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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
##### Citations:
Zbl 0641.14013; Zbl 0665.10019; Zbl 0531.10031; Zbl 0686.10019
Full Text:
##### References:
 [1] Bateman, H. : Higher Transcendental Functions . II. New York-Toronto-London, McGraw-Hill, 1953. [2] Böcherer, S. and Raghavan, S. : On Fourier coefficients of Siegel modular forms , J. reine angew. Math. 384 (1988), 80-101. · Zbl 0636.10022 [3] Eichler, M. and Zagier, D. : The Theory of Jacobi Forms (Progress in Maths. vol. 55), Birkhäuser, Boston, 1985. · Zbl 0554.10018 [4] Fomenko, O.M. : Fourier coefficients of Siegel cusp forms of genus n , J. Soviet. Math. 38 (1987), 2148-2157. · Zbl 0624.10022 [5] Gross, B. , Kohnen, W. and Zagier, D. : Heegner points and derivatives of L-series. II . Math. Ann. 278 (1987), 497-562. · Zbl 0641.14013 [6] Iwaniec, H. : Fourier coefficients of modular forms of half-integral weight . Invent Math. 87 (1987), 385-401. · Zbl 0606.10017 [7] Kitaoka, Y. : Fourier coefficients of Siegel cusp forms of degree two , Nagoya Math. J. 93 (1984), 149-171. · Zbl 0531.10031 [8] Kohnen, W. and Skoruppa, N. - P.: A certain Dirichlet series attached to Siegel modular forms of degree two . Invent. Math. 95 (1989), 541-558. · Zbl 0665.10019 [9] Landau, E. : Uber die Anzahl der Gitterpunkte in Gewissen Bereichen. II. Göttinger Nachr . (1915), 209-243.(Collected works, vol. 6, pp. 308-342. Essen: Thales, 1986.) · JFM 45.0312.02 [10] Raghavan, S. and Weissauer, R. : Estimates for Fourier coefficients of cusp forms . In M. M. Dodson and J. A. G. Vickers eds., Number Theory and Dynamical Systems , pp. 87-102. London Math. Soc. Lect. Not. ser. vol. 134. Cambridge University Press, Cambridge, 1989. · Zbl 0686.10019 [11] Sato, M. and Shintani, T. : On zeta functions associated with prehomogeneous vector spaces , Ann. of Math. 100 (1974), 131-170. · Zbl 0309.10014
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