On classical Saito-Kurokawa liftings

There exist two different generalizations of the classical Saito–Kurokawa lifting to modular forms with (square-free) level; one lifting produces modular forms with respect to Γ0(m), the other one with respect to the paramodular group Γ(m). We shall give an alternative and unified construction of both liftings using group theoretic methods. The construction shows that a single elliptic modular form may in fact have many Saito–Kurokawa liftings. We also obtain precise information about the spin L–function of the resulting Siegel modular forms.