Advertisement

On the Nature of the “Explicit Formulas” in Analytic Number Theory — A Simple Example

  • Christopher Deninger
Part of the Developments in Mathematics book series (DEVM, volume 8)

Abstract

We interpret the “explicit formulas” in the sense of analytic number theory for the zeta function of an elliptic curve over a finite field as a transversal index theorem on a 3-dimensional laminated space.

Keywords

Explicit formula elliptic curve transversal index transversally elliptic operator foliation arithmetic topology 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J.A. Alvarez López, Y. Kordyukov, Distributional Betti numbers of transitive foliations of codimension one. Preprint 2000Google Scholar
  2. [2]
    M.F. Atiyah, Elliptic operators and compact groups. Springer LNM 401, 1974Google Scholar
  3. [3]
    K. Barner, On A. Weil’s explicit formula. J. Reine Angew. Math. 323 (1981), 139–152MathSciNetzbMATHGoogle Scholar
  4. [4]
    C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces. Doc. Math. J. DMV. Extra Volume ICM I (1998), 23–46Google Scholar
  5. [5]
    C. Deninger, Number theory and dynamical systems on foliated spaces. In: Jber. d. dt. Math.-Verein 103 (2001), 79–100Google Scholar
  6. [6]
    C. Deninger, W. Singhof, A note on dynamical trace formulas. In: M.L. Lapidus, M. van Frankenhuysen (eds.), Dynamical Spectral and Arithmetic Zeta-Functions. In: AMS Contemp. Math. 290, 41–55Google Scholar
  7. [7]
    M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hamburg 14 (1941), 197–272MathSciNetCrossRefGoogle Scholar
  8. [8]
    Y. Ihara, On (co x p)-adic coverings of curves (the simplest example). Trudy Mat. Inst. Steklov 132 (1973), 133–148MathSciNetGoogle Scholar
  9. [9]
    Y. Ihara, On congruence monodromy problems. Vols 1,2, Lecture Notes, nos. 1,2, Dept. of Mathematics, Univ. of Tokyo, Tokyo 1968, 1969. MR # 6706, # 6707Google Scholar
  10. [10]
    Y. Ihara, Non-abelian classfields over function fields in special cases. Proc. Internat. Congress Math. (Nice 1970 ), vol 1, Gauthier-Villars, Paris 1971, 381390Google Scholar
  11. [11]
    Y. Ihara, Congruence relations and Shimura curves. Proceedings of Symp. Pure Math. 33 (1979) part 2, 291–311Google Scholar
  12. [12]
    C. Lazarov, Transverse index and periodic orbits. GAFA 10 (2000), 124–159MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    C.C. Moore, C. Schochet, Global analysis on foliated spaces. MSRI Publications 9, Springer 1988Google Scholar
  14. [14]
    A. Neske, F. Zickermann, The index of transversally elliptic complexes. Proceedings of the 13th winter school on abstract analysis (Srni, 1985 ). Rend. Circ. Mat. Palermo (2) Suppl. No. 9 (1986), 165–175Google Scholar
  15. [15]
    F. Oort, Lifting an endomorphism of an elliptic curve to characteristic zero. Indag. Math. 35 (1973), 466–470MathSciNetGoogle Scholar
  16. [16]
    Robocop 3, Orion pictures 1993Google Scholar
  17. [17]
    J.H. Silverman, The arithmetic of elliptic curves. Springer GTM 106, 1986Google Scholar
  18. [18]
    A.S. Sikora, Analogies between group actions on 3-manifolds and number fields. Preprint arXiv:math. GT/0107210, 29. Juli 2001Google Scholar
  19. [19]
    I.M. Singer, Index theory for elliptic operators, Proc. Symp. Pure Math. 28 (1973), 11–31Google Scholar
  20. [20]
    D. Sullivan, Linking the universalities of Milnor-Thurston Feigenbaum and Ahlfors-Bers. In: Topological methods in modern mathematics. Publish or perish 1993, 543–564Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Christopher Deninger
    • 1
  1. 1.Mathematisches InstitutWWU MünsterMünsterGermany

Personalised recommendations