Skip to main content
SpringerLink
Log in

On congruence modules related to Hilbert Eisenstein series

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute their constant terms by computing local integrals. In the second part, we prove a control theorem for one-variable ordinary \(\Lambda \)-adic Hilbert modular forms following Hida’s work on the space of multivariable ordinary \(\Lambda \)-adic Hilbert cusp forms. In part three, we compute congruence modules related to Hilbert Eisenstein series through an analog of Ohta’s methods.

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. Betina, A., Dimitrov, M., Pozzi, A.: On the failure of Gorensteinness at weight \(1\) Eisenstein points of the eigencurve. arXiv:1804.00648

  2. Bump, D.: Automorphic forms and representations. In: Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1997)

  3. Casselman, W.: On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973)

    Article  MathSciNet  Google Scholar 

  4. Cassels, J.W.S., Frohlich, A.: Algebraic Number Theory. Academic Press, New York (1968)

    Google Scholar 

  5. Deligne, P., Ribet, K.A.: Values of abelian \(L\)-functions at negative integers over totally real fields. Invent. Math. 59, 227–286 (1980)

    Article  MathSciNet  Google Scholar 

  6. Diamond, F., Shurman, J.: A first course in modular forms. In: Graduate Texts in Mathematics , vol. 228. Springer, New York (2005)

  7. Dimitrov, M.: Compactifications arithmétiques des variétés de Hilbert et forms modulaires de Hilbert pour \(\Gamma _1(\mathfrak{c},\mathfrak{n})\), geometric aspects of Dwork theory. In: Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N., Loeser, F. (eds.) Walter de Gruyter, Berlin, pp. 525–551 (2004)

  8. Dimitrov, M., Tilouine, J.: Variétés et forms modulaires de Hilbert arithétiques pour \(\Gamma _1(\mathfrak{c},\mathfrak{n})\). Geometric aspects of Dwork theory. In: Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N., Loeser, F. (eds.) Walter de Gruyter, Berlin, pp. 553–609 (2004)

  9. Dimitrov, M., Wiese, G.: Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p. J. Inst. Math. Jussieu 1, 1–26 (2018)

    Google Scholar 

  10. Emerton, M.: The Eisenstein ideal in Hida’s ordinary Hecke algebra. Int. Math. Res. Not. 15, 793–802 (1999)

    Article  MathSciNet  Google Scholar 

  11. Goren, E.Z.: Lectures on Hilbert modular varieties and modular forms. In: CRM Monograph Series (2001)

  12. Hida, H.: On \(p\)-adic Hecke algebras for GL\(_2\) over totally real fields. Ann. Math. 128, 295–384 (1988)

    Article  MathSciNet  Google Scholar 

  13. Hida, H.: Elementary theory of \(L\)-functions and Eisenstein series. In: London Mathematical Society Student Texts, vol. 26. Cambridge University Press, Cambridge (1993)

  14. Hida, H.: \(p\)-adic automorphic forms on Shimura varieties. In: Springer Monographs in Mathematics. Springer, New York (2004)

  15. Hsieh, M.-L.: Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields. J. Am. Math. Soc. 27–3, 753–862 (2014)

    Article  MathSciNet  Google Scholar 

  16. Katz, N.: \(p\)-adic \(L\)-function for CM fields. Invent. Math. 49, 199–297 (1978)

    Article  MathSciNet  Google Scholar 

  17. Lafferty, M.: Eichler-Shimura cohomology groups and the Iwasawa main conjecture. Ph.D. thesis (2015)

  18. Mazur, B., Wiles, A.: Class fields of abelian extension of \(\mathbb{Q}\). Invent. Math. 76, 179–330 (1984)

    Article  MathSciNet  Google Scholar 

  19. Neukirch, J., Schappacher, N.: Algebra Number Theory, vol. 9. Springer, Berlin (1999)

    Book  Google Scholar 

  20. Ohta, M.: Ordinary \(p\)-adic étale cohomology groups attached to towers of elliptic modular curves. Compos. Math. 115, 241–301 (1999)

    Article  Google Scholar 

  21. Ohta, M.: Ordinary \(p\)-adic étale cohomology groups attached to towers of elliptic modular curves II. Math. Ann. 318, 557–583 (2000)

    Article  MathSciNet  Google Scholar 

  22. Ohta, M.: Congruence modules related to Eisenstein series. Ann. Sci. École Norm Sup. (4) 36, 225–269 (2003)

    Article  MathSciNet  Google Scholar 

  23. Ozawa, T.: Constant terms of Eisenstein series over a totally real field. Int. J. Number Theory 13, 309–324 (2017)

    Article  MathSciNet  Google Scholar 

  24. Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45, 637–679 (1978)

    Article  MathSciNet  Google Scholar 

  25. Schmidt, R.: Some remarks on local newforms for GL(2). J. Ramanujan Math. Soc. 17(2), 115–147 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Sharifi, R.: A reciprocity map and the two-variable \(p\)-adic \(L\)-function. Ann. Math. 173, 251–300 (2011)

    Article  MathSciNet  Google Scholar 

  27. Tate, J.: Number theoretic background, automorphic forms, representations and L-functions. Proc. Sympos. Pure Math. (Oregon State Univ., Corvallis, Ore) 2, 3–26 (1977)

    Google Scholar 

  28. Wang, J.: On the Jacobi sums mod \(P^n\). J. Number Theory 39, 50–64 (1991)

    Article  MathSciNet  Google Scholar 

  29. Wiles, A.: On \(p\)-adic representations for totally real fields. Ann. Math. 123, 407–456 (1986)

    Article  MathSciNet  Google Scholar 

  30. Wiles, A.: On ordinary \(\lambda \)-adic representations associated to modular forms. Invent. Math. 94, 539–573 (1988)

    Article  MathSciNet  Google Scholar 

  31. Wiles, A.: The Iwasawa conjecture for totally real fields. Ann. Math. 131, 493–540 (1990)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The results of this paper are a part of the author’s Ph.D. thesis in University of Arizona. The author would like to thank his advisor, Romyar Sharifi, for his guidance, support, and suggesting this problem. Also, the author would like to thank Adel Betina, Mladen Dimitrov, Haruzo Hida, Ming-Lun Hsieh, and Hang Xue for helpful suggestions during preparation of this article. The author was partially supported by National Science Foundation under Grants No. DMS-1360583 and No. DMS-1401122, by the Labex CEMPI under Grant No. ANR-11-LABX-0007-01, and by I-SITE ULNE under Grant No. ANR-16-IDEX-0004. Finally, the author is grateful to the referees for a careful reading and valuable suggestions for improvement.

Author information

Authors and Affiliations

  1. UMR 8524-Laboratoire Paul Painlevé, CNRS, University of Lille, 59000, Lille, France

    Sheng-Chi Shih

Authors
  1. Sheng-Chi Shih
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sheng-Chi Shih.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shih, SC. On congruence modules related to Hilbert Eisenstein series. Math. Z. 296, 1331–1385 (2020). https://doi.org/10.1007/s00209-020-02486-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-020-02486-9

Search

Navigation