Explicit formulae of Siegel Eisenstein series
Let \(\Gamma_m\) denote the Siegel modular group of degree \(m\). The author investigates the Eisenstein series \[ E^{(m)}_k(Z,s):= \text{det}(\text{Im }z)^s\sum_{(\begin{smallmatrix} a\\ c\end{smallmatrix} \begin{smallmatrix} b\\ d\end{smallmatrix}):(\begin{smallmatrix} *\\ 0\end{smallmatrix} \begin{smallmatrix} *\\ *\end{smallmatrix})\backslash\Gamma_m}\text{det}(cz+ d)^{-k}|\text{det}(cz+ d)|^{-2s}, \] which converges absolutely for \(\text{Re}(s)>(m+1-k)/2\). As is known, \(E^{(m)}_k(z,s)\) has a meromorphic continuation to the whole \(s\)-plane and satisfies a functional equation. The author quotes explicit formulae of the Fourier expansion of \(E^{(m)}_k(z,s)\) from \textit{S. Mizumoto} [Math. Ann. 297, 581-625 (1993; Zbl 0786.11024)]. The orders (as functions of \(s\)) of the Fourier coefficients at \(s=0\) are investigated. They turn out to be holomorphic except for \(k=(m+2)/2\). In the main part, functional equations for some pairs of Fourier coefficients are given. This leads to explicit formulae. Thus, the author is able to reprove \textit{R. Weissauer's} results [Stabile Modulformen und Eisensteinreihen, Lect. Notes Math. 1219, Springer-Verlag, Berlin (1986; Zbl 0596.10023)] on the holomorphy of \(E^{(m)}_k(Z,s)\) at \(s=0\) and the structure of the Fourier coefficients.
scientific article; zbMATH DE number 980720