Eisenstein series on quaternion half-space of degree 2
The author considers the Eisenstein series of Siegel type on the quaternion half-space of degree 2 \[ E_ k(Z,s):=(\text{det} Y)^{s/2}\sum_{\textstyle {{*\;*\choose C D}:{*\;*\choose 0\;*}\backslash\Gamma_ 2}}|\text{det}(CZ+D)|^{- s}\text{det}(CZ+D)^{-k}, \] where \(\Gamma_ 2\) denotes the modular group over the Hurwitz order [for details cf. the reviewer, Modular forms on half-spaces of quaternions (Lect. Notes Math. 1143) (Springer, Berlin (1985; Zbl 0564.10032)]). By calculating the Siegel series explicitly the meromorphic continuation and functional equation of \(E_ 0(Z,s)\) under \(s\mapsto 6-s\) and \(Z\mapsto{^ tZ}\) are proved. Moreover, the Fourier expansion of \(E_ k(Z,0)\), \(k\geq 4\) even, is determined, which involves divisor sums only. A part of these results were independently communicated by \textit{H. H. Kim} [Manuscr. Math. 76, 85-104 (1992); see the review above].
scientific article; zbMATH DE number 97939