Remarks on generating series for special cycles on orthogonal Shimura varieties
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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 signatureFor each n, 1 ≤ n ≤ 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 OF .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 1 ≤ n ≤ m.
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