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Estimates for Fourier coefficients of Siegel cusp forms of degree two. (English) Zbl 0783.11023
The article introduces a new idea into the study of the Fourier coefficients of Siegel modular forms, namely, to use the Fourier-Jacobi expansion of the form. In a first step, the Fourier coefficients of Jacobi cusp forms are estimated, using Poincaré series and estimates for certain linear combinations of Kloosterman sums from earlier work of the author with B. H. Gross and D. B. Zagier [Math. Ann. 278, 497-562 (1987; Zbl 0641.14013)]. The Petersson norm of the Jacobi form appears as a factor in this estimate and has to be estimated in a second step for the Fourier-Jacobi coefficients of a Siegel cusp form. This is done using the analytic properties of the Dirichlet series \(\zeta(2s- 2k+4)\sum_{m\geq 1} \|\varphi_ m\|^ 2 m^{-s}\) where the \(\varphi_ m\) are the Fourier-Jacobi coefficients of a Siegel cusp form \(F\) of weight \(k\). The analytic properties of this Dirichlet series have been studied by the author and N. Skoruppa [Invent. Math. 95, 541- 558 (1989; Zbl 0665.10019)].
The estimate \(a(T)\ll_{\varepsilon,F} (\min T)^{5/18+\varepsilon} (\text{det} T)^{(k-1)/2+ \varepsilon}\) achieved for Siegel cusp forms of degree 2 by the new method improves on the results of Y. Kitaoka [Nagoya Math. J. 93, 149-171 (1984; Zbl 0531.10031)] and of S. Raghavan and R. Weissauer [Number theory and dynamical systems, Lond. Math. Soc. Lect. Note Ser. 134, 87-102 (1989; Zbl 0686.10019)]. The present article restricts attention to the case of modular forms for the full modular group and to Fourier coefficients at matrices \(T\) of fundamental discriminant. The latter restriction is lifted in Part II of this article [Nagoya Math. J. 128, 171-176 (1992)]. The case of Siegel modular forms of arbitrary degree is investigated in joint work of the author and S. Böcherer [Math. Ann. 297, 499-517 (1993)].

MSC:
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F30 Fourier coefficients of automorphic forms
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References:
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