Advertisement

The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds

  • Andreas-Stephan Elsenhans
  • Jörg Jahnel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)

Abstract

For the families ax 3 = by 3 + z 3 + v 3 + w 3, a, b = 1, ... ,100, and ax 4 = by 4 + z 4 + v 4 + w 4, a, b = 1, ... ,100, of projective algebraic threefolds, we test numerically the conjecture of Manin (in the refined form due to Peyre) about the asymptotics of points of bounded height on Fano varieties.

Keywords

Rational Point Complete Intersection Diophantine Equation Rational Line Fano Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ap]
    Apéry, F.: Models of the real projective plane. Vieweg, Braunschweig (1987)Google Scholar
  2. [BT]
    Batyrev, V.V., Tschinkel, Y.: Rational points on some Fano cubic bundles. C. R. Acad. Sci. Paris 323, 41–46 (1996)MATHMathSciNetGoogle Scholar
  3. [Bv]
    Beauville, A.: Complex algebraic surfaces. LMS Lecture Note Series, vol. 68. Cambridge University Press, Cambridge (1983)MATHGoogle Scholar
  4. [Be]
    Bernstein, D.J.: Enumerating solutions to p(a) + q(b) = r(c) + s(d). Math. Comp. 70, 389–394 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. [Bi]
    Birch, B.J.: Forms in many variables. Proc. Roy. Soc. Ser. A 265, 245–263 (1961/1962)Google Scholar
  6. [CG]
    Clemens, C.H., Griffiths, P.A.: The intermediate Jacobian of the cubic threefold. Ann. of Math. 95, 281–356 (1972)CrossRefMathSciNetGoogle Scholar
  7. [CLR]
    Cormen, T., Leiserson, C., Rivest, R.: Introduction to algorithms. MIT Press and McGraw-Hill, Cambridge and New York (1990)MATHGoogle Scholar
  8. [De]
    Deligne, P.: La conjecture de Weil I. Publ. Math. IHES 43, 273–307 (1974)MathSciNetGoogle Scholar
  9. [EJ1]
    Elsenhans, A.-S., Jahnel, J.: The Diophantine equation x 4 + 2y 4 = z 4 + 4w 4. Math. Comp. 75, 935–940 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. [EJ2]
    Elsenhans, A.-S., Jahnel, J.: The Diophantine equation x 4 + 2y 4 = z 4 + 4w 4 — a number of improvements (preprint)Google Scholar
  11. [Fo]
    Forster, O.: Algorithmische Zahlentheorie. Vieweg, Braunschweig (1996)Google Scholar
  12. [FMT]
    Franke, J., Manin, Y.I., Tschinkel, Y.: Rational points of bounded height on Fano varieties. Invent. Math. 95, 421–435 (1989)MATHCrossRefMathSciNetGoogle Scholar
  13. [Ha1]
    Hartshorne, R.: Ample subvarieties of algebraic varieties. Lecture Notes Math., vol. 156. Springer, Berlin-New York (1970)MATHCrossRefGoogle Scholar
  14. [Ha2]
    Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York-Heidelberg (1977)MATHGoogle Scholar
  15. [H-B]
    Heath-Brown, D.R.: The density of zeros of forms for which weak approximation fails. Math. Comp. 59, 613–623 (1992)MATHCrossRefMathSciNetGoogle Scholar
  16. [Hi]
    Hirzebruch, F.: Der Satz von Riemann-Roch in Faisceau-theoretischer Formulierung: Einige Anwendungen und offene Fragen. In: Erven, P., Noordhoff, N.V. (eds.) Proc. Int. Cong., 1954, vol. III, pp. 457–473. North-Holland, Groningen and Amsterdam (1956)Google Scholar
  17. [Kr]
    Kress, R.: Numerical analysis. Graduate Texts in Mathematics, vol. 181. Springer, New York (1998)MATHGoogle Scholar
  18. [Mu]
    Murre, J.P.: Algebraic equivalence modulo rational equivalence on a cubic threefold. Compositio Math. 25, 161–206 (1972)MATHMathSciNetGoogle Scholar
  19. [Pe1]
    Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79, 101–218 (1995)MATHCrossRefMathSciNetGoogle Scholar
  20. [Pe2]
    Peyre, E.: Points de hauteur bornée et géométrie des variétés (d’après Y. Manin et al.). Séminaire Bourbaki 2000/2001, Astérisque 282, 323–344 (2002)MathSciNetGoogle Scholar
  21. [PT]
    Peyre, E., Tschinkel, Y.: Tamagawa numbers of diagonal cubic surfaces, numerical evidence. Math. Comp. 70, 367–387 (2001)MATHCrossRefMathSciNetGoogle Scholar
  22. [SchSt]
    Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971)MATHCrossRefGoogle Scholar
  23. [Se]
    Sedgewick, R.: Algorithms. Addison-Wesley, Reading  (1983)MATHGoogle Scholar
  24. [S-D]
    Swinnerton-Dyer, S.P.: Counting points on cubic surfaces II. In: Geometric methods in algebra and number theory, Progr. Math., vol. 235, pp. 303–309. Birkhäuser, Boston (2005)Google Scholar
  25. [Za]
    Zak, F.L.: Tangents and secants of algebraic varieties. AMS Translations of Mathematical Monographs, vol. 127, Providence (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Andreas-Stephan Elsenhans
    • 1
  • Jörg Jahnel
    • 1
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenGermany

Personalised recommendations