On Doi–Naganuma and Shimura liftings

Abstract

In this paper, we extend the Doi–Naganuma lifting to higher levels by following the methods of Zagier and Kohnen. We prove that there is a Hecke-equivariant linear map from the space of elliptic cusp forms of integer weight k, level $$N, ((N,D)=1)$$N,((N,D)=1) to Hilbert cusp forms of weight k, level N associated to a real quadratic field of discriminant D ($$D\\equiv 1\\pmod {4}$$D≡1(mod4)) with class number one. The above lifting is obtained by computing the explicit image of Poincaré series of weight k, level N for the cusp at $$\\infty $$∞. Finally, we see that the above lifting is closely related to the Dth Shimura lift on the Kohnen plus space.