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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

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
Full Text: DOI EuDML
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