On the Petersson norm of certain Siegel modular forms (Q1417917)

Let \(f\) be a normalized Hecke eigenform of weight \(2k\) with respect to \(\text{SL}_2(\mathbb{Z})\), and let \(F\) be its Ikeda lift of genus \(2n\), which is a Hecke eigenform of weight \(k+n\) with respect to the Siegel modular group \(\text{Sp}_{2n}(\mathbb{Z})\). [See \textit{T. Ikeda}, Ann. Math. (2) 154, 641--681 (2001; Zbl 0998.11023)]. The authors show that if \(k> 3n\), then the quotient \(\langle F,F\rangle/\langle f,f\rangle^n\) belongs to the field \(K_f\) which is generated by the Fourier coefficients of \(f\). Here, \(\langle\;,\;\rangle\) denotes the Petersson inner product. The proof uses results from \textit{S. Böcherer} [Math. Z. 189, 81--110 (1985; Zbl 0558.10022)] on special values of standard zeta-functions of modular forms. If \(n=1\), then \(F\) is the Saito-Kurokawa lift of \(f\); in this case the result was proved by \textit{M. Furusawa} [Math. Ann. 267, 543--548 (1984; Zbl 0519.10020)] and \textit{W. Kohnen} [J. Reine Angew. Math. 357, 96--100 (1985; Zbl 0547.10026)].