Summary
For general cubic surfaces, we test numerically the conjecture of Manin (in the refined form due to E. Peyre) about the asymptotics of points of bounded height on Fano varieties. We also study the behavior of the height of the smallest rational point versus the Tamagawa type number introduced by Peyre.
2000 Mathematics Subject Classifications: 14G05, 11656 (Primary); 11Y50, 14J26 (Secondary)
The computer part of this work was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientific Computing at the Göttingen Mathematical Institute. Both authors are grateful to Prof. Y. Tschinkel for permission to use these machines as well as to the system administrators for their support.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Birch, B. J.: Forms in many variables, Proc. Roy. Soc. Ser. A 265 (1961/1962), 245–263.
Cassels, J. W. S.: Bounds for the least solutions of homogeneous quadratic equations, Proc. Cambridge Philos. Soc. 51 (1955) 262–264.
Cohen, H.: A course in computational algebraic number theory, Springer, Berlin, Heidelberg 1993.
Colliot-Thélène, J.-L. and Sansuc, J.-J.: On the Chow groups of certain rational surfaces: A sequel to a paper of S. Bloch, Duke Math. J. 48 (1981) 421–447.
Dickson, L. E.: Determination of all the subgroups of the known simple group of order \(25\,920\), Trans. Amer. Math. Soc. 5 (1904) 126–166.
Ekedahl, T.: An effective version of Hilbert's irreducibility theorem, in: Séminaire de Théorie des Nombres, Paris 1988–1989, Prog. Math. 91, Birkhäuser, Boston 1990, 241–249.
Elsenhans, A.-S. and Jahnel, J.: The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds, in: Algorithmic number theory, Lecture Notes in Computer Science 4076, Springer, Berlin 2006, 317–332.
Elsenhans, A.-S. and Jahnel, J.: On the smallest point on a diagonal quartic threefold, J. Ramanujan Math. Soc. 22 (2007) 189–204.
Franke, J., Manin, Y. I., and Tschinkel, Y.: Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989) 421–435.
Handbook of numerical analysis, edited by P. G. Ciarlet and J. L. Lions, Vols. II–IV, North-Holland Publishing Co., Amsterdam 1991–1996.
Hermann, G.: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926) 736–788.
Manin, Yu. I.: Cubic forms, algebra, geometry, arithmetic, North-Holland Publishing Co. and American Elsevier Publishing Co., Amsterdam, London, and New York 1974.
Peyre, E.: Points de hauteur bornée et géométrie des variétés (d'après Y. Manin et al.), Séminaire Bourbaki 2000/2001, Astérisque 282 (2002) 323–344.
Pohst, M. and Zassenhaus, H.: Algorithmic Algebraic Number Theory, Cambridge University Press, Cambridge 1989.
Schwarz, H. A.: Sur une définition erronée de l'aire d'une surface courbe, Communication faite à M. Charles Hermite, 1881–82, in: Gesammelte Mathematische Abhandlungen, Zweiter Band, Springer, Berlin 1890, 309–311.
Siegel, C. L.: Normen algebraischer Zahlen, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II 1973 (1973) 197–215.
Sims, C. C.: Computational methods in the study of permutation groups, in: Computational Problems in Abstract Algebra (Proc. Conf., Oxford 1967), Pergamon, Oxford 1970, 169–183.
Swinnerton-Dyer, Sir P.: Counting points on cubic surfaces II, in: Geometric methods in algebra and number theory, Progr. Math. 235, Birkhäuser, Boston 2005, 303–309.
Tate, J.: Global class field theory, in: Algebraic number theory, Edited by J. W. S. Cassels and A. Fröhlich, Academic Press and Thompson Book Co., London and Washington 1967
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to Yuri Ivanovich Manin on the occasion of his 70th birthday
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Elsenhans, AS., Jahnel, J. (2009). Experiments with General Cubic Surfaces. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_14
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4745-2_14
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4744-5
Online ISBN: 978-0-8176-4745-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)