Journal of the Mathematical Society of Japan

On the Siegel Eisenstein series of degree two for low weights

Keiichi GUNJI

Full-text: Open access


In this paper, we give the Fourier coefficients of Siegel Eisenstein series of degree 2, level $p$, in order to calculate the dimensions of the space of Eisenstein series for low weights. The main methods of the calculation is to compute the Siegel series of level $p$ directly, following the similar way to that of Kaufhold.

Article information

J. Math. Soc. Japan, Volume 67, Number 3 (2015), 1043-1067.

First available in Project Euclid: 5 August 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F30: Fourier coefficients of automorphic forms

Siegel modular forms Eisenstein series


GUNJI, Keiichi. On the Siegel Eisenstein series of degree two for low weights. J. Math. Soc. Japan 67 (2015), no. 3, 1043--1067. doi:10.2969/jmsj/06731043.

Export citation


  • A. N. Andrianov, Quadratic forms and Hecke operators, Grundl. math. Wiss., 286, Springer-Verlag, 1987.
  • S. Böcherer and C.-G. Schmidt, $p$-adic measures attached to Siegel modular forms, Ann. Inst. Fourier (Grenoble), 50 (2000), 1375–1443.
  • K. Gunji, The dimension of the space of Siegel Eisenstein series of weight one, Math. Z., 260 (2008), 187–201.
  • E. Hecke, Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. Math. Sem. Univ. Hamburg, 5 (1927), 199–224; Mathematische Werke, Göttingen Vandenhoeck & Ruprecht, 1970, 461–486.
  • H. Katsurada, An explicit formula for Siegel series, Amer. J. Math., 121 (1999), 415–452.
  • G. Kaufhold, Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades, Math. Ann., 137 (1959), 454–476.
  • H. Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Math., 216, Springer-Verlag, Berlin-New York, 1971.
  • T. Miyake, Modular forms, Springer-Verlag, Berlin, 1989.
  • Y. Mizuno, An explicit arithmetic formula for the Fourier coefficients of Siegel-Eisenstein series of degree two and square-free odd levels, preprint.
  • B. Schoeneberg, Elliptic Modular Functions, Grundle. math. Wiss., 203, Springer-Verlag, 1974.
  • G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann., 260 (1982), 269–302.
  • G. Shimura, On Eisenstein series, Duke Math. J., 50 (1983), 417–476.
  • B. Srinivasan, The character of the finite symplectic group $Sp(4,q)$, Trans. Amer. Math. Soc., 131 (1968), 488–525.
  • S. Takemori, $p$-adic Siegel Eisenstein series of degree two, J. Number Theory, 132 (2012), 1203–1264.