Mathematical Notes

, Volume 99, Issue 3–4, pp 397–405 | Cite as

Classification of zeta functions of bielliptic surfaces over finite fields

Article

Abstract

Let S be a bielliptic surface over a finite field, and let an elliptic curve B be the Albanese variety of S; then the zeta function of the surface S is equal to the zeta function of the direct product P1 × B. Therefore, the classification problem for the zeta functions of bielliptic surfaces is reduced to the existence problem for surfaces of a given type with a given Albanese curve. In the present paper, we complete this classification initiated in [1].

Keywords

finite field zeta function elliptic curve bielliptic surface 

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References

  1. 1.
    S. Yu. Rybakov, “Zeta functions of bi-elliptic surfaces over finite fields,” Mat. Zametki 83 2, 273–285 (2008) [Mat. Zametki 83 (1–2), 246–256 (2008)].MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. A. Tsfasman, “Nombre de points des surfaces sur un corps fini,” in Arithmetic, Geometry and Coding Theory (Walter de Gruyter, Berlin, 1996), pp. 209–224.Google Scholar
  3. 3.
    S. Rybakov, “Zeta functions of conic bundles and Del Pezzo surfaces of degree 4 over finite fields,” Mosc. Math. J. 5 4, 919–926 (2005).MathSciNetMATHGoogle Scholar
  4. 4.
    S. Rybakov, “Finite group subschemes of abelian varieties over finite fields,” Finite Fields Appl. 29, 132–150 (2014); arXiv: 1006. 5959.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    E. Bombieri and D. Mumford, “Enriques’ classification of surfaces in char. p. II,” in Complex Analysis and Algebraic Geometry (Iwanami Shoten, Tokyo, 1977), pp. 23–42.CrossRefGoogle Scholar
  6. 6.
    M. Demazure, Lectures on p-Divisible Groups, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1972), Vol. 302.Google Scholar
  7. 7.
    M. Deuring, “Die Typen der Multiplikatorenringe elliptischer Funktionenkörper,” Abh. Math. Sem. Univ. Hamburg 14 1, 197–272 (1941).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    W. C. Waterhouse, “Abelian varieties over finite fields,” Ann. Sci. école. Norm. Sup. 2, 521–560 (1969).MathSciNetMATHGoogle Scholar
  9. 9.
    M. A. Tsfasman, “Group of points of an elliptic curve over a finite field,” in Theory of Numbers and Its Applications (Tbilisi, 1985), pp. 286–287.Google Scholar
  10. 10.
    S. G. Vladut, D. Yu. Nogin, and M. A. Tsfasman, Algebraic Geometry Codes. Basic Notions (MTsNMO, Moscow, 2003; AMS, Providence, RI, 2007).MATHGoogle Scholar
  11. 11.
    L. Badescu, Algebraic Surfaces, in Universitext (Springer-Verlag, New York, 2001).Google Scholar
  12. 12.
    D. Mumford, Abelian Varieties (Oxford University Press, London, 1970; Mir, Moscow, 1971).MATHGoogle Scholar
  13. 13.
    R. Hartshorne, Algebraic Geometry (Springer-Verlag, New York–Heidelberg, 1977;Mir, Moscow, 1981).MATHGoogle Scholar
  14. 14.
    J. Milne, Abelian Varieties (2008), http://www. jmilne. org/math/CourseNotes/avhtml.MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia
  2. 2.Laboratoire PonceletIndependent University of MoscowMoscowRussia
  3. 3.Laboratory of Algebraic Geometry and Its ApplicationsNational Research University Higher School of EconomicsMoscowRussia

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