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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:
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