Piatetski-Shapiro, Ilya I.; Rallis, S. A new way to get Euler products. (English) Zbl 0651.10021 J. Reine Angew. Math. 392, 110-124 (1988). A new principle is introduced that allows one to obtain Euler products in some situations where there is no uniqueness of “Whittaker models”. Namely we prove a certain local integral is independent of the choice of a certain Whittaker model. Then using induction we prove that this local statement implies an Euler product for the standard \(L\)-function of automorphic forms on the symplectic group. From this we then show a necessary and sufficient condition for the existence of the largest pole of this \(L\)-function in terms of the theta correspondence. Reviewer: I. Piatetski-Shapiro Cited in 2 ReviewsCited in 29 Documents MSC: 11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations 22E30 Analysis on real and complex Lie groups 11F27 Theta series; Weil representation; theta correspondences Keywords:Euler products; local integral; Whittaker model; L-function of automorphic forms; symplectic group; pole; theta correspondence PDFBibTeX XMLCite \textit{I. I. Piatetski-Shapiro} and \textit{S. Rallis}, J. Reine Angew. Math. 392, 110--124 (1988; Zbl 0651.10021) Full Text: DOI Crelle EuDML