Advertisement

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
Article

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Böcherer, S.: Über gewisse Siegelsche Modulformen zweiten Grades. Math. Ann. 261, 23–41 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Böcherer, S., Funke, J., Schulze-Pillot, R.: Trace operator and theta series. J. Number Theory 78, 119–139 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Böcherer, S., Kikuta, T.: On mod \(p\) singular modular forms. Forum Math. 28, 1051–1065 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Böcherer, S., Nagaoka, S.: On mod \(p\) properties of Siegel modular forms. Math. Ann. 338, 421–433 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Freitag, E.: Siegelsche Modulfunktionen Grundlehren der mathematischen Wissenschaften, vol. 254. Springer, Berlin (1983)zbMATHGoogle Scholar
  14. 14.
    Ichikawa, T.: Vector-valued \(p\)-adic Siegel modular forms. J. Rein. Angew. Math. 690, 35–49 (2014)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klingen, H.: Bemerkungen über Kongruenzuntergruppen der Modulgruppe \(n\)-ten Grades. Archiv der Math. 10, 113–122 (1959)CrossRefzbMATHGoogle Scholar
  18. 18.
    Klingen, H.: Introductory Lectures on Siegel Modular Forms. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nagaoka, S.: On the mod \(p\) kernel of the theta operator. Proc. Am. Math. Soc. 143, 4237–4244 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nagaoka, S.: Note on mod \(p\) Siegel modular forms. Math. Z. 235, 405–420 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Peters, M.: Ternäre und quaternäre quadratische Formen und Quaternionenalgebren. Acta Arithm. 15, 329–365 (1969)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zagier, D.B.: Zetafunktionen und quadratische Körper. Springer, Berlin (1981)CrossRefzbMATHGoogle 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

Personalised recommendations