Skip to main content
SpringerLink
Log in

Kato’s Euler system and rational points on elliptic curves I: A p-adic Beilinson formula

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic étale cohomology, this leads to an alternate proof of the main results of [Br2] and [Ge] which is independent of Kato’s explicit reciprocity law.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. A. Beilinson, Higher regulators of modular curves, in Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), Contemporary Mathematics, Vol. 55, American Mathematical Society, Providence, RI, 1986, pp. 1–34.

    Google Scholar 

  2. M. Bertolini, H. Darmon and K. Prasanna, Generalised Heegner cycles and p-adic Rankin L-series, Duke Mathematical Journal 162 (2013), 1033–1148.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Bertolini and H. Darmon, Hida families and rational points on elliptic curves, Inventiones Mathematicae 168 (2007), 371–431.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Bertolini and H. Darmon, Kato’s Euler system and rational points on elliptic curves II: The explicit reciprocity law, in preparation.

  5. M. Bertolini and H. Darmon, Kato’s Euler system and rational points on elliptic curves III: The conjecture of Perrin-Riou, in preparation.

  6. M. Bertolini, H. Darmon and V. Rotger, Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series, submitted.

  7. A. Besser, Syntomic regulators and p-adic integration. I. Rigid syntomic regulators, in Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), Israel Journal of Mathematics 120 (2000), 291–334.

  8. A. Besser, Syntomic regulators and p-adic integration. II. K 2 of curves, in Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), Israel Journal of Mathematics 120 (2000), 335–359.

    MATH  MathSciNet  Google Scholar 

  9. K. Bannai and G. Kings, p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure, American Journal of Mathematics 132 (2010), 1609–1654.

    MATH  MathSciNet  Google Scholar 

  10. S. J. Bloch, Higher Regulators, Algebraic K-theory, and Zeta Functions of Elliptic Curves, CRM Monograph Series, Vol. 11, American Mathematical Society, Providence, RI, 2000.

    MATH  Google Scholar 

  11. F. Brunault, Valeur en 2 de fonctions L de formes modulaires de poids 2: théorème de Beilinson explicite, Bulletin de la Société Mathématique de France 135 (2007), 215–246.

    MATH  MathSciNet  Google Scholar 

  12. F. Brunault, Régulateurs p-adiques explicites pour le K 2 des courbes elliptiques, Actes de la Conférence “Fonctions L et Arithmétique”, Publ. Math. Besançon Algèbre Théorie Nr., Lab. Math. Besançon, Besançon, 2010, pp. 29–57.

    Google Scholar 

  13. R. F. Coleman and E. de Shalit, p-adic regulators on curves and special values of p-adic L-functions, Inventiones Mathematicae 93 (1988), 239–266.

    MATH  MathSciNet  Google Scholar 

  14. R. F. Coleman, A p-adic Shimura isomorphism and p-adic periods of modular forms, in p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemporary Mathematics, Vol. 165, American Mathematical Society, Providence, RI, 1994, pp. 21–51.

    Google Scholar 

  15. P. Colmez, Fonctions L p-adiques, Séminaire Bourbaki, Vol. 1998/99, Astérisque No. 266 (2000), Exp. No. 851, 3, 21–58.

  16. P. Colmez, La conjecture de Birch et Swinnerton-Dyer p-adique, (French) Astérisque No. 294 (2004), ix, 251–319.

    Google Scholar 

  17. H. Darmon and V. Rotger, Diagonal cycles and Euler systems I: A p-adic Gross-Zagier formula, Annales Scientifiques de l’École Normale Supérieure, to appear.

  18. M. Gealy, On the Tamagawa number conjecture for motives attached to modular forms, PhD Thesis, California Institute of Technology, 2006.

  19. H. Hida, Elementary Theory of L-functions and Eisenstein Series, London Mathematical Society Student Texts, Vol.26, Cambridge University Press, Cambridge, 1993.

    MATH  Google Scholar 

  20. K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, in Cohomologies p-adiques et applications arithmétiques. III, Astérisque No. 295 (2004), ix, 117–290.

    Google Scholar 

  21. K. Kitagawa, On standard p-adic L-functions of families of elliptic cusp forms, in p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemporary Mathematics, Vol. 165, American Mathematical Society, Providence, RI, 1994, pp. 81–110.

    Google Scholar 

  22. M. Niklas, Rigid syntomic regulators and the p-adic L-function of a modular form, Regensburg PhD Thesis, 2010, available at http://epub.uni-regensburg.de/19847/

  23. B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones Mathematicae 84 (1986), 1–48.

    MATH  MathSciNet  Google Scholar 

  24. B. Perrin-Riou, Fonctions L p-adiques d’une courbe elliptique et points rationnels, Annales de l’Institut Fourier (Grenoble) 43 (1993), 945–995.

    MATH  MathSciNet  Google Scholar 

  25. B. Perrin-Riou, Théorie d’Iwasawa des représentations p-adiques sur un corps local, with an appendix by Jean-Marc Fontaine, Inventiones Mathematicae 115 (1994), 81–161.

    MATH  MathSciNet  Google Scholar 

  26. G. Shimura, The special values of the zeta functions associated with cusp forms, Communications on Pure and Applied Mathematics 29 (1976), 783–804.

    MATH  MathSciNet  Google Scholar 

  27. G. Shimura, On a class of nearly holomorphic automorphic forms, Annals of Mathematics 123 (1986), 347–406.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartmento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133, Milano, Italy

    Massimo Bertolini

  2. Mathematics Department, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    Henri Darmon

Authors
  1. Massimo Bertolini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henri Darmon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Massimo Bertolini.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bertolini, M., Darmon, H. Kato’s Euler system and rational points on elliptic curves I: A p-adic Beilinson formula. Isr. J. Math. 199, 163–188 (2014). https://doi.org/10.1007/s11856-013-0047-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-013-0047-2

Keywords

Search

Navigation