Eisenstein series on quaternion half-space
The author investigates the non-analytic Eisenstein series of Siegel type on the quaternion half-space of degree 2 \[ E(Z,s):=\sum_{\textstyle{{*\;*\choose C D}:{*\;*\choose 0\;*}\backslash\Gamma_ 2}}|\text{det}(CZ+D)|^{-s}, \] where \(\Gamma_ 2\) denotes the modular group over the Hurwitz order [cf. the reviewer, Modular forms on half-spaces of quaternions (Lect. Notes Math. 1143) (Springer, Berlin 1985; Zbl 0564.10032)]. Set \[ \Omega(s,Z):=2^{s/2}(1-2^{2-s})(s- 2)(\text{det} Y)^{s/2}\xi(s)\xi(s-2)E(Z,s) , \] \(\xi(s):=\pi^{- s/2}\Gamma\left({s\over 2}\right)\zeta(s)\). Then \(\Omega(s,Z)\) is shown to be continued meromorphically to the whole \(s\)-plane with simple poles at \(s=0,1,5,6\) and the functional equation \(\Omega(s,Z)=\Omega(6-s,{^ tZ})\). The proof proceeds by calculating the singular series and applying Shimura's results on the confluent hypergeometric function. A part of these results were independently communicated by \textit{S. Nagaoka} [Proc. Japan Acad., Ser. A 68, 101-104 (1992); see the review below].
scientific article; zbMATH DE number 97884