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On Fourier coefficients of Siegel modular forms. (English) Zbl 0636.10022
The authors deal with estimates of Fourier coefficients of Siegel modular forms. The second author [Math. Gottingensis 75 (1986; Zbl 0597.10022)] had obtained the estimate \[ a_ f(T)=O((\det T)^{k-(n+1)/2} (\min T)^{1-k/2}) \] for a Siegel modular form f of degree n and half- integral weight \(k>2n\). In the paper under review the estimate is improved, whenever the arithmetical minimum min T is fixed and det T tends to infinity. The proof is based on the fact that for even \(k>2n\) the space of Siegel modular forms is spanned by Klingen-Eisenstein series, whose Fourier coefficients were determined by the first author [Math. Z. 183, 21-46 (1983; Zbl 0497.10020)].
The application to theta series yields a remarkable theorem of Tartakovskij-type on the number of primitive representations of a positive definite quadratic form by a given even unimodular positive definite quadratic form. Finally the well-known estimate \(a_ f(T)=O((\det T)^{k/2})\) for a cusp form f is improved.
Reviewer: A.Krieg

11F27 Theta series; Weil representation; theta correspondences
11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11E16 General binary quadratic forms
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