On Doi-Naganuma lifting

. 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 i D which maps the m th Poincare series of weight k , level M and character w D ¼ : D (cid:1) (cid:2) 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 i (cid:1) D with respect to the Petersson inner product.