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Über die Fourierkoeffizienten der Siegelschen Eisensteinreihen. (German) Zbl 0533.10023
Let \(\psi_ n^ k(Z)=\sum_{C,D}\det(CZ+D)^{-k}\) be the Siegel-Eisenstein series of degree \(n\) and weight \(k\). \(\psi_ n^ k(Z)\) can be expanded to \[ \psi^ k_ n(Z)=\sum_{T}a^ k_ n(T)e^{2\pi i \text{trace}(TZ)} \] where \(T\) runs over all positive semi-definite semi-integral symmetric matrices of degree \(n\). By Siegel’s formula, \(a^ k_ n(T)\) can be expressed as \[ a^ k_ n(T)=A^ k_ n \det(T)^{k-(n+1)/2}\prod_{p}S_ p(T). \] In the above, \(A^ k_ n\) is a certain constant involving values of the gamma function, and \(S_ p(T)\) is given by \[ S_ p(T)=\lim_{a\to \infty}(p^ a)^{n(n+1)/2- 2kn}\quad A_{p^ a}(J^{2k},T), \] where \(A_{p^ a}(J^{2k},T)\) is the number of matricial solutions of a given congruence (say(C)).
In this paper, the author introduces a quantity \(b^ k_ n(T)\), which is also expressed by the infinite product formula: \[ b^ k_ n(T)=A^ k_ n \det(T)^{k-(n+1)/2}\prod_{p}\hat S_ p(T), \] where \(\hat S_ p(T)\) is given by \[ \hat S_ p(T)=\lim_{a\to \infty}(p^ a)^{n(n+1)/2-2kn}\quad B_{p^ a}(J^{2k},T). \] \(B_{p^ a}(J^{2k},T)\) is the number of primitive matricial solutions of the congruence (C). At first a relation between \(a^ k_ n(T)\) and \(b^ k_ n(T)\) is proved, and the values of \(b^ k_ n(T)\) and \(a^ k_ n(T)\) are shown to coincide in very special cases of \(T\). Then a study of \(b^ k_ n(T)\) is done, and as an application the common denominator of all \(a^ k_ n(T)\), where \(T\) is non-degenerate and \(n\) is fixed, is determined.
Reviewer: M. Ozeki

MSC:
11F30 Fourier coefficients of automorphic forms
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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References:
[1] Andrianov, A.N.: Euler decompositions of theta-transformations of Siegel modular forms of genus n. Math. USSR-Sb. 34, 259-300 (1978) · Zbl 0412.10021 · doi:10.1070/SM1978v034n03ABEH001160
[2] Andrianov, A.N.: The multiplicative arithmetic of Siegel modular forms. Russian Math. Surveys 34, 75-148 (1979) · Zbl 0427.10017 · doi:10.1070/RM1979v034n01ABEH002872
[3] Böcherer,S.: Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. Erscheint in Math. Z. · Zbl 0497.10020
[4] Brandt, H.: Zur Zahlentheorie der quadratischen Formen. Jahresber. Deutsch. Math.-Verein. 47, 149-159 (1937) · JFM 63.0119.02
[5] Carlitz, L.: Arithmetic properties of generalized Bernoulli numbers. J. Reine Angew. Math. 202, 174-182 (1959) · Zbl 0125.02202 · doi:10.1515/crll.1959.202.174
[6] Cassels, J.W.S.: Rational quadratic forms. L.M.S. Monographs 13. London-New York-San Francisco: Academic Press 1978 · Zbl 0395.10029
[7] Hermann,O.: Numerische Untersuchungen an Fourierkoeffizienten Siegelscher Modulformen dritten Grades. In: Anwendungen automorpher Funktionen auf Zahlentheorie. Tagungsbericht 32/1979, Mathematisches Forschungsinstitut Oberwolfach.
[8] Kitaoka, Y.: Modular forms of degree n and representation by quadratic forms. Nagoya Math. J. 74, 95-122 (1979) · Zbl 0404.10014
[9] Leopoldt, H.W.: Eine Verallgemeinerung der Bernoullischen Zahlen. Abh. Math. Sem. Univ. Hamburg 131-140 (1958) · Zbl 0080.03002
[10] Maaß, H.: Die Fourierkoeffizienten der Eisensteinreihen zweiten Grades. Mat. Fys. M?dd. Danske Vid. Selsk. 34, Nr. 7 (1964) · Zbl 0244.10023
[11] Maaß, H.: Über die Fourierkoeffizienten der Eisensteinreihen zweiten Grades. Mat. Fys. Medd. Danske Vid. Selsk. 38, Nr. 14 (1972) · Zbl 0244.10023
[12] Ozeki, M.: Explicit formulas for the Fourier coefficients of Eisenstein series of degree 3. In: Siegeische Modulfunktionen. Tagungsbericht 34/1980. Mathematisches Forschungsinstitut Oberwolfach.
[13] Siegel, C.L.: Über die analytische Theorie der quadratischen Formen. Ann. of Math. 36, 527-606 (1935) · JFM 61.0140.01 · doi:10.2307/1968644
[14] Siegel, C.L.: Einführung in die Theorie der Modulfunk-tionen n-ten Grades. Math. Ann. 116, 617-657 (1939) · Zbl 0021.20302 · doi:10.1007/BF01597381
[15] Siegel, C.L.: Über die Fourierschen Koeffizienten der Eisensteinschen Reihen. Mat. Fys. Medd. Danske Vid. Selsk. 34, Nr.6 (1964) · Zbl 0132.06401
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