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On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

  • Masataka Chida
  • Hidenori Katsurada
  • Kohji MatsumotoEmail author
Article

Abstract

We prove a formula of Petersson’s type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficients. The method in this paper is essentially a generalization of Kitaoka’s previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.

Keywords

Siegel modular form Fourier coefficients Petersson formula 

Mathematics Subject Classification

11F30 11F46 

References

  1. 1.
    Christian, U.: Über Hilbert-Siegelsche Modulformen und Poincarésche Reihen. Math. Ann. 148, 257–307 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Duke, W.: The critical order of vanishing of automorphic \(L\)-functions with large level. Invent. Math. 119, 165–174 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Hua, L.K.: Introduction to Number Theory. Springer-Verlag, NY (1982)zbMATHGoogle Scholar
  4. 4.
    Iwaniec, H.: Topics in classical automorphic forms, graduate studies in mathematics 17. American Mathematical Society, Providence (1997)Google Scholar
  5. 5.
    Kamiya, Y.: Certain mean values and non-vanishing of automorphic \(L\)-functions with large level. Acta Arith. 93, 157–176 (2000)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Kitaoka, Y.: Fourier coefficients of Siegel cusp forms of degree two. Nagoya Math. J. 93, 149–171 (1984)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Klingen, H.: Introductory Lectures on Siegel Modular Forms, Cambridge studies in advanced mathematics 20. Cambridge University Press, London (1990)CrossRefGoogle Scholar
  8. 8.
    Kowalski, E., Saha, A., Tsimerman, J.: Local spectral equidistribution for Siegel modular forms and applications. Compositio Math. 148, 335–384 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Masataka Chida
    • 1
  • Hidenori Katsurada
    • 2
  • Kohji Matsumoto
    • 3
    Email author
  1. 1.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyoto Japan
  2. 2.Muroran Institute of TechnologyMuroran Japan
  3. 3.Graduate School of MathematicsNagoya UniversityNagoya Japan

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