Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields

In this paper we consider the Eisenstein series for the Hilbert modular group of a general number field. We compute the Fourier expansion at each cusp explicitly. The Fourier coefficients are given in terms of completed partial Hecke -series, and from their functional equations, we get the functional equation for the Eisenstein vector. That is, we identify the scattering matrix. When we compute the determinant of the scattering matrix in the principal case, the Dedekind -function of the Hilbert class field shows up. A proof in the imaginary quadratic case was given in Efrat and Sarnak, and for totally real fields with class number one a proof was given in Efrat.