Kocher-Maaß-series for modular forms of quaternions
In previous work [``Modular Forms on Half-Spaces of Quaternions'', Lect. Notes Math. 1143 (1985; Zbl 0564.10032)] the author created a theory of modular forms of quaternions following closely the classical lines of the theory of Siegel modular forms. As in the Siegel case one can attach to a modular form of quaternions a Dirichlet series by a generalized Mellin transform. In the paper under review the author proves meromorphic continuation and functional equation of such Dirichlet series along the lines of \textit{H. Maa}'s work [``Siegel's Modular Forms and Dirichlet Series'', Lect. Notes Math. 216 (1971; Zbl 0224.10028)] on the Siegel case. To determine poles and residues the author uses a method of \textit{T. Arakawa} [Math. Ann. 238, 157-173 (1978; Zbl 0373.10017)]. Surprisingly (and in contrast to the Siegel case) this Dirichlet series has an Euler product expansion, if the modular form in question is a Hecke eigenform in the Maa space.
scientific article; zbMATH DE number 4144057
scientific article