On the Petersson inner products of Fourier–Jacobi coefficients and Hecke eigenvalues of Siegel cusp forms
Volume 197 / 2021
Abstract
Let $F$ and $G$ be Siegel cusp forms of weight $k$ and degree $n \gt 1$ with Fourier–Jacobi coefficients $f_m$ and $g_m$ respectively for all $m \in \mathbb N $. Assume that the Petersson inner products $\langle f_m, g_m \rangle $ are real for all $m \in \mathbb N $. We prove that if $ \langle F, G \rangle = 0$ and not all $\langle f_m, g_m \rangle $ are zero, the sequence $ \{ \langle f_m, g_m \rangle \}_{m \in \mathbb N }$ changes sign infinitely often. When $\langle F, G \rangle \ne 0$, we show that $ | \langle f_m, g_m \rangle | \gt c m^{k-1}$ for infinitely many $m\in \mathbb N $, where $c \gt 0$ is a constant depending on $F$ and $G$. This generalizes a result of Kohnen. We also investigate similar properties of these Petersson inner products in arithmetic progressions. In this case, our results strengthen a result of Gun and Kumar. Finally, we study simultaneous non-vanishing of the Hecke eigenvalues of Siegel cusp forms of degree $2$.
