A remark on the growth of Fourier coefficients of Siegel cusp forms
About 25 years ago \textit{H. L. Resnikoff} and \textit{R. L. Saldana} [J. Reine Angew. Math. 265, 90-109 (1974; Zbl 0278.10028)] conjectured that the Fourier coefficients \(a(T)\) of a Siegel cusp form \(F\) of genus \(g\) and weight \(k\) satisfy \[ a(T)\ll_{\varepsilon,F} (\det T)^{\frac k2- \frac{g+1}{4}+ \varepsilon} \qquad (\varepsilon> 0), \tag \(*\) \] which generalizes the Ramanujan-Petersson conjecture in the case \(g=1\). It is known that this estimate does not hold for cusp forms \(F\) in the Maaß space. The authors construct the first explicit examples of such forms satisfying \((*)\) in the case \(g>1\). They are given by theta series with spherical harmonics.
scientific article; zbMATH DE number 1336886