
Journal of Lie Theory 22 (2012), No. 3, 769801 Copyright Heldermann Verlag 2012 Picard Groups of Siegel Modular 3Folds and θLiftings Hongyu He Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. hongyu@math.lsu.edu Jerome William Hoffman Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. hoffman@math.lsu.edu [Abstractpdf] \def\R{{\Bbb R}} 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 $\tilde{\rm SL}_2(\R)$ to automorphic forms on ${\rm Sp}_4(\R)$. (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. Keywords: Siegel modular threefold, Picard group, theta lifting. MSC: 14G35; 11F46, 11F27, 14C22, 11F23 [ Fulltextpdf (458 KB)] for subscribers only. 