manuscripta mathematica

, Volume 156, Issue 1–2, pp 149–169 | Cite as

On the kernel of the theta operator mod p

  • Siegfried Böcherer
  • Hirotaka Kodama
  • Shoyu Nagaoka


We construct many examples of level one Siegel modular forms in the kernel of theta operators mod p by using theta series attached to positive definite quadratic forms.

Mathematics Subject Classification

11F33 11F30 


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  1. 1.
    Andrianov, A.N.: Quadratic Forms and Hecke operators. In: (ed.) Grundlehren Math. Wiss, vol. 286. Springer, Berlin (1987)Google Scholar
  2. 2.
    Aoki, H., Ibukiyama, T.: Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Int. J. Math. 16, 249–279 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Böcherer, S.: Über gewisse Siegelsche Modulformen zweiten Grades. Math. Ann. 261, 23–41 (1982)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Böcherer, S., Funke, J., Schulze-Pillot, R.: Trace operator and theta series. J. Number Theory 78, 119–139 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Böcherer, S., Kikuta, T.: On mod \(p\) singular modular forms. Forum Math. 28, 1051–1065 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Böcherer, S., Nagaoka, S.: On mod \(p\) properties of Siegel modular forms. Math. Ann. 338, 421–433 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Böcherer, S., Nagaoka, S.: On Siegel modular forms of level \(p\) and their properties mod \(p\). Manuscr. math. 132, 501–515 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Böcherer, S., Nagaoka, S.: On \(p\)-adic properties of Siegel modular forms. In: Heim, B., et al. (eds.) Automorphic Forms, Springer Proceedings in Mathematics and Statistics, vol. 115, pp. 47–66. Springer, Berlin (2014)Google Scholar
  9. 9.
    Böcherer, S., Nebe, G.: On theta series attached to maximal lattices and their adjoints. J. Ramanujan Math. Soc. 25(3), 265–284 (2000)MathSciNetMATHGoogle Scholar
  10. 10.
    Böcherer, S., Kikuta, T., Takemori, S.: Weights of mod \(p\) kernels of theta operators. arXiv: 1606.06390v1 [math.NT] (to appear in Canadian J. Math.)
  11. 11.
    Böcherer, S., Schulze-Pillot, R.: Siegel modular forms and theta series attached to quaternion algebras. Nagoya Math. J. 121, 35–96 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Choi, D., Choie, Y., Kikuta, T.: Sturm type theorem for Siegel modular forms of genus 2 modulo \(p\). Acta Arith. 158, 129139 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Freitag, E.: Siegelsche Modulfunktionen Grundlehren der mathematischen Wissenschaften, vol. 254. Springer, Berlin (1983)MATHGoogle Scholar
  14. 14.
    Ichikawa, T.: Vector-valued \(p\)-adic Siegel modular forms. J. Rein. Angew. Math. 690, 35–49 (2014)MathSciNetMATHGoogle Scholar
  15. 15.
    Igusa, J.-I.: On the ring of modular forms of degree two over \(\varvec {Z}\). Am. J. Math. 101, 149–183 (1979)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kikuta, T., Kodama, H., Nagaoka, S.: Note on Igusa’s cusp form of weight 35. Rocky Mt. J. Math. 45, 963–972 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Klingen, H.: Bemerkungen über Kongruenzuntergruppen der Modulgruppe \(n\)-ten Grades. Archiv der Math. 10, 113–122 (1959)CrossRefMATHGoogle Scholar
  18. 18.
    Klingen, H.: Introductory Lectures on Siegel Modular Forms. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  19. 19.
    Kodama, H., Nagaoka, S.: A congruence relation satisfied by Siegel cusp form of odd weight. J. School Sci. Eng. Kinki Univ. 49, 9–15 (2013)Google Scholar
  20. 20.
    Krieg, A.: Hecke Algebras, Memoirs of AMS, 435, (1990)Google Scholar
  21. 21.
    Maaß, H.: Konstruktion von Spitzenformen beliebigen Grades mit Hilfe von Thetareihen. Math. Ann. 226, 275–284 (1977)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Maaß, H.: Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics 216, Springer, Berlin (1971)Google Scholar
  23. 23.
    Mizumoto, S.: On integrality of certain algebraic numbers associated with modular forms. Math. Ann. 265, 119–135 (1983)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Nagaoka, S.: On the mod \(p\) kernel of the theta operator. Proc. Am. Math. Soc. 143, 4237–4244 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nagaoka, S.: Note on mod \(p\) Siegel modular forms. Math. Z. 235, 405–420 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Peters, M.: Ternäre und quaternäre quadratische Formen und Quaternionenalgebren. Acta Arithm. 15, 329–365 (1969)CrossRefMATHGoogle Scholar
  27. 27.
    Scharlau, W.: Quadratic and Hermitian Forms, Grundlehren der mathematischen Wissenschaften, vol. 270. Springer, Berlin (1985)CrossRefGoogle Scholar
  28. 28.
    Shimura, G.: On the Fourier coefficients of modular forms in several variables. Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch–Physikalische Klasse 1975, 261-268 (=[75d] in Collected Papers II)Google Scholar
  29. 29.
    Waldspurger, J.-L.: Engendrement par des séries thêta de certains espaces de formes modulaires. Invent. Math. 50, 135–168 (1978)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zagier, D.B.: Zetafunktionen und quadratische Körper. Springer, Berlin (1981)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.FreiburgGermany
  2. 2.Academic Support CenterKogakuin UniversityHachiojiJapan
  3. 3.Department MathematicsKindai UniversityHigashi-OsakaJapan

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