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Mat. Sb. (N.S.), 1982, Volume 119(161), Number 2(10), Pages 248–277 (Mi msb2847)  

This article is cited in 13 scientific papers (total in 13 papers)

The action of modular operators on the Fourier–Jacobi coefficients of modular forms

V. A. Gritsenko

Abstract: The author studies the imbedding of the Hecke $p$-ring $L_p^{n+1}$ of the modular group $\mathrm{Sp}_{n+1}(\mathbf{Z})$ of genus $n+1$ in the Hecke ring $L_p^{n,1}$ of the group $\Gamma_{n,1}$ given by
$$ \Gamma_{n,1}=\{\begin{pmatrix} A&0&B&*
0&0&0&* \end{pmatrix}\in\mathrm{Sp}_{n+1}(\mathbf{Z})\}. $$
It is proved that the Hecke polynomial $Q_{n,1}^{(n+1)}(z)$ of $L_p^{n+1}$ splits over $L_p^{n,1}$, and the coefficients of the factors can be written explicitly in terms of the coefficients of the Hecke polynomial $Q^{(n)}(z)$ of genus $n$ and “negative” powers of a particular element $\Lambda$ of $L_p^{n,1}$. The “$-1$ power” of $\Lambda$ is computed and a formula for $\Lambda^{-2}$ is presented. The results that are obtained permit one to describe a large class of power series constructed from the Fourier–Jacobi coefficients by means of eigenfunctions with denominators depending only on the eigenvalues.
Bibliography: 19 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 47:1, 237–268

Bibliographic databases:

UDC: 519.4
MSC: Primary 10D05, 10D20; Secondary 10D24, 10D40
Received: 02.02.1982

Citation: V. A. Gritsenko, “The action of modular operators on the Fourier–Jacobi coefficients of modular forms”, Mat. Sb. (N.S.), 119(161):2(10) (1982), 248–277; Math. USSR-Sb., 47:1 (1984), 237–268

Citation in format AMSBIB
\by V.~A.~Gritsenko
\paper The action of modular operators on the Fourier--Jacobi coefficients of modular forms
\jour Mat. Sb. (N.S.)
\yr 1982
\vol 119(161)
\issue 2(10)
\pages 248--277
\jour Math. USSR-Sb.
\yr 1984
\vol 47
\issue 1
\pages 237--268

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    This publication is cited in the following articles:
    1. V. A. Gritsenko, “Expantion of Hecke polynomials of classical groups”, Math. USSR-Sb., 65:2 (1990), 333–356  mathnet  crossref  mathscinet  zmath
    2. C. Ziegler, “Jacobi forms of higher degree”, Abh Math Semin Univ Hambg, 59:1 (1989), 191  crossref  mathscinet  zmath
    3. Kohnen W. Skoruppa N., “A Certain Dirichlet Series Attached to Siegel Modular-Forms of Degree-2”, Invent. Math., 95:3 (1989), 541–558  crossref  mathscinet  zmath  adsnasa  isi
    4. Kohnen W., “A Note on Eigenvalues of Hecke Operators on Siegel Modular-Forms of Degree 2”, Proc. Amer. Math. Soc., 113:3 (1991), 639–642  crossref  mathscinet  zmath  isi
    5. Rolf Berndt, “On local Whittaker models for the Jacobi group of degree one”, manuscripta math, 84:1 (1994), 177  crossref  mathscinet  zmath  isi
    6. Rolf Berndt, “L-functions for Jacobi forms à la Hecke”, manuscripta math, 84:1 (1994), 101  crossref  mathscinet  zmath  isi
    7. V. A. Gritsenko, V. V. Nikulin, “Igusa modular forms and 'the simplest' Lorentzian Kac–Moody algebras”, Sb. Math., 187:11 (1996), 1601–1641  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. Rolf Berndt, Jost Homrighausen, “On automorphicL-functions for the Jacobi group of degree one and a relation withL-functions for Jacobi forms”, manuscripta math, 92:1 (1997), 223  crossref  mathscinet  zmath  isi
    9. Y. Martin, “L-Functions for Jacobi Forms of Arbitrary Degree”, Abh Math Semin Univ Hambg, 68:1 (1998), 45  crossref  mathscinet  zmath
    10. Gritsenko, VA, “Automorphic forms and Lorentzian Kac-Moody algebras. Part II”, International Journal of Mathematics, 9:2 (1998), 201  crossref  isi  elib
    11. Choie Y., Dougherty S., “Codes, Lattices and Modular Forms”, 2003 IEEE Information Theory Workshop, Proceedings, IEEE, 2003, 259–262  crossref  isi
    12. Choie Y., Dougherty S., “Codes Over Rings, Complex Lattices and Hermitian Modular Forms”, Eur. J. Comb., 26:2 (2005), 145–165  crossref  mathscinet  zmath  isi
    13. Shuichi Hayashida, “Fourier-Jacobi expansion and the Ikeda lift”, Abh. Math. Semin. Univ. Hambg, 2011  crossref
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