Hilbert cusp forms and special values of Dirichlet series of Rankin type

Source abstract

Let K be a totally real number field of degree n over ℚ and let c be an integral ideal of a maximal order of K . Given a nonnegative integer j and a Hecke character on the group of ideles of K , let denote the space of Hilbert cusp forms of holomorphic type on ℋ n of weight j , level c and character ψ where ℋ n is the n -th power of the Poincaré upper half plane ℋ.Let g be an element of , where 1 is the trivial character. If u ∈ S k (c, ψ), then the product gu is an element of S k+l (c, ψ), and therefore we can consider the linear map sending u to gu . Let be the adjoint of the linear map Φ g with respect to the Petersson inner product.