Skip to main content
SpringerLink
Log in

A Note on Critical p-adic L-functions

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable p-adic L-functions around critical points via Emerton’s representation theoretic approach.

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

    Article  MathSciNet  Google Scholar 

  2. Bellaïche, J., Chenevier, G.: Families of Galois representations and Selmer groups. Asterisqué, 324, 1–314 (2009)

    MATH  Google Scholar 

  3. Bergdall, J.: Ordinary modular forms and companion points on the eigencurve. Journal of Number Theory, 134, 226–239 (2014)

    Article  MathSciNet  Google Scholar 

  4. [4] Breuil, C: Remarks on some locally Qp-analytic representations of GL2(F) in the crystalline case. In: Non-abelian Fundamental Groups and Iwasawa Theory, London Math. Soc. Lecture Note Ser., Vol. 393, Cambridge Univ. Press, 212–238, 2010

    MathSciNet  MATH  Google Scholar 

  5. [5] Breuil, C: Correspondance de langlands p-adique, compatibilité local-global et applications. Séminaire Bourbaki, 1031, 119–147 (2011)

    MATH  Google Scholar 

  6. Breuil, C., Emerton, M.: Représentations p-adiques ordinaires de GL2(Qp) et compatibilite local-global. Asterisque, 331, 255–315 (2010)

    MATH  Google Scholar 

  7. Breuil, C., Hellmann, E., Schraen, B.: Smoothness and classicality on eigenvarieties. Inventiones mathematicae, 209(1), 197–274 (2017)

    Article  MathSciNet  Google Scholar 

  8. Breuil, C., Hellmann, E., Schraen, B.: Une interpretation modulaire de la variete trianguline. Mathematis- che Annalen, 367(3–4), 1587–1645 (2017)

    Article  MathSciNet  Google Scholar 

  9. Chenevier, G.: Familles p-adiques de formes automorphes pour GLn. J. Reine Angew. Math, 570, 143–217 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Chenevier, G.: Une correspondance de Jacquet-Langlands p-adique. Duke Mathematical Journal, 126(1), 161–194 (2005)

    Article  MathSciNet  Google Scholar 

  11. Chenevier, G.: On the infinite fern of Galois representations of unitary type. Ann. Sci. Éc. Norm. Supér. (4), 44(6), 963–1019 (2011)

    Article  MathSciNet  Google Scholar 

  12. Colmez, P.: Representations triangulines de dimension 2. Asterisque, 319, 213–258 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Ding, Y. W.: L-invariants and local-global compatibility for the group GL2/F. Forum of Mathematics, Sigma, 4, e13, 49 pp., (2016)

    Google Scholar 

  14. Ding, Y. W.: Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global. Memoires de la SMF, No. 155, 2017

  15. Emerton, M.: p-adic L-functions and unitary completions of representations of p-adic reductive groups. Duke Mathematical Journal, 130, 353–392 (2005)

    Article  MathSciNet  Google Scholar 

  16. Emerton, M.: Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties. Annales Scientifiques de l’Ecole Normale Superieure, 39(5), 775–839 (2006)

    Article  MathSciNet  Google Scholar 

  17. Emerton, M.: On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. Inventiones Mathematicae, 164, 1–84 (2006)

    Article  MathSciNet  Google Scholar 

  18. Emerton, M.: Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction. to appear in J. Institut Math. Jussieu, 2007

    Google Scholar 

  19. Emerton, M.: Locally analytic representation theory of p-adic reductive groups: A summary of some recent developments. In: L-Functions and Galois Representations, London Mathematical Society Lecture Note Series, Vol. 320, 407–437, 2007

    Article  MathSciNet  Google Scholar 

  20. Emerton, M.: Local-global compatibility in the p-adic Langlands programme for GL2/Q. preprint, 2011

    Google Scholar 

  21. Emerton, M.: Locally analytic vectors in representations of locally p-adic analytic groups. Memoirs of the Amer. Math. Soc, 248(1175), (2017)

    Google Scholar 

  22. Grothendieck, A., Dieudonne, J.: Éléments de gómétrie algébrique iv: étude locale des schemas et des morphismes de schemas (premiere partie). Pub. Math. I.H.É.S., 20, 5–259 (1964)

    Article  Google Scholar 

  23. Kedlaya, K., Pottharst, J., Xiao, L.: Cohomology of arithmetic families of φ, Г)-modules. Journal of the American Mathematical Society, 27(4), 1043–1115 (2014)

    Article  MathSciNet  Google Scholar 

  24. Kisin, M.: Overconvergent modular forms and the Fontaine-Mazur conjecture. Inventiones Mathematicae, 153(2), 373–454 (2003)

    Article  MathSciNet  Google Scholar 

  25. Lei, A., Loeffler, D., Zerbes, S. L.: Critical slope p-adic L-functions of CM modular forms. Israel Journal of Mathematics, 198(1), 261–282 (2013)

    Article  MathSciNet  Google Scholar 

  26. Liu, R.: Triangulation of refined families. Commentarii Mathematici Helvetici, 90(4), 831–904 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Pollack, R., Stevens, G.: Critical slope p-adic L-functions. Journal of the London Mathematical Society, 87(2), 428–452 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I want to thank Matthew Emerton for suggesting the problem of extending his adjunction formula to critical points on the eigencurve, that led to the note. I thank Daniel Barrera Salazar, John Bergdall, Xin Wan, Shanwen Wang for helpful discussions or remarks. I also thank the anonymous referee for the reading and helpful suggestions.

Author information

Authors and Affiliations

  1. Bejing International Center for Mathematical Research, Peking University, Beijing, 100871, P.R. China

    Yi Wen Ding

Authors
  1. Yi Wen Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Wen Ding.

Additional information

Supported by EPSRC (Grant No. EP/L025485/1) and (Grant No. 7101500268) from Peking University

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ding, Y.W. A Note on Critical p-adic L-functions. Acta. Math. Sin.-English Ser. 37, 121–141 (2021). https://doi.org/10.1007/s10114-020-8396-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-020-8396-3

Keywords

MR(2010) Subject Classification

Search

Navigation