Picard Groups of Siegel Modular 3-Folds and θ-Liftings

We show that the Humbert surfaces rationally generate the Picard groups of Siegel modular threefolds.This involves three ingredients: (1) R. Weissauer's determination of these Picard groups in terms of theta lifting from cusp forms of weight 5/2 on SL 2 (R) to automorphic forms on Sp 4 (R) .( 2) The theory of special cycles due to Kudla/Millson and Tong/Wang relating cohomology defined by automorphic forms to that defined by certain geometric cycles.(3) Results of R. Howe about the structure of the oscillator representation in this situation.