Note on Siegel-Eisenstein series of low weight
The author mainly proves the results announced in [Proc. Japan. Acad., Ser. A 70, 115-117 (1994; Zbl 0815.11026)]. Let \(\Gamma_n(K)\) denote the Hermitian modular group of degree \(n\) over the imaginary quadratic number field \(K\) of class number 1. The author considers the Siegel-Eisenstein series \[ E^{(n)}_k(Z,s):= \sum_{(\begin{smallmatrix} A\\ C\end{smallmatrix}\begin{smallmatrix} B\\ D\end{smallmatrix}):(\begin{smallmatrix} *\\ 0\end{smallmatrix} \begin{smallmatrix} *\\ *\end{smallmatrix})\backslash\Gamma_n(K)}\text{det}(CZ+ D)^{-k}|\text{det}(CZ+ D)|^{-s}, \] which converges absolutely for \(\text{Re}(s)+ k>2n\). At first he calculates the Fourier expansion expressing the coefficients in terms of Koecher-Maaß zeta-functions. It is shown that \(E^{(n)}_k(Z,s)\) possesses a meromorphic continuation to the whole \(s\)-plane and satisfies a functional equation under \(s\mapsto 2n-s\). Now Shimura's results are applied in order to derive conditions when Eisenstein series of low weight are holomorphic in \(Z\) as well as identities with theta series.
scientific article; zbMATH DE number 977864
scientific article