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A new way to get Euler products. (English) Zbl 0651.10021

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.

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
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