Tamagawa numbers of diagonal cubic surfaces, numerical evidence

Peyre, Emmanuel; Tschinkel, Yuri

doi:10.1090/S0025-5718-00-01189-3

Tamagawa numbers of diagonal cubic surfaces, numerical evidence
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by Emmanuel Peyre and Yuri Tschinkel;

Math. Comp. 70 (2001), 367-387

DOI: https://doi.org/10.1090/S0025-5718-00-01189-3

Published electronically: June 12, 2000

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Abstract:

A refined version of Manin’s conjecture about the asymptotics of points of bounded height on Fano varieties has been developed by Batyrev and the authors. We test numerically this refined conjecture for some diagonal cubic surfaces.

References

V. V. Batyrev and Yu. I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Ann. 286 (1990), no. 1-3, 27–43 (French). MR 1032922, DOI 10.1007/BF01453564
Victor V. Batyrev and Yuri Tschinkel, Rational points of bounded height on compactifications of anisotropic tori, Internat. Math. Res. Notices 12 (1995), 591–635. MR 1369408, DOI 10.1155/S1073792895000365
Victor V. Batyrev and Yuri Tschinkel, Rational points on some Fano cubic bundles, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 1, 41–46 (English, with English and French summaries). MR 1401626
—, Tamagawa numbers of polarized algebraic varieties, Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 299–340.
D. J. Bernstein, Enumerating solutions to $p(a)+q(b)=r(c)+s(d)$, Preprint (1998).
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
Jean-Louis Colliot-Thélène, Hilbert’s Theorem 90 for $K_{2}$, with application to the Chow groups of rational surfaces, Invent. Math. 71 (1983), no. 1, 1–20. MR 688259, DOI 10.1007/BF01393336
J.-L. Colliot-Thélène, The Hasse principle in a pencil of algebraic varieties, Number theory (Tiruchirapalli, 1996) Contemp. Math., vol. 210, Amer. Math. Soc., Providence, RI, 1998, pp. 19–39. MR 1478483, DOI 10.1090/conm/210/02782
Jean-Louis Colliot-Thélène, Dimitri Kanevsky, and Jean-Jacques Sansuc, Arithmétique des surfaces cubiques diagonales, Diophantine approximation and transcendence theory (Bonn, 1985) Lecture Notes in Math., vol. 1290, Springer, Berlin, 1987, pp. 1–108 (French). MR 927558, DOI 10.1007/BFb0078705
Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La descente sur une variété rationnelle définie sur un corps de nombres, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 19, A1215–A1218 (French, with English summary). MR 447250
Jens Franke, Yuri I. Manin, and Yuri Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), no. 2, 421–435. MR 974910, DOI 10.1007/BF01393904
D. R. Heath-Brown, The density of zeros of forms for which weak approximation fails, Math. Comp. 59 (1992), no. 200, 613–623. MR 1146835, DOI 10.1090/S0025-5718-1992-1146835-5
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
Gilles Lachaud, Une présentation adélique de la série singulière et du problème de Waring, Enseign. Math. (2) 28 (1982), no. 1-2, 139–169 (French). MR 670501
Y. I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars Éditeur, Paris, 1971, pp. 401–411. MR 427322
Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR 833513
Emmanuel Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), no. 1, 101–218 (French). MR 1340296, DOI 10.1215/S0012-7094-95-07904-6
—, Terme principal de la fonction zêta des hauteurs et torseurs universels, Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 259–298.
P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties, Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 91–258.
P. Salberger and A. N. Skorobogatov, Weak approximation for surfaces defined by two quadratic forms, Duke Math. J. 63 (1991), no. 2, 517–536. MR 1115119, DOI 10.1215/S0012-7094-91-06322-2
Peter Swinnerton-Dyer, Counting rational points on cubic surfaces, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 371–379. MR 1272709, DOI 10.1090/conm/162/01543