Euler products and Fourier coefficients of automorphic forms on symplectic groups (Q1320060)

The standard \(L\)-function attached to a Siegel modular form of degree \(n\) (or more generally an automorphic form on the adelic symplectic group of rank \(n\)) has been the subject of a great deal of work; a survey of the history of this can be found in the introduction of this article. In particular one is interested in proving analytic continuation and functional equation of the \(L\)-function from a suitable integral representation on it. In the present paper the author considers this problem in the very general situation of holomorphic modular cusp forms with respect to a group of \(\Gamma_ 0 (N)\)-type on the symplectic group over a totally real number field. To obtain the integral representation the cusp form is integrated against the product of an Eisenstein series and a theta series. The resulting Dirichlet series is then decomposed into an Euler product by proving the rationality of a formal power series over the local Hecke ring and giving an explicit factorization of it.