Journal of Lie Theory 22 (2012), No. 3, 769--801
Copyright Heldermann Verlag 2012
Picard Groups of Siegel Modular 3-Folds and θ-Liftings
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
[Abstract-pdf]
\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
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