On Doi-Naganuma lifting

Abstract

In this paper, we extend the Doi-Naganuma lifting as suggested by Kudla [4], on the lines of Zagier's work [6]. For each fundamental discriminant D associated with a real quadratic field, we prove that there exists a Hecke-equivarient map ι_D which maps the mth Poincare series of weight k, level M and character χ_D = (./D) into a Hilbert cusp form of weight k, level M/D associated with the real quadratic field of discriminant D of class number one. Through this, we get its adjoint ι^*_D with respect to the Petersson inner product.