A Siegel modular threefold and Saito-Kurokawa type lift to S ₃(Γ_1,3(2))

Hulek and others conjectured that the unique differential three-form F (up to scalar) on the Siegel threefold associated to the group Γ _1,3(2) comes from the Saito-Kurokawa lift of the elliptic newform h of weight 4 for Γ₀(6). This F have been already constructed as a Borcherds product (cf. Gritsenko and Hulek in Int Math Res Notices 17:915-937, 1999). In this paper, we prove this conjecture by using the Yoshida lift and we settle a conjecture which relates our theorem. A remarkable fact is that the Yoshida lift using the usual test function cannot give the Saito-Kurokawa type lift of weight 3 associated to the group Γ_1,3(2). So important task is to find special test functions for the Yoshida lift at the bad primes 2 and 3.