On Hermitian Eisenstein series
Let \(\Gamma_ n(K)\) denote the Hermitian modular group of degree \(n\) in the sense of \textit{H. Braun} [Ann. Math., II. Ser. 50, 827-855 (1949; Zbl 0038.238)] associated with an imaginary quadratic number field \(K\) of class number 1. The author considers the Eisenstein series \[ E_ k^{(n)} (Z,s) : =\sum_{{AB \choose CD} : {** \choose 0*} \backslash \Gamma_ n (K)} \text{det} (CZ + D)^{-k} |\;\text{det} (CZ + D) |^{-s}, \] which converges absolutely if and only if \(\text{Re} (s) + k>2n\). At first the author shows that \(E_ k^{(n)}(Z,s)\) can be meromorphically continued to the whole \(s\)-plane and satisfies a functional equation under \(s \mapsto 2n - s\). Now the author makes use of \textit{G. Shimura}'s work [Duke Math. J. 50, 417-476 (1983; Zbl 0519.10019)] and the theory of singular modular forms. As an application he obtains conditions when Eisenstein series of low weight are holomorphic in \(Z\) as well as certain identities with theta series e.g. \[ E_ 4^{(4)} (Z,0) = 2 \theta^{(4)} (Z;I), \] where \(I\) denotes the Iyanaga form.
scientific article; zbMATH DE number 701463