Non-vanishing of scalar products of Fourier-Jacobi coefficients of Siegel cusp forms
Let \(F\) and \(G\) be nonzero Siegel cusp forms of integral weight \(k\) w.r.t. the Siegel modular group \(\Gamma_n\) of degree \(n\geq 2\), with Fourier-Jacobi coefficients \(f_m\) and \(g_m\), where \(m\in{\mathbb{N}}\). In this paper, the authors prove that if \(N\in{\mathbb{N}}\), then, for any given \(a\in{\mathbb Z}\) with \((a,N)=1\), there exist infinitely many \(m\in{\mathbb{N}}\) such that \(f_m\not=0\) and \(m \equiv a \pmod{N}\). They also prove a similar (but a little weaker and conditional) result on the nonvanishing of the Petersson scalar products \(\langle f_{m},g_{m}\rangle\). The proof is mainly based on analytic properties of the Rankin-Dirichlet series \(D_{F,G}(s)=\sum_{m\geq 1}\langle f_m,g_m \rangle m^{-s}\) and its character twists, studied in [\textit{W. Kohnen} and \textit{N. P. Skoruppa}, Invent. Math. 95, 541-558 (1989; Zbl 0665.10019)], [\textit{A. Krieg}, Acta Arith. 59, 243-259 (1991; Zbl 0731.11028)], [\textit{W. Kohnen}, Mem. Fac. Sci., Kyushu Univ. Ser. A 47, 103-117 (1993; Zbl 0783.11024)] and [\textit{W. Kohnen, A. Krieg}, and \textit{A. Sengupta}, Manuscr. Math. 87, 489-499 (1995; Zbl 0879.11021)]. As an application, in the special case of \(n=2\) and \(k\) even, the authors derive results on the nonvanishing of Hecke eigenvalues of a simultaneous Hecke eigenform \(F\) whose first Fourier-Jacobi coefficient \(f_1\) is nonzero.
scalar products of Fourier-Jacobi coefficients
scientific article; zbMATH DE number 1336969
scientific article