On p-Adic Properties of Siegel Modular Forms

  • Siegfried BöchererEmail author
  • Shoyu Nagaoka
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 115)


We show that Siegel modular forms of level \(\Gamma _{0}(p^{m})\) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre [18] closely; his proofs however do not generalize to the Siegel case or need some modifications.

Mathematics Subject Classification 2010

Primary 11F33 Secondary 11F55. 



Crucial work on this paper was done during our stay at the Mathematisches Forschungsinstitut Oberwolfach under the programme “Research in Pairs”; we continued our work during research visits at Kinki University and Universität Mannheim (respectively); a final revision was done, when the first author held a guest professorship at the University of Tokyo. We thank these institutions for the support. We also thank Dr.Kikuta for pointing out some gaps in our presentation and Professor T.Ichikawa for discussions about p-adic modular forms.


  1. 1.
    A. Aizenbud, D. Gurevitch, Some regular symmetric pairs. Trans. AMS 362, 3757–3777 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    S. Böcherer, On the Hecke operator U(p). J. Math. Kyoto Univ. 45, 807–829 (2005)zbMATHMathSciNetGoogle Scholar
  3. 3.
    S. Böcherer, S. Das, On holomorphic differential operators equivariant for \(Sp(n,\mathbb{R})\hookrightarrow U(n,n)\). Int. Math. Res. Notices. doi:10.1093/imrn/rns116Google Scholar
  4. 4.
    S. Böcherer, S. Nagaoka, On mod p properties of Siegel modular forms. Math. Ann. 338, 421–433 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    S. Böcherer, S. Nagaoka, On Siegel modular forms of level p and their properties mod p. Manuscripta Math. 132, 501–515 (2010)Google Scholar
  6. 6.
    S. Böcherer, S. Nagaoka, On vector-valued p-adic Siegel modular forms (in preparation)Google Scholar
  7. 7.
    S. Böcherer, J. Funke, R. Schulze-Pillot, Trace operator and theta series. J. Number Theory 78, 119–139 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    S. Böcherer, H. Katsurada, R. Schulze-Pillot, On the basis problem for Siegel modular forms with level, in Modular Forms on Schiermonnikoog (Cambridge University Press, Cambridge, 2008)Google Scholar
  9. 9.
    W. Eholzer, T. Ibukiyama, Rankin-Cohen type differential operators for Siegel modular forms. Int. J. Math. 9, 443–463 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    G. Faltings, C.L. Chai, Degeneration of Abelian Varieties (Springer, New York, 1990)CrossRefzbMATHGoogle Scholar
  11. 11.
    E. Freitag, Siegelsche Modulfunktionen (Springer, New York, 1983)CrossRefzbMATHGoogle Scholar
  12. 12.
    P.B. Garrett, On the arithmetic of Hilbert-Siegel cusp forms: Petersson inner products and Fourier coefficients. Invent. Math. 107, 453–481 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    T. Ibukiyama, On differential operators on automorphic forms and invariant pluri-harmonic polynomials. Commentarii Math. Univ. St. Pauli 48, 103–118 (1999)zbMATHMathSciNetGoogle Scholar
  14. 14.
    T. Ichikawa, Congruences between Siegel modular forms. Math. Ann. 342, 527–532 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    T. Ichikawa, Vector-valued p-adic Siegel modular forms (preprint)Google Scholar
  16. 16.
    N.M. Katz, p-adic properties of modular schemes and modular forms, in Modular Functions of One Variable III. Lecture Notes in Mathematics, vol. 350 (Springer, New York, 1973), pp. 69–190Google Scholar
  17. 17.
    K. Koike, I. Terada, Young diagrammatic methods for the representation theory of the classical groups of type B n, C n, D n. J. Algebra 107, 466–511 (1978)CrossRefMathSciNetGoogle Scholar
  18. 18.
    J.-P. Serre, Formes modulaires et fonctions z\(\hat{\text{e}}\) ta p-adiques, in Modular Functions of One Variable III. Lecture Notes in Mathematics, vol. 350 (Springer, New York, 1973), pp. 191–268Google Scholar
  19. 19.
    J.-P. Serre, Divisibilité de certaines fonctions arithmétiques. L’Ens. Math. 22, 227–260 (1976)zbMATHGoogle Scholar
  20. 20.
    G. Shimura, On the Fourier coefficients of modular forms in several variables. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 17, 261–268 (1975)MathSciNetGoogle Scholar
  21. 21.
    G. Shimura, Nearly holomorphic functions on hermitian symmetric spaces. Math. Ann. 278, 1–28 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    A. Sofer, p-adic aspects of Jacobi forms. J. Number Theory 63, 191–202 (1997)Google Scholar
  23. 23.
    J. Sturm, The critical values of zeta functions associated to the symplectic group. Duke Math. J. 48, 327–350 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    H.P.F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, in Modular Functions of One Variable III. Lecture Notes in Mathematics, vol. 350 (Springer, New York, 1973), pp. 1–55Google Scholar
  25. 25.
    R. Weissauer, Stabile Modulformen und Eisensteinreihen. Lecture Notes in Mathematics, vol. 1219 (Springer, New York, 1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.FreiburgGermany
  2. 2.Department of MathematicsKinki UniversityOsakaJapan

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