Journal of Lie Theory

no. 3
Picard Groups of Siegel Modular 3-Folds and θ-Liftings
Hongyu He1; Jerome William Hoffman1
1Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
Journal of Lie Theory, Volume 22 (2012) no. 3, pp. 769-801
\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.
DOI: 10.5802/jolt.691
Classification: 14G35, 11F46, 11F27, 14C22, 11F23
Keywords: Siegel modular threefold, Picard group, theta lifting

Hongyu He  1 ; Jerome William Hoffman  1

1 Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. 
Hongyu He; Jerome William Hoffman. Picard Groups of Siegel Modular 3-Folds and θ-Liftings. Journal of Lie Theory, Volume 22 (2012) no. 3, pp. 769-801. doi: 10.5802/jolt.691
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