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Mathematics of the USSR-Sbornik, 1982, Volume 43, Issue 3, Pages 299–321
DOI: https://doi.org/10.1070/SM1982v043n03ABEH002450
(Mi sm2400)
 

This article is cited in 8 scientific papers (total in 8 papers)

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

S. A. Evdokimov
References:
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.
Received: 15.12.1980
Russian version:
Matematicheskii Sbornik. Novaya Seriya, 1981, Volume 115(157), Number 3(7), Pages 337–363
Bibliographic databases:
UDC: 511.61
MSC: Primary 10D20; Secondary 10D07
Language: English
Original paper language: Russian
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/sm2400}
\mathscinet{http://mathscinet.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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  • https://doi.org/10.1070/SM1982v043n03ABEH002450
  • https://www.mathnet.ru/eng/sm/v157/i3/p337
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    This publication is cited in the following 8 articles:
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    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    References:56
     
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