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 Mat. Sb. (N.S.), 1975, Volume 96(138), Number 4, Pages 584–593 (Mi msb3410)

On the connection of the eigenvalues of Hecke operators and the Fourier coefficients of eigenfunctions for Siegel's modular forms of genus $n$

N. A. Zharkovskaya

Abstract: Let $f(z)=\sum_{N\geqslant0}a(N)\exp2\pi i\sigma(NZ)$ be Siegel's modular form of genus $n$ which is an eigenfunction for all operators in the $p$-component of a Hecke ring; in particular, $T_{p^\delta}f(Z)=\lambda_f(p^\delta)f(Z)$. This paper examines the series $\sum_{\delta=0}^\infty a(p^\delta N)t^\delta$ ($p$ does not divide $N$). It is proved that each such series is a rational function, where the degree of the numerator of this function does not exceed $2^n-2$ and the denominator coincides with the denominator of the series $\sum_{\delta=0}^\infty \lambda_f(p^\delta)t^\delta$.
Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 25:4, 549–557

Bibliographic databases:

UDC: 517.863
MSC: 10D20, 42A16

Citation: N. A. Zharkovskaya, “On the connection of the eigenvalues of Hecke operators and the Fourier coefficients of eigenfunctions for Siegel's modular forms of genus $n$”, Mat. Sb. (N.S.), 96(138):4 (1975), 584–593; Math. USSR-Sb., 25:4 (1975), 549–557

Citation in format AMSBIB
\Bibitem{Zha75} \by N.~A.~Zharkovskaya \paper On~the connection of the eigenvalues of~Hecke operators and the Fourier coefficients of eigenfunctions for Siegel's modular forms of genus~$n$ \jour Mat. Sb. (N.S.) \yr 1975 \vol 96(138) \issue 4 \pages 584--593 \mathnet{http://mi.mathnet.ru/msb3410} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=379383} \zmath{https://zbmath.org/?q=an:0313.10028} \transl \jour Math. USSR-Sb. \yr 1975 \vol 25 \issue 4 \pages 549--557 \crossref{https://doi.org/10.1070/SM1975v025n04ABEH002462} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. N. Andrianov, “On factorization of Hecke polynomials for the sympletic group of genus $n$”, Math. USSR-Sb., 33:3 (1977), 343–373
2. Shin-ichiro Mizumoto, “On integrality of Eisenstein liftings”, manuscripta math, 90:1 (1996), 267
3. Stefan Breulmann, Winfried Kohnen, “On hecke eigenforms of degree n”, Abh Math Semin Univ Hambg, 70:1 (2000), 119
4. H. Katsurada, “Euler factor of a certain Dirichlet series attached to Siegel Eisenstein Series”, Abh Math Semin Univ Hambg, 71:1 (2001), 81
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