The Ramanujan Journal

, Volume 27, Issue 2, pp 151–162 | Cite as

Pullbacks of Siegel Eisenstein series and weighted averages of critical L-values

  • Nadine Amersi
  • Jeffrey Beyerl
  • Jim Brown
  • Allison Proffer
  • Larry Rolen


In this paper we obtain a weighted average formula for special values of L-functions attached to normalized elliptic modular forms of weight k and full level. These results are obtained by studying the pullback of a Siegel Eisenstein series and working out an explicit spectral decomposition.


Special values of L-functions Pullbacks of Eisenstein series 

Mathematics Subject Classification (2000)

11F67 11F46 11F30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Böcherer, S.: Uber das Verhalten der Fourierentwicklung bei Liflung von Modulformen. PhD thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, Freiburg (1981) Google Scholar
  2. 2.
    Böcherer, S.: Über die Funktionalgleichung automorpher L-Funktionen zur Siegelscher Modulgruppe. J. Reine Angew. Math. 362, 146–168 (1985) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brown, J.: On the cuspidality of pullbacks of Siegel Eisenstein series to Sp(2m)×Sp(2n). J. Number Theory 131, 106–119 (2011) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cohen, H.: Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann. 217, 271–285 (1975) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Prog. in Math, vol. 55. Birkhäuser, Boston (1985) MATHGoogle Scholar
  6. 6.
    Garrett, P.: Pullbacks of Eisenstein series; Applications. In: Automorphic Forms of Several Variables, Katata, 1983. Prog. in Math., vol. 46, pp. 114–137. Birkhäuser, Boston (1984) Google Scholar
  7. 7.
    Garrett, P.: Decomposition of Eisenstein series: Rankin triple products. Math. Ann. 125(2), 209–235 (1987) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Geer, G.: Siegel modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms. Universitext, vol. 44, pp. 181–245. Springer, Berlin (2008) CrossRefGoogle Scholar
  9. 9.
    Heim, B.: Congruences for the Ramanujan function and generalized class numbers. Math. Comput. 78(265), 431–439 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Klingen, H.: Introductory Lectures on Siegel Modular Forms. Cambridge Studies in Advanced Mathematics, vol. 20. Cambridge Univ. Press, Cambridge (1996) Google Scholar
  11. 11.
    Kohnen, W.: A Short Course on Siegel Modular Forms. POSTECH Lecture Series (2007) Google Scholar
  12. 12.
    Lanphier, D.: Values of symmetric cube L-functions and Fourier coefficients of Siegel Eisenstein series of degree-3. Math. Comput. (2010). doi: 0025-5718(10)02350-1 Google Scholar
  13. 13.
    Manin, J.I.: Periods of parabolic forms and p-adic Hecke series. Math. USSR Sb. 21(3), 371–393 (1973) CrossRefGoogle Scholar
  14. 14.
    Mizumoto, S.: Fourier coefficients of generalized Eisenstein series of degree two. Invent. Math. 65, 115–135 (1981) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Shimura, G.: On the periods of modular forms. Math. Ann. 229, 211–221 (1977) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Zagier, D.: Modular Forms Whose Fourier Coefficients Involve Zeta-Functions of Quadratic Fields. Lect. Notes in Math., vol. 627. Springer, Berlin (1977) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Nadine Amersi
    • 1
  • Jeffrey Beyerl
    • 2
  • Jim Brown
    • 2
  • Allison Proffer
    • 3
  • Larry Rolen
    • 4
  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA
  3. 3.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA
  4. 4.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations