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

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
$$\mathfrak M^n_k(\Gamma,\mu)=\bigoplus^n_{r=0}\mathfrak M^{n,r}_k(\Gamma,\mu) \tag{1}$$
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

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
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
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
4. Siegfried Böcherer, Soumya Das, “Characterization of Siegel cusp forms by the growth of their Fourier coefficients”, Math. Ann, 2013
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