Journal of the Mathematical Society of Japan

On Siegel-Eisenstein series attached to certain cohomological representations


Full-text: Open access


We introduce a Siegel-Eisenstein series of degree 2 which generates a cohomological representation of Saito-Kurokawa type at the real place. We study its Fourier expansion in detail, which is based on an investigation of the confluent hypergeometric functions with spherical harmonic polynomials. We will also consider certain Mellin transforms of the Eisenstein series, which are twisted by cuspidal Maass wave forms, and show their holomorphic continuations to the whole plane.

Article information

J. Math. Soc. Japan, Volume 63, Number 2 (2011), 599-646.

First available in Project Euclid: 25 April 2011

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: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F30: Fourier coefficients of automorphic forms

real analytic Eisenstein series cohomological representations confluent hypergeometric functions Dirichlet series


MIYAZAKI, Takuya. On Siegel-Eisenstein series attached to certain cohomological representations. J. Math. Soc. Japan 63 (2011), no. 2, 599--646. doi:10.2969/jmsj/06320599.

Export citation


  • T. Arakawa, I. Makino and F. Sato, Converse theorem for not necessarily cuspidal Siegel modular forms of degree 2 and Saito-Kurokawa lifting, Comment. Math. Univ. St. Pauli, 50 (2001), 197–234.
  • H. Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann., 217 (1975), 271–285.
  • W. Duke and Ö. Imamo\=glu, A converse theorem and the Saito-Kurokawa lift, Internat. Math. Res. Notices, 7 (1996), 347–355.
  • Y. Hasegawa and T. Miyazaki, Twisted Mellin transforms of a real analytic residue of Siegel-Eisenstein series of degree 2, Internat. J. of Math, 20 (2009), 1011–1027.
  • S. Katok and P. Sarnak, Heegner points, cycles and Maass forms, Israel J. of Math., 84 (1993), 193–227.
  • W. Kohnen and D. Zagier, Values of $L$-series of modular forms at the center of the critical strip, Invent. Math., 64 (1981), 175–198.
  • G. Kaufhold, Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2, Grades, Math. Ann., 137 (1959), 454–476.
  • S. T. Lee, Degenerate principal series representations of $Sp(2n,\R)$, Composit. Math., 103 (1996), 123–151.
  • H. Maass, Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik, Math. Ann., 138 (1959), 287–315.
  • W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, third edition, Die Grundlehren der mathematischen Wissenschaften, 52, Springer-Verlag New York, Inc., New York, 1966.
  • T. Miyazaki, On Saito-Kurokawa lifting to cohomological Siegel modular forms, manuscripta math., 114 (2004), 139–163.
  • R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982.
  • S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J., 56 (1974), 147–161.
  • G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann., 260 (1982), 269–302.
  • G. Shimura, On differential operators attached to certain representations of classical groups, Invent. Math., 77 (1984), 463–488.
  • G. Shimura, On modular forms of half integral weight, Ann. of Math., 97 (1973), 440–481.
  • T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J., 58 (1975), 83–126.
  • N. J. Vilenkin, Special Functions and the Theory of Group Representations, Translated from the Russian by V. N. Singh, Translations of Mathematical Monographs, 22, American Mathematical Society, Providence, R. I., 1968.
  • D. Vogan and G. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math., 53 (1984), 51–90.