Manuscripta Mathematica

, Volume 142, Issue 1–2, pp 245–255 | Cite as

Iterated period integrals and multiple Hecke L-functions

  • YoungJu Choie
  • Kentaro IharaEmail author


In this paper we express the multiple Hecke L-function in terms of a linear combination of iterated period integrals associated with elliptic cusp forms, which is introduced by Manin around 2004. This expression generalizes the classical formula of Hecke L-function obtained by the Mellin transformation of a cusp form. Also the expression gives a way of the analytic continuation of the multiple Hecke L-function.

Mathematics Subject Classification (2010)

Primary 11E45 Secondary 11M32 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Deitmar A., Diamantis N.: Automorphic forms of higher order. J. Lond. Math. Soc. 80, 18–34 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Goncharov, A.B.: Multiple ζ-values, Galois groups and geometry of modular varieties. Progr. Math. 201, 361–392 (2001)Google Scholar
  3. 3.
    Goncharov, A.B.: Multiple polylogarithms and mixed Tate motives, preprint, math.AG/0103059 (2001)Google Scholar
  4. 4.
    Ichikawa, T.: Motives for elliptic modular groups, arXiv:0909.5277, 29, Sep (2009)Google Scholar
  5. 5.
    Ihara K.: Special values of multiple Hecke L-functions. J. Pure Appl. Algebra 216, 192–201 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Knopp M.: Some new results on the Eichler cohomology of automorphic forms. Bull. Am. Math. Soc. 80, 607–632 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kohnen, W., Zagier, D.: Modular forms with rational periods, Modular forms (Durham, 1983), 197–249, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester (1984)Google Scholar
  8. 8.
    Manin, Y.I.: Iterated integrals of modular forms and noncommutative modular symbols. Progr. Math. 253 (2006)Google Scholar
  9. 9.
    Manin Y.I.: Iterated Shimura integrals. Mosc. Math. J. 5, 869–881 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Matsumoto K., Tanigawa Y.: The analytic continuation and the order estimate of multiple Dirichlet series. J. Theor. Nombres Bordeaux 15, 267–274 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Razar M.: Modular forms for Γ0(N) and Dirichlet series. Trans. AMS 231, 489–495 (1977)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Shokurov, V.: Shimura integrals of cusp forms. Math. USSR Izvestiya, 603–646 (1981)Google Scholar
  13. 13.
    Zagier, D.: Values of zeta functions and their applications. Progr. Math. 120, 497–512 (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Pohang Mathematics Institute, POSTECH HyojaPohangKorea
  2. 2.Department of MathematicsInternational College, Osaka UniversityToyonakaJapan

Personalised recommendations