Skip to main content

Critical slope p-adic L-functions of CM modular forms

Abstract

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the critical-slope L-function arising from Kato’s Euler system and comparing this with results of Bellaïche on the critical-slope L-function defined using overconvergent modular symbols.

This is a preview of subscription content, access via your institution.

References

  1. Y. Amice and J. Vélu, Distributions p-adiques associées aux séries de Hecke, in Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Astérisque, Vol. 24-25, Société Mathématique de France, Paris, 1975, pp. 119–131.

    Google Scholar 

  2. J. Bellaïche, Critical p-adic L-functions, Inventiones Mathematicae 189 (2012), 1–60.

    MathSciNet  Article  MATH  Google Scholar 

  3. J. Bellaïche, P-adic L-functions of critical CM forms, preprint, 2011.

  4. P. Colmez, La conjecture de Birch et Swinnerton-Dyer p-adique, Astérisque, Vol. 294, 2004, 251–319.

  5. E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication, Perspectives in Mathematics, Vol. 3, Academic Press Inc., Boston, MA, 1987.

    MATH  Google Scholar 

  6. P. Deligne, Formes modulaires et représentations ℓ-adiques, Séminaire Bourbaki 11 (1968/69), Exp. No. 355, 139–172.

    Google Scholar 

  7. B. H. Gross, On the factorization of p-adic L-series, Inventiones Mathematicae 57 (1980), 83–95.

    MathSciNet  Article  MATH  Google Scholar 

  8. H. Hida and J. Tilouine, Anti-cyclotomic Katz p-adic L-functions and congruence modules, Annales Scientifiques de l’École Normale Supérieure (Quatrième Série) 26 (1993), 189–259.

    MathSciNet  MATH  Google Scholar 

  9. K. Kato, P-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 Cohomologies p-adiques et applications arithmétiques. III (2004), 117–290.

    Google Scholar 

  10. D. Loeffler and S. L. Zerbes, Wach modules and critical slope p-adic L-functions, Journal für die Reine und Angewandte Mathematik (2012), published online March 2012, print version to appear. DOI 10.1515/crelle.2012.012.

  11. D. Loeffler and S. L. Zerbes, Iwasawa theory and p-adic L-functions over Z p 2-extensions, preprint (arxiv:1108.5954), 2012.

  12. B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Inventiones Mathematicae 25 (1974), 1–61.

    MathSciNet  Article  MATH  Google Scholar 

  13. B. Perrin-Riou, Fonctions L p-adiques des représentations p-adiques, Astérisque 229 (1995), 1–198.

    Google Scholar 

  14. R. Pollack and G. Stevens, Overconvergent modular symbols and p-adic L-functions, Annales Scientifiques de l’École Normale Supérieure (Quatrième Série) 44 (2011), 1–42.

    MathSciNet  MATH  Google Scholar 

  15. R. Pollack and G. Stevens, Critical slope p-adic L-functions, to appear in Journal of the London Mathematical Society, 2012. DOI 10.1112/jlms/jds057.

  16. M. Višik, Nonarchimedean measures associated with Dirichlet series, Matematicheskiĭ Sbornik (Novaya Seriya) 99(141) (1976), 248–260, 296.

    Google Scholar 

  17. A. Weil, On a certain type of characters of the idèle-class group of an algebraic number-field, in Proceedings of the International Symposium on Algebraic Number Theory, Tokyo & Nikko, 1955, Science Council of Japan, Tokyo, 1956, pp. 1–7.

    Google Scholar 

  18. R. Yager, On two variable p-adic L-functions, Annals of Mathematics 115 (1982), 411–449.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antonio Lei.

Additional information

The first author is grateful for the support of a CRM-ISM postdoctoral fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lei, A., Loeffler, D. & Zerbes, S.L. Critical slope p-adic L-functions of CM modular forms. Isr. J. Math. 198, 261–282 (2013). https://doi.org/10.1007/s11856-013-0020-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-013-0020-0

Keywords

  • Modular Form
  • Galois Group
  • Dirichlet Character
  • Modular Symbol
  • Iwasawa Theory