A Note on Critical p-adic L-functions


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.

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  1. [1]

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

    MathSciNet  Article  Google Scholar 

  2. [2]

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

    MATH  Google Scholar 

  3. [3]

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

    MathSciNet  Article  Google Scholar 

  4. [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]

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

    MATH  Google Scholar 

  6. [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. [7]

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

    MathSciNet  Article  Google Scholar 

  8. [8]

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

    MathSciNet  Article  Google Scholar 

  9. [9]

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

    MathSciNet  MATH  Google Scholar 

  10. [10]

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

    MathSciNet  Article  Google Scholar 

  11. [11]

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

    MathSciNet  Article  Google Scholar 

  12. [12]

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

    MathSciNet  MATH  Google Scholar 

  13. [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. [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. [15]

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

    MathSciNet  Article  Google Scholar 

  16. [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)

    MathSciNet  Article  Google Scholar 

  17. [17]

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

    MathSciNet  Article  Google Scholar 

  18. [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. [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

    MathSciNet  Article  Google Scholar 

  20. [20]

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

    Google Scholar 

  21. [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. [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. [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)

    MathSciNet  Article  Google Scholar 

  24. [24]

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

    MathSciNet  Article  Google Scholar 

  25. [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)

    MathSciNet  Article  Google Scholar 

  26. [26]

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

    MathSciNet  Article  Google Scholar 

  27. [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)

    MathSciNet  Article  Google Scholar 

  28. [28]

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

    MathSciNet  Article  Google Scholar 

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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.

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Corresponding author

Correspondence to Yi Wen Ding.

Additional information

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

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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

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  • p-adic L-function
  • eigencurve
  • critical p-stabilization
  • Jacquet-Emerton functor

MR(2010) Subject Classification

  • 11F67
  • 11S80