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• W2969369112 abstract "In this note, we consider special algebraic cycles on the Shimura variety S associated to a quadratic space V over a totally real field F, |F:Q|=d, of signature ((m,2)^{d_+},(m+2,0)^{d-d_+}), 1le d_+<d. For each n, 1le nle m, there are special cycles Z(T) in S, of codimension nd_+, indexed by totally positive semi-definite matrices with coefficients in the ring of integers O_F. The generating series for the classes of these cycles in the cohomology group H^{2nd_+}(S) are Hilbert-Siegel modular forms of parallel weight m/2+1. One can form analogous generating series for the classes of the special cycles in the Chow group CH^{nd_+}(S). For d_+=1 and n=1, the modularity of these series was proved by Yuan-Zhang-Zhang. In this note we prove the following: Assume the Bloch-Beilinson conjecture on the injectivity of Abel-Jacobi maps. Then the Chow group valued generating series for special cycles of codimension nd_+ on S is modular for all n with 1le nle m." @default.
• W2969369112 created "2019-08-29" @default.
• W2969369112 creator A5020427762 @default.
• W2969369112 date "2019-08-22" @default.
• W2969369112 modified "2023-09-27" @default.
• W2969369112 title "Remarks on generating series for special cycles" @default.
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