Estimates for Fourier coefficients of Siegel cusp forms of degree three (Q1377078)

Let \(f\) be a Siegel cusp form of integral weight \(k\) and degree \(g\geq 2\) on the Siegel modular group \(\Gamma_g\). For any \(g\)-rowed half-integral matrix \(S=S^t>0\) let \(a(S)\) denote the corresponding Fourier coefficient of \(f\). For arbitrary \(g\geq 2\) and \(k>g+1\) the best estimate so far for \(a(S)\) is due to \textit{S. Böcherer} and \textit{W. Kohnen} [Math. Ann. 297, 499-517 (1993; Zbl 0787.11017)]. The aim of the present paper is to improve upon the latter result in the special case \(g=3\). Omitting here the more technical estimate of the author's theorem, we only quote the shortest estimate from the Corollary: \[ a(S)\ll_\varepsilon (\text{det }S)^{\frac k2-\frac 14+\varepsilon} \tag \(*\) \] for any \(\varepsilon>0\). The proof is based on a clever modification of the method of Böcherer and Kohnen (loc. cit.) which is possible in the special case \(g=3\). -- The work under review is an abridged version of the author's thesis. (Heidelberg 1996; Zbl 0865.11038). See also the review Zbl 0894.11025 below.