A generalization of Kohnen's estimates for Fourier coefficients of Siegel cusp forms
In [Compos. Math. 87, 231-240 (1993; Zbl 0783.11023)], \textit{W. Kohnen} has applied the Eichler-Zagier theory of Jacobi forms to obtain estimates on the Fourier coefficients of Siegel cusp forms of weight \(k\) on the full Siegel modular group \(\Gamma_2= \text{Sp}_2(\mathbb{Z})\) of degree 2. Suppose the Siegel cusp form on \(\Gamma_2\) is given in terms of its expansion \[ f(Z)= \sum_T a(T) e^{2\pi i\text{ tr}(TZ)},\tag{1} \] where \(Z\) is in \(H_2\), the Siegel upper half-space of degree 2, and \(T\) runs through all positive definite integral \(2\times 2\) matrices. Kohnen's estimate is \[ a(T)= O\Biggl((\text{det }T)^{{k\over 2}-{13\over 36}+\varepsilon}\Biggr),\quad \varepsilon>0.\tag{2} \] This article extends Kohnen's method to the (Hecke) congruence subgroups \(\Gamma_2(N)\), \(N\in\mathbb{Z}^+\), defined by: \(\Gamma_2(N)= \left\{\left(\begin{smallmatrix} A & B\\ C & D\end{smallmatrix}\right)\in \Gamma_2\mid C\equiv 0\pmod N\right\}\). The author states the result in the form \[ a(T)= O\Biggl((\min T)^{{5\over 18}+ \varepsilon}(\text{det }T)^{{k- 1\over 2}+\varepsilon}\Biggr),\quad \varepsilon> 0,\tag{3} \] where \(\min T\) is the smallest positive integer represented by \(T\). In view of the standard estimate, \(\min T= O((\text{det T})^{1/2})\), (3) can be rewritten in the form (2).
Fourier coefficients of Siegel cusp forms
scientific article; zbMATH DE number 1114162