Skip to main content

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

  • Chapter

Part of the book series: Developments in Mathematics ((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.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.A. Alvarez López, Y. Kordyukov, Distributional Betti numbers of transitive foliations of codimension one. Preprint 2000

    Google Scholar 

  2. M.F. Atiyah, Elliptic operators and compact groups. Springer LNM 401, 1974

    Google Scholar 

  3. K. Barner, On A. Weil’s explicit formula. J. Reine Angew. Math. 323 (1981), 139–152

    MathSciNet  MATH  Google Scholar 

  4. C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces. Doc. Math. J. DMV. Extra Volume ICM I (1998), 23–46

    Google Scholar 

  5. C. Deninger, Number theory and dynamical systems on foliated spaces. In: Jber. d. dt. Math.-Verein 103 (2001), 79–100

    Google Scholar 

  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, 41–55

    Google Scholar 

  7. M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hamburg 14 (1941), 197–272

    Article  MathSciNet  Google Scholar 

  8. Y. Ihara, On (co x p)-adic coverings of curves (the simplest example). Trudy Mat. Inst. Steklov 132 (1973), 133–148

    MathSciNet  Google Scholar 

  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, # 6707

    Google Scholar 

  10. Y. Ihara, Non-abelian classfields over function fields in special cases. Proc. Internat. Congress Math. (Nice 1970 ), vol 1, Gauthier-Villars, Paris 1971, 381390

    Google Scholar 

  11. Y. Ihara, Congruence relations and Shimura curves. Proceedings of Symp. Pure Math. 33 (1979) part 2, 291–311

    Google Scholar 

  12. C. Lazarov, Transverse index and periodic orbits. GAFA 10 (2000), 124–159

    Article  MathSciNet  MATH  Google Scholar 

  13. C.C. Moore, C. Schochet, Global analysis on foliated spaces. MSRI Publications 9, Springer 1988

    Google Scholar 

  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), 165–175

    Google Scholar 

  15. F. Oort, Lifting an endomorphism of an elliptic curve to characteristic zero. Indag. Math. 35 (1973), 466–470

    MathSciNet  Google Scholar 

  16. Robocop 3, Orion pictures 1993

    Google Scholar 

  17. J.H. Silverman, The arithmetic of elliptic curves. Springer GTM 106, 1986

    Google Scholar 

  18. A.S. Sikora, Analogies between group actions on 3-manifolds and number fields. Preprint arXiv:math. GT/0107210, 29. Juli 2001

    Google Scholar 

  19. I.M. Singer, Index theory for elliptic operators, Proc. Symp. Pure Math. 28 (1973), 11–31

    Google Scholar 

  20. D. Sullivan, Linking the universalities of Milnor-Thurston Feigenbaum and Ahlfors-Bers. In: Topological methods in modern mathematics. Publish or perish 1993, 543–564

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Deninger, C. (2002). On the Nature of the “Explicit Formulas” in Analytic Number Theory — A Simple Example. In: Kanemitsu, S., Jia, C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3675-5_7

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5239-4

  • Online ISBN: 978-1-4757-3675-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics