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Experiments with General Cubic Surfaces

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Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 269))

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.

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References

  1. Birch, B. J.: Forms in many variables, Proc. Roy. Soc. Ser. A 265 (1961/1962), 245–263.

    MathSciNet  Google Scholar 

  2. Cassels, J. W. S.: Bounds for the least solutions of homogeneous quadratic equations, Proc. Cambridge Philos. Soc. 51 (1955) 262–264.

    Article  MathSciNet  Google Scholar 

  3. Cohen, H.: A course in computational algebraic number theory, Springer, Berlin, Heidelberg 1993.

    MATH  Google Scholar 

  4. 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.

    Article  MATH  MathSciNet  Google Scholar 

  5. 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.

    MATH  MathSciNet  Google Scholar 

  6. 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.

    Google Scholar 

  7. 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.

    Google Scholar 

  8. Elsenhans, A.-S. and Jahnel, J.: On the smallest point on a diagonal quartic threefold, J. Ramanujan Math. Soc. 22 (2007) 189–204.

    MATH  MathSciNet  Google Scholar 

  9. Franke, J., Manin, Y. I., and Tschinkel, Y.: Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989) 421–435.

    Article  MATH  MathSciNet  Google Scholar 

  10. Handbook of numerical analysis, edited by P. G. Ciarlet and J. L. Lions, Vols. II–IV, North-Holland Publishing Co., Amsterdam 1991–1996.

    Google Scholar 

  11. Hermann, G.: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926) 736–788.

    Article  MATH  MathSciNet  Google Scholar 

  12. Manin, Yu. I.: Cubic forms, algebra, geometry, arithmetic, North-Holland Publishing Co. and American Elsevier Publishing Co., Amsterdam, London, and New York 1974.

    MATH  Google Scholar 

  13. 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.

    MathSciNet  Google Scholar 

  14. Pohst, M. and Zassenhaus, H.: Algorithmic Algebraic Number Theory, Cambridge University Press, Cambridge 1989.

    Book  MATH  Google Scholar 

  15. 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.

    Google Scholar 

  16. Siegel, C. L.: Normen algebraischer Zahlen, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II 1973 (1973) 197–215.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. 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.

    Book  Google Scholar 

  19. 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

    Google Scholar 

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Author information

Authors and Affiliations

  1. Universität Göttingen, Mathematisches Institut, Bunsenstraße 3–5, D-37073, Göttingen, Germany

    Andreas-Stephan Elsenhans & Jörg Jahnel

Authors
  1. Andreas-Stephan Elsenhans
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  2. Jörg Jahnel
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Corresponding author

Correspondence to Andreas-Stephan Elsenhans .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

  2. Eberly College of Science, Dept. Mathematics, Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Yuri Zarhin

Additional information

Dedicated to Yuri Ivanovich Manin on the occasion of his 70th birthday

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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

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