\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.
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, Tome 22 (2012) no. 3, pp. 769-801. http://geodesic.mathdoc.fr/item/JOLT_2012_22_3_a7/
@article{JOLT_2012_22_3_a7,
author = {Hongyu He and Jerome William Hoffman},
title = {Picard {Groups} of {Siegel} {Modular} {3-Folds} and {\ensuremath{\theta}-Liftings}},
journal = {Journal of Lie Theory},
pages = {769--801},
year = {2012},
volume = {22},
number = {3},
url = {http://geodesic.mathdoc.fr/item/JOLT_2012_22_3_a7/}
}
TY - JOUR
AU - Hongyu He
AU - Jerome William Hoffman
TI - Picard Groups of Siegel Modular 3-Folds and θ-Liftings
JO - Journal of Lie Theory
PY - 2012
SP - 769
EP - 801
VL - 22
IS - 3
UR - http://geodesic.mathdoc.fr/item/JOLT_2012_22_3_a7/
ID - JOLT_2012_22_3_a7
ER -
%0 Journal Article
%A Hongyu He
%A Jerome William Hoffman
%T Picard Groups of Siegel Modular 3-Folds and θ-Liftings
%J Journal of Lie Theory
%D 2012
%P 769-801
%V 22
%N 3
%U http://geodesic.mathdoc.fr/item/JOLT_2012_22_3_a7/
%F JOLT_2012_22_3_a7
