Assuming the Beilinson-Bloch conjecture, generating series of special cycles of codimension er on these Shimura varieties are Hilbert-Siegel modular forms of genus r and weight 1+n/2.
Generating series of a new class of orthogonal Shimura varieties
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abstract
For a new class of Shimura varieties of orthogonal type over a totally real number field, we construct special cycles and show the the modularity of Kudla's generating series in the cohomology group.
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The modularity of special cycles on orthogonal Shimura varieties over totally real fields under the Beilinson-Bloch conjecture
Assuming the Beilinson-Bloch conjecture, generating series of special cycles of codimension er on these Shimura varieties are Hilbert-Siegel modular forms of genus r and weight 1+n/2.