On Saito-Kurokawa descent for congruence subgroups

The conjecture made by H. Saito and N. Kurokawa states the existence of a “lifting” from the space of elliptic modular forms of weight 2k?2 (for the full modular group) to the subspace of the space of Siegel modular forms of weightk (for the full Siegel modular group) which is compatible with the action of Hecke operators. (The subspace is the so called “Maaß spezialschar” defined by certain identities among Fourier coefficients). This conjecture was proved (in parts) by H. Maaß, A.N. Andrianov and D. Zagier. The purpose of this paper is to prove a generalised version of the conjecture for cusp forms of odd squarefree level.