On the growth of Fourier coefficients of certain special Siegel cusp forms
Let \(S_k(\Gamma_n)\) denote the space of Siegel cusp forms of weight \(k\) and genus \(n\). A conjecture of \textit{H.~L. Resnikoff} and \textit{R. L. Saldaña} [J. Reine Angew. Math. 265, 90--109 (1974; Zbl 0278.10028)] claims that the Fourier coefficients \(A(T)\) of any \(F\in S_k(\Gamma_n)\) satisfy \[ A(T)\ll_{\varepsilon,F}(\text{det\,}T)^{{k\over 2}-{n+1\over 5}+\varepsilon}\qquad (\varepsilon> 0)\tag{1} \] for all half-integral symmetric matrices \(T> 0\) of size \(n\). This conjecture is true for \(n= 1\) by Deligne's theorem (previously the Ramanujan-Petersson conjecture) whereas for \(n> 1\) counterexamples are known. The author adds to the list of counterexamples and proves: Let \(n\equiv 1\pmod 4\), \(n\equiv k\mod 2\) and let \(F\in S_{k+n}(\Gamma_{2n})\) be a Hecke eigenform that is an Ikeda lift of a normalized Hecke eigenform \(f\in S_{2k}(\Gamma_1)\). Then estimate (1) does not hold true.
scientific article; zbMATH DE number 2131982