Skip to main content
SpringerLink
Log in

Cuspidal class number of the tower of modular curves X 1(Np n)

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We consider a cuspidal class number, which is the order of a subgroup of the full cuspidal divisor class group of X 1(Np n) with \({p\nmid N}\) and n ≥ 1. By studying the second generalized Bernoulli numbers, we obtain results similar to ones (Ferrero and Washington in Ann Math (2) 109(2):377–395, 1979; Washington in Invent Math 49:87–97, 1978) about the relative class numbers of cyclotomic \({\mathbb{Z}_p}\)-extension of an abelian number field.

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. Ferrero B., Washington L.: The Iwasawa invariant μ p vanishes for abelian number fields. Ann. Math. (2) 109(2), 377–395 (1979)

    Article  MathSciNet  Google Scholar 

  2. Iwasawa K.: Lectures on p-adic L-functions. Ann. Math. Stud. 74. Princeton University Press, New Jersey (1972)

    Google Scholar 

  3. Kubert D., Lang S.: Distributions on toroidal groups. Math. Z. 148, 33–51 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kubert D., Lang S.: The index of Stickelberger ideals of order 2 and cuspidal class numbers. Math. Ann. 237, 213–232 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kubert D., Lang S.: Cartan–Bernoulli numbers as values of L-Series. Math. Ann. 240, 21–26 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Lang, S.: Introduction to modular forms, Grundlehren der Mathematischen Wissenschaften, 222. Springer, Berlin (1976)

  7. Oesterlé, J.: Travaux de Ferrero et Washington sur le Nombre de Classes d’Idéaux des Corps Cyclotomiques, Séminaire Bourbaki, 31e année, 1978/79, n. 535

  8. Sun H.-S.: Homological interpretation of a Theorem of Washington. J. Number Theory 127(1), 47–63 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Washington L.: The non-p-part of the class number in a cyclotomic \({\mathbb{Z}_p}\) -extension. Invent. Math. 49, 87–97 (1978)

    Article  MathSciNet  Google Scholar 

  10. Washington, L.: Introduction to cyclotomic fields, 2nd edn. Graduate Texts in Mathematics, vol. 83. Springer, New York (1997)

  11. Yang Y.: Modular units and cuspidal divisor class groups of X 1(N). J. Algebra 322, 514–553 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Yu J.: A cuspidal class number formula for the modular curves X 1(N). Math. Ann. 252, 197–216 (1980)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Hae-Sang Sun

    Present address: Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul, 130-722, Korea

Authors and Affiliations

  1. Max-Planck Institut für Mathematik Bonn, Vivatsgasse 7, 53111, Bonn, Germany

    Hae-Sang Sun

Authors
  1. Hae-Sang Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hae-Sang Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, HS. Cuspidal class number of the tower of modular curves X 1(Np n). Math. Ann. 348, 909–927 (2010). https://doi.org/10.1007/s00208-010-0505-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0505-7

Keywords

Search

Navigation