nauchnykh Seminarov (POMI) 377 (2010), 26–43.Google Scholar
[FI85]Friedlander, J. B. and Iwaniec, H.Incomplete Kloosterman sums and a divisor problem. Ann. of Math. (2) 121 (2) (1985), 319–350. With an appendix by B. J. Birch and E. Bombieri.CrossRefGoogle Scholar
[FMT89]Franke, J., Manin, Y. I. and Tschinkel, Y.Rational points of bounded height on Fano varieties. Invent. Math. 95 (2) (1989), 421–435.CrossRefGoogle Scholar
[Fou98]Fouvry, E.Sur la hauteur des points d'une certaine surface cubique singulière. Astérisque (251) (1998), 31–49. Nombre et répartition de points de hauteur bornée (Paris, 1996).Google Scholar
[Fra09]Franz, M. Convex - a Maple package for convex geometry, version 1.1 (2009).Google Scholar
[HB86]Heath–Brown, D. R.The divisor function d 3(n) in arithmetic progressions. Acta Arith. 47 (1) (1986), 29–56.CrossRefGoogle Scholar
[HB03]Heath–Brown, D. R.The density of rational points on Cayley's cubic surface. In Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, vol. 360, p. 33 (University Bonn, 2003.Google Scholar
[HBM99]Heath–Brown, D. R. and Moroz, B. Z.The density of rational points on the cubic surface X 03 = X 1X 2X 3. Math. Proc. Camb. Phil. Soc. 125 (3) (1999), 385–395.Google Scholar
[HT04]Hassett, B. and Tschinkel, Y.Universal torsors and Cox rings. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr. Math., vol. 226, pp. 149–173. (Birkhäuser Boston, 2004).CrossRefGoogle Scholar
[Joy08]Joyce, M.Rational points on the E 6 cubic surface. PhD. thesis, Brown University (2008).Google Scholar
[LB12]Le Boudec, P.Manin's conjecture for two quartic del Pezzo surfaces with 3 A1 and A1 + A2 singularity types. Acta Arith. 151 (2) (2012), 109–163.CrossRefGoogle Scholar
[Lou10]Loughran, D.Manin's conjecture for a singular sextic del Pezzo surface. J. Théor. Nombres Bordeaux 22 (3) (2010), 675–701.CrossRefGoogle Scholar
[Pey95]Peyre, E.Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79 (1) (1995), 101–218.CrossRefGoogle Scholar
[Sal98]Salberger, P.Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque 251 (1998), 91–258. Nombre et répartition de points de hauteur bornée (Paris, 1996).Google Scholar
[Sch63]Schläfli, L.On the distribution of surfaces of the third order into species, in reference to the absence or presence of singular points, and the reality of their lines. Phil. Trans. Roy. Soc. 153 (1863), 193–241.Google Scholar
[Smi79]Smith, R. A.On n-dimensional Kloosterman sums. J. Number Theory 11 (3 S. Chowla Anniversary Issue) (1979), 324–343.CrossRefGoogle Scholar
[Wei81]Weinstein, L.The hyper-Kloosterman sum. Enseign. Math. (2) 27(1–2) (1981), 29–40.Google Scholar