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Mat. Sb. (N.S.), 1981, Volume 115(157), Number 3(7), Pages 337–363 (Mi msb2400)  

This article is cited in 4 scientific papers (total in 4 papers)

A basis of eigenfunctions of Hecke operators in the theory of modular forms of genus $n$

S. A. Evdokimov


Abstract: Let $\mathfrak M^n_k(\Gamma,\mu)$, where $n,k>0$ are integers, $\Gamma$ is some congruence subgroup of $\Gamma^n=\operatorname{Sp}_n(\mathbf Z)$ and $\mu\colon\Gamma\to\mathbf C^*$ is a congruence-character of $\Gamma$, be the space of all Siegel modular forms of genus $n$, weight $k$ and character $\mu$ with respect to $\Gamma$. In this paper, for a very broad class of congruence subgroups $\Gamma$ of $\Gamma^n$, including all those previously investigated and practically all those groups encountered in applications, the author constructs a sufficiently large commutative ring of Hecke operators, acting on $\mathfrak M^n_k(\Gamma,\mu)$, a canonical decomposition
\begin{equation} \mathfrak M^n_k(\Gamma,\mu)=\bigoplus^n_{r=0}\mathfrak M^{n,r}_k(\Gamma,\mu) \tag{1} \end{equation}
and a canonical inner product $( {,} )_\Gamma$ on $\mathfrak M^n_k(\Gamma,\mu)$. It is shown that the Hecke operators preserve the canonical decomposition (1) and that they are normal with respect to the canonical inner product $( {,} )_\Gamma$.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Sbornik, 1982, 43:3, 299–321

Bibliographic databases:

UDC: 511.61
MSC: Primary 10D20; Secondary 10D07
Received: 15.12.1980

Citation: S. A. Evdokimov, “A basis of eigenfunctions of Hecke operators in the theory of modular forms of genus $n$”, Mat. Sb. (N.S.), 115(157):3(7) (1981), 337–363; Math. USSR-Sb., 43:3 (1982), 299–321

Citation in format AMSBIB
\Bibitem{Evd81}
\by S.~A.~Evdokimov
\paper A~basis of eigenfunctions of Hecke operators in the theory of modular forms of genus~$n$
\jour Mat. Sb. (N.S.)
\yr 1981
\vol 115(157)
\issue 3(7)
\pages 337--363
\mathnet{http://mi.mathnet.ru/msb2400}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=628215}
\zmath{https://zbmath.org/?q=an:0465.10019}
\transl
\jour Math. USSR-Sb.
\yr 1982
\vol 43
\issue 3
\pages 299--321
\crossref{https://doi.org/10.1070/SM1982v043n03ABEH002450}


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    Erratum

    This publication is cited in the following articles:
    1. V. A. Gritsenko, “The action of modular operators on the Fourier–Jacobi coefficients of modular forms”, Math. USSR-Sb., 47:1 (1984), 237–268  mathnet  crossref  mathscinet  zmath
    2. Evdokimov S., “Dirichlet Series, Proportional to Andrianov Zeta-Functions, in the Theory of Siegel Modular-Forms of the 3rd Degree”, 277, no. 1, 1984, 25–29  mathscinet  zmath  isi
    3. A. Yu. Nenashev, “Multiplicative properties of the Fourier coefficients of Siegel modular forms for principal congruence subgroups of the group $\operatorname{Sp}(n,\mathbf Z)$”, Math. USSR-Sb., 60:1 (1988), 237–254  mathnet  crossref  mathscinet  zmath
    4. Siegfried Böcherer, Soumya Das, “Characterization of Siegel cusp forms by the growth of their Fourier coefficients”, Math. Ann, 2013  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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